\section{INTRODUCTION}
Glitches (sudden ``spin-up'' of the star) are detected commonly in pulsars, including both normal pulsars and anomalous X-ray pulsars / soft gamma-ray repeaters (AXPs/SGRs). Stellar oscillations are likely excited during a glitch, which would propagate into the magnetosphere to affect the properties of the inner gap of the pulsar. \citet{Morozova2010, Zanotti2012} have studied the oscillations effect on the magnetospheric activity within the framework of the space-charge-limited flow (SCLF) model.
On the other hand, drifting sub-pulses in pulsar radio emission suggests that there are sub-beams (sparks) generated in the pulsar polar cap region, which can be naturally interpreted as the existence of a inner vacuum gap \citep{Ruderman1975} or a partially screened inner gap \citep{Gil2003,Gil2006}.
The binding energy problem faced by the inner gap model (which was the motivation of the SCLF model) is alleviated if pulsars have strong surface magnetic field and low temperature \citep{Medin2007,Gil2006}, and can be completely solved if pulsars are bare strange quark stars \citep{Xu1999}. Therefore there is a strong motivation to study the oscillation effect on magnetospheric activities within the framework of the inner gap model, which is the subject of this paper.

Here we consider toroidal stellar oscillations. Similar to rotation, toroidal oscillations can also produce a voltage across pulsar polar cap due to unipolar induction, which would modify the Goldreich-Julian (1969) charge density and affect gap properties and particle acceleration processes. The sign of voltage produced by oscillations varies periodically. One may roughly estimate the period of an oscillation as the size of star over the speed of sound. The time scale of particle acceleration in the inner gap, on the other hand, is approximately the height of gap over speed of light. Since the size of star is much larger than the gap height and the speed of sound is much less than the speed of light, the time scale for particle acceleration is much shorter than the period of oscillation. Therefore, the periodic variation of the oscillation direction would not affect the ability of the inner gap to accelerate particles. Thanks to the additional voltage provided by oscillations, the properties of the inner gap, such as height and voltage, would be modified. In this paper, we present a general solution to determine the properties of the inner gap with oscillations. We consider three modes of pair plasma production: curvature radiation \citep[CR,][]{Ruderman1975}, inverse Compton scattering \citep[ICS,][]{Zhang1996}, and two-photon annihilation (2$\gamma$) processes \citep{Zhang1998}. We also discuss mode changing due to variation of the oscillation amplitude.

Interesting observational properties are observed to be associated with glitches in AXPs/SGRs.
AXPs/SGRs are isolated neutron stars with long periods ($2-10\,{\rm s}$) and high period derivatives ($10^{-13}-10^{-11}$). They are radio quiet in the quiescent state, which was considered as a distinctive characteristic with respective to normal radio pulsars \citep{Mereghetti2008}.
Recent observations suggest that some of them can become radio loud after some transient activities. Up to now, four AXPs/SGRs have been detected in the radio band: 1E 1547.0-5408, PSR J1622-4950, XTE J1810-197 and SGR J1745-2900 \citep{Olausen2014}\footnote{See McGill Online Magnetar Catalog:\\ http://www.physics.mcgill.ca/~pulsar/magnetar/main.html}, all of which behave as transient AXPs/SGRs. Pulsed radio emission from these objects was detected after some outbursts, which decays in a time scale of months to years\footnote{Transient, unpulsed, radio emission was also observed after the giant flares of SGR 1900+14 \citep{Frail1999} and SGR 1806-20 \citep{Gaensler2005,Cameron2005}, but those are believed to be the ``afterglow'' of the flaring material interacting with the ambient medium \citep[e.g.][]{Wang2005}.}. The origin of this transient radio emission of AXPs/SGRs is still a mystery, and some ideas to interpret it have been proposed \citep[e.g.][]{Morozova2012}. Since the outbursts of AXPs/SGRs sometimes are associated with glitches \citep{Kaspi2003,Dib2014}, it is possible that toroidal oscillations are excited during the outbursts, which modify the magnetospheric structure to allow radio emission to be produced. A direct motivation of our study is to explore this possibility.

In Section 2, we present the inner gap model under stellar oscillations. The general theory of the inner gap for three different modes are presented. In Section 3, we perform calculations of the gap properties by adopting typical parameters of radio pulsars and AXPs/SGRs. In Section 4, we apply the model to pulsars and AXPs/SGRs, and in particular, interpret the transient radio emission phenomenology of AXPs/SGRs.
Conclusions are drawn in Section 5 with some discussion.

\section{THE MODEL}

\subsection{Goldreich-Julian charge density and toroidal oscillation modification}
    Pulsars are rapidly rotating, highly magnetized compact objects.
Strong electric fields are generated
due to unipolar induction. \citet{Goldreich1969} found that in order to have the pulsar magnetosphere co-rotates with the star, the spatial electric charges have to follow a certain distribution, such that at every point in the magnetosphere the ${\bf E \times B}$ drift of the charges just maintain corotation of the magnetosphere. This charge density is called $\rho_{\rm GJ}$.
    Without the effect of oscillation and considering the general relativisty effect,  for a small-angle approximation $\theta \ll 1$, this density can be written as \citep{Muslimov1992,Morozova2010}
    \begin{align} \label{rho_GJ_rot}
        \rho_{\rm GJ,rot} &= -\frac{\Omega B}{2\pi c} \frac{1}{N\tilde{r}^3} \frac{f(\tilde{r})}{f(1)} \nonumber\\
        & \left[(1-\frac{\kappa}{\tilde{r}^3})\cos\alpha + \frac{3}{2}H(\tilde{r})\theta\sin\alpha\cos\phi \right].
    \end{align}
Here $\Omega$ is the rotation angular velocity of the pulsar, $B$ is the magnetic field strength at the pulsar surface, $c$ is the velocity of light, $\tilde{r}\equiv\frac{r}{R}$ is the reduced radial coordinate, $R$ is the radius of the pulsar, $\theta,\ \phi$ denote the angular coordinates in a spherical coordinate system with $z$-axis corresponding to the spin axis, $\alpha$ is the magnetic axis inclination angle of the pulsar, $N=\sqrt{1-\frac{2GM}{c^2r}}$ is the lapse function, $\kappa\equiv\frac{2GM}{Rc^2}\frac{I}{MR^2}$, and $M$ and $I$ are the mass and moment of inertia of pulsar, respectively.
    The function $f(\tilde{r})$ reads
    \begin{equation}\label{f}
        f(\tilde{r})=-3\left(\frac{\tilde{r}}{\varepsilon}\right)^3 \left[ \ln(1-\frac{\varepsilon}{\tilde{r}}) + \frac{\varepsilon}{\tilde{r}}(1+\frac{\varepsilon}{2\tilde{r}}) \right],
    \end{equation}
    where $\varepsilon=2GM/Rc^2$, and the function $H(\tilde{r})$ reads
    \begin{equation}
        H(\tilde{r}) = \frac{1}{\tilde{r}}\left(\varepsilon-\frac{\kappa}{\tilde{r}^2}\right) + \left(1-\frac{3}{2}\frac{\varepsilon}{\tilde{r}}+\frac{1}{2}\frac{\kappa}{\tilde{r}^3}\right) \left[f(\tilde{r})(1-\frac{\varepsilon}{\tilde{r}})\right]^{-1}.
    \end{equation}

    We now consider the effect of oscillation. In general, there are two types of oscillations: spherical and toroidal. Only the toroidal oscillations can effectively modify the spin of the star, and therefore affect the Goldrich-Julian charge density and magnetospheric configurations. Spherical oscillations, on the other hand, would not modify the magnetosphere structure significantly.
Therefore here we only consider toroidal oscillations. The veolocity components of toroidal oscillations can be written as \citep{Unno1989}
    \begin{equation} \label{oscillation}
        \delta v^{\hat{i}} = \left[0,\  \frac{1}{\sin\theta}\partial_{\phi} Y_{lm}(\theta,\phi),\  -\partial_\theta Y_{lm}(\theta,\phi) \right] \tilde{\eta}(\tilde{r})e^{-i\omega t},
    \end{equation}
    where
    $Y_{lm}$ is the spherical harmonic function, $\tilde{\eta}(\tilde{r})$ is a parameter denoting the amplitude of the oscillations, and $\omega$ is the angular frequency of the oscillations. Generally speaking, the axis of toroidal oscillations should be close to the spin axis itself, so the $z$-axis of the spherical coordinate system is likely the spin axis.
Because the height of a pulsar inner gap is much smaller than the pulsar radius, in our calculations we take $\tilde{\eta}(\tilde{r})=\tilde{\eta}(1)$.

    \citet{Morozova2010} obtained the modifed G-J charge density caused by oscillations (their Eq. (13)). However, there is a mistake in their derivation from their Eq. (11) to Eq. (13). The factor $1/\Theta^2(\tilde{r})$ ($\Theta$ is the co-latitude of the last closed magnetic field line) should not appear in the second term of their Eq. (13). Within the polar cap region, where the co-latitude with respective to the magnetic axis $\vartheta$ is small, the correct expression should read
    \begin{equation} \label{rho_GJ_osc}
        \rho_{\rm GJ,osc} = -K \frac{\Omega B}{4\pi c} \frac{1}{\tilde{r}^4} \frac{1}{N} \frac{f(\tilde{r})}{f(1)} l(l+1) Y_{lm}(\theta,\phi) \cos\vartheta e^{-i\omega t},
    \end{equation}
    where $K=\tilde{\eta}(1)/\Omega R$ denotes the ratio of oscillation velocity and rotation velocity in the pulsar surface, which represents the intensity of the oscillations.
    Since
$\Theta$ is a small quantity at the polar cap region, the factor $1/\Theta^2(\tilde{r})$ is a large quantity. As a result, the $\rho_{\rm GJ,osc}$ value obtained in their paper is significantly over-calculated. After correcting the error, one gets the ratio
    \begin{equation} \label{rho_GJ_ratio}
        \frac{\rho_{\rm GJ,osc}}{\rho_{\rm GJ,rot}} = \frac{K}{2\tilde{r}} \frac{l(l+1)Y_{lm}(\theta,\phi)\cos\vartheta}{1-\kappa/\tilde{r}^3} e^{-i\omega t}.
    \end{equation}

With $z$ axis fixed to the spin axis, the pulsar polar cap region would have $\theta \sim \alpha$, where $\alpha$ is the magnetic axis inclination angle of the pulsar. For a non-extreme $\alpha$ in the realistic cases, given a same value of $K$, the oscillation modes with a same $l$ but different $m$ values would have $\rho_{\rm GJ}$ values of the same order of magnitude. For different $l$ values, $\rho_{\rm GJ}$ increases by a factor $Kl(l+1)Y_{lm}$, and the oscillation velocity increase as a factor $\sim KlY_{lm}$. As a result, when $l$ is large, a large $\rho_{\rm GJ}$ would be obtained with a relatively smaller oscillation velocity.

It is unclear in reality which mode(s) would be excited and be dominant. Practically, introducing the inclination angle $\alpha$ would significantly complicate the calculations, so in the following calculations we assume $\alpha=0$ for simplification, so that $\theta=\vartheta$. At this configuration, the $m=0$ modes have similar amplitudes as the cases with an arbitrary $\alpha$ except those close to 90 degree, and we only calculate $l=2,\ m=0$ mode as an illustration\footnote{Since $Y_{lm}(\theta,\phi)$ has a factor of $\sin^m\theta$, the $m>1$ modes are much suppressed when $\theta \sim 0$, so that $\alpha=0$ is a bad approximation.}. As discussed above, this example may be regarded as a typical example of $l=2$ for arbitrary $\alpha$ values except those close to 90 degree and arbitrary $m$ values, and the cases for different $l$ values may be inferred through the simple scaling relations.

    \subsection{Basic physical picture of inner gap}
The picture of the vacuum gap model (RS75) is the following. Due to a centrifugal force, particles in a pulsar magnetosphere continuously streams out the light cylinder in the open field line regions. At the polar cap of an ``anti-parallel'' rotator, if the ion binding energy at the pulsar surface is large enough, no charge can be provided to sustain a continuous outflow, and a gap, in which the charge density $\rho \ll \rho_{\rm GJ}$, is formed. An electric field component parallel to the magnetic field, $E_\parallel$, is developed.
Seed $\gamma$-rays in the gap would materialize in the strong $B$ field and generate electron-positron pairs. These pairs are accelerated in the gap to ultra-relativistic energies and produce high-energy photons through CR or ICS. High-energy photons undergo pair production through $\gamma-B$ or two-photon processes, and the newly produced positrons/electrons are accelerated and radiate again. This leads to a run-away pair creation avalanch, which ultimately leads to discharge of the gap potential and breakdown of the gap.  Denoting the gap height as $h$, under one-dimensional approximation for $h\ll r_p$ (where $r_p=R(\frac{\Omega R}{c})^{1/2}$ is the polar cap radius), one can write the electric field in the gap
    \begin{equation}
        E(s)=-4\pi\int_h^s\rho_{\rm GJ}(s){\rm d}s,
    \end{equation}
    where $s$ denotes the distance from the surface of the star. The corresponding electric potential difference  along the gap is
    \begin{equation}
        \Delta V(h) = -\int_0^h E(s) {\rm d}s.
    \end{equation}

    There is a maximum voltage across the polar cap provided by the unipolar induction, denoted as $\Delta V_{\rm max}$. When $\Delta V$ approaches $\Delta V_{\rm max}$, the 1D approximation above would fail.
The criterion
    \begin{equation} \label{death line}
        \Delta V(h) < \Delta V_{\rm max}
    \end{equation}
ensures that pair production screens $E_\parallel$ and limits the gap potential. Since pair production is the necessary condition for coherent radio emission,
Eq. (\ref{death line}) defines the radio death line of pulsars \citep[e.g. RS75,][]{Zhang2000}.

    For $\alpha=0$, the maximum voltage across the polar cap can be written as (see derivation in the Appendix)
    \begin{align}\label{V_max_m=0}
        \Delta V_{\rm max} & \approx -\frac{\Omega R^2NB_0}{c} \frac{\Theta_0^2}{2} \nonumber\\
        & \left[(1-\kappa) + K\frac{l(l+1)}{2}Y_{lm}(\Theta_0,\phi)\right],~{\rm for}~ m=0;
    \end{align}
    \begin{align}\label{V_max_m!=0}
        \Delta V_{\rm max} & \approx -\frac{\Omega R^2NB_0}{c} \nonumber\\
        & \left[(1-\kappa)\frac{\sin^2\Theta_0}{2} - KY_{lm}(\Theta_0,\phi)\right],
        {\rm for}\ m\neq0.
    \end{align}
    And the polar angle of the last open magnetic field line $\Theta_0$ is determined by
    \begin{equation} \label{polar_angle_m=0}
        \Theta_0 \approx \frac{\sqrt[4]{N}}{f(1)} \sqrt{\frac{2\Omega R}{c} \left[(1-\kappa)+K\frac{l(l+1)}{2}Y_{l0}\right]},\ {\rm for}\ m=0;
    \end{equation}
    \begin{align} \label{polar_angle_m!=0}
        \frac{1}{2}f^4(1)\Theta_0^6 \approx \frac{\Omega^2R^2N}{c^2} \left[(1-\kappa)\frac{\sin^2\Theta_0}{2}-KY_{lm}(\Theta_0,\phi)\right] \nonumber\\ \left[(1-\kappa)+K\frac{l(l+1)}{2}Y_{lm}(\Theta_0,\phi)\right],\ {\rm for}\ m\neq0.
    \end{align}

    \subsection{Different gap modes}

    For a pulsar inner gap, several processes operate simultaneously. Gamma-rays are produced via both CR and ICS by electrons/positrons, and pair productions proceed through both $\gamma-B$ and $2\gamma$ processes. The height and potential of the gap is determined by the process which takes the shortest distance to generate enough pairs to shut down the gap \citep{Zhang1997}. Based on physical conditions, one of the following three modes may play a dominant role to define gap parameters: CR, resonant ICS, and $2\gamma$ processes. In the following, we discuss these modes in turn.

\subsubsection{CR mode}

    In RS75, only the CR mode was considered, and the height of gap is determined by $h\sim l_\gamma$, where $l_\gamma$ is the mean free path of typical $\gamma$ photons, which reads
    \begin{eqnarray}\label{l_gamma}
        l_\gamma = \frac{4.4}{e^2/\hbar c} \frac{\hbar}{m_e c} \frac{B_q}{B_\bot}\exp(\frac{4}{3\chi}), \
        \chi \ll 1, \\
        \chi \equiv \frac{E_\gamma}{2m_e c^2} \frac{B_\bot}{B_q} \equiv \frac{E_{\gamma\bot}}{2m_e c^2} \frac{B}{B_q}.
    \end{eqnarray}
    where $e$ and $m_e$ are the electric charge and the mass of an electron, respectively, $c$ is speed of light, $\hbar$ is reduced Planck's constant, $B_q=m^2 c^3/e\hbar=4.4\times10^{33}\,{\rm G}$ is the critical magnetic field, $B_\bot$ is the perpendicular magnetic component with respect to the direction of the $\gamma$-ray photon in the co-rotating frame, $E_\gamma$ is the energy of the $\gamma$-ray photon, and $E_{\gamma\bot}$ is the perpendicular energy of the $\gamma$ photon with respect to the direction of the magnetic field in the co-rotation frame. According to RS75, a photon emitted parallel to the direction of the local magnetic field line and propagates along a distance $h$ would see a perpendicular magnetic field
    \begin{equation}
        B_\bot=\frac{h}{\rho}B,
    \end{equation}
    where $\rho$ is the curvature radius of the magnetic field line.
    For a dipolar magnetic field, one has
    \begin{equation}
        \rho \simeq \frac{4}{3} \sqrt{Rc/\Omega} \simeq 9.21\times10^7(P R_6)^{1/2}\,{\rm cm},
    \end{equation}
    where $P$ is the rotational period of pulsar, $R_6$ is pulsar's radius in units of $10^6\,{\rm cm}$.
    For multipole magnetic fields, one may adopt a typical curvature radius $\rho\sim10^6\,{\rm cm}$, comparable to the size of the pulsar. We consider both possibilities in the following calculations.

    In order to break a gap, a lepton needs to at least release one photon within a typical mean free path $l_e$, and the photon needs to attenuate in the magnetic field to produce pairs within a mean free path $l_\gamma$ (Eq.(\ref{l_gamma})). For the CR process, $l_e \ll l_\gamma$ is usually satisfied \citep{Zhang2000}, one may then take $h \sim l_\gamma$. Notice that in Eq. (\ref{l_gamma}), $l_\gamma$ decreases with increasing $E_\gamma$, so the largest Lorentz factor of electrons/positrons $\gamma_e$ determines the height of gap. Under the assumption that electrons/positrons can be accelerated to the largest Lorentz factor in the gap $\gamma_e m_e c^2=\Delta V(h)$, one can solve the height of gap with Eq. (\ref{l_gamma}) (RS75), denoted as $h_{\rm min}$.

    However, in some situations, especially when the ICS mode is invoked,
$l_e$ often exceeds $h_{min}$ (see more detail below). As a result, $l_\gamma$ related to the largest $\gamma_e$ is no longer a good estimate of the gap height $h$, since at this distance, an electron on average has not produced a photon yet. The photon-pair cascade chain is not launched. In this case,
we determine the gap height via
    \begin{equation}\label{h_equal}
        l_e(\gamma_e) = l_\gamma(\gamma_e) = h(\gamma_e).
    \end{equation}
    When oscillations are considered, in some cases, even the CR mode also has $l_e > l_\gamma$.

    For the CR mode, we have
    \begin{equation}
        E_{\gamma,{\rm CR}}=\hbar\omega_{c,{\rm CR}}=\hbar\frac{3}{2}\frac{\gamma_e^3 c}{\rho},
    \end{equation}
    \begin{equation}
        l_{e,{\rm CR}} \sim c \left( \frac{P_{\rm CR}}{\hbar\omega_{c,{\rm CR}}} \right)^{-1}
        = \frac{9}{4}\frac{\hbar\rho c}{\gamma_e e^2},
    \end{equation}
    where $\gamma_e$ is the Lorentz factor of electrons/positrons.

\subsubsection{ICS mode}\label{sec:ICS}

    For the ICS mode, we mainly focus on ICS at resonance. The scattering cross section of ICS becomes very large when the incident photon energy in the electron rest frame equals to the energy difference between electron's Laudau level interval from the ground state to the first excited state \citep{Xia1985}.
The large scattering cross section of this resonant mode makes $l_e$ reduce significantly so that pair cascade in the gap becomes possible \citep{Zhang1996}. There is another typical ICS energy ($\gamma_e^2 kT$), which is related to boosting the thermal peak of the source photons to high energies. This mode, called the thermal mode \citep{Zhang1997}, was later found to have too large a $l_e$, which may not play an important role \citep{Zhang2000}. We do not consider this mode in the following calculations.

    In the resonant ICS mode, when the magnetic field is below critical magnetic field, we have \citep{Zhang1997}
    \begin{equation} \label{E_gamma_res}
        E_{\gamma,{\rm res}} \sim 2\gamma_e^2\hbar\omega_{\rm res}(1-\beta_e\mu_i) = 2\hbar\gamma_e\omega_B,
    \end{equation}
    where $\beta_e=(1-\gamma_e^{-2})^{1/2}$ is the dimensionless velocity of the relativistic particle, $\mu_i$ is the cosine of the angle between incident photon and the direction of electron's velocity, $\omega_B=eB/mc$ is the cyclotron frequency of an electron/positron in the magnetic field, and $\omega_{\rm res}=\omega_B/[\gamma_e(1-\beta_e\mu_i)]$ is the so-called resonant frequency of the ICS process.
    When the magnetic field exceeds $B_q$, the resonant ICS is in the Klein-Nishina regime, so that the largest energy of scattered photon is essentially the total energy of electron
    \begin{equation}
        E_{\gamma,{\rm max}} \simeq \gamma_e m_e c^2.
    \end{equation}
    For given energy of scattered photon $\epsilon_s$ (in units of $m_e c^2$), the incident photon's energy (in units of $m_e c^2$) is
    \begin{equation}
        \epsilon_i=\epsilon_s/[2\gamma_e^2(1-\beta_e\mu_i)].
    \end{equation}
    Then the mean free path of an electron can be written as
    \begin{align}
        l_{e,{\rm ICS}}(\epsilon_s) &\sim
        [ \int_0^1 \sigma^\prime(\epsilon_s, \mu_i) (1-\beta_e\mu_i) n_{\rm ph}(\epsilon_s, \mu_i) \nonumber\\
        & \times\frac{\epsilon_s}{2\gamma_e^2} \frac{\beta_e}{(1-\beta_e\mu_i)^2} d\mu_i ]^{-1}
    \end{align}
    where
    \begin{equation}
        n_{\rm ph}(\epsilon_i)d\epsilon_i = \frac{8\pi}{\lambda_c^3} \frac{\epsilon_i^2}{\exp(\epsilon_i/\theta_{\rm T})-1} d\epsilon_i
    \end{equation}
    represents the photon number density distribution of a semi-isotropic blackbody radiation, in which  $\theta_{\rm T}=k_B T/m_e c^2$, $k_B$ is Boltzmann's constant, and $\lambda_c=\hbar/m_e c \simeq 2.426\times10^{-10}\,{\rm cm}$ is the electron Compton wavelength.
    Here $\sigma^\prime$ is the scattering cross section in the electron rest frame (ERF), which is calculated using the Sokolov \& Ternov cross section in \citet{Baring2011}
    \begin{align}
        \frac{d\sigma^\prime_{\rm ST}}{d(\cos\theta_f)} &=  \frac{3\sigma_T}{64} \frac{\omega_f^2e^{-\kappa}}{\omega_i[2\omega_i-\omega_f-\zeta]\varepsilon_\perp^3} \nonumber\\
        &\times \sum_{s=\pm1}\frac{(\varepsilon_\perp-s)^2 \Lambda_s}{(\omega_i-B/B_q)^2+(\Gamma_s/2)^2}.
    \end{align}
    Here $\theta_f$ is the scattered angle in the ERF, $\sigma_T$ is the Thomson cross section, $\omega_i=\gamma_e\epsilon_i(1-\beta_e\mu_i)$ is the incident photon energy in units of $m_e c^2$ in the ERF, $\omega_f$ is the energy of scattered photon in the ERF written as
    \begin{equation}
        \omega_f = \frac{2\omega_i r}{1+\sqrt{1-2\omega_i r^2 \sin^2\theta_f}}, \ r=\frac{1}{1+\omega_i(1-\cos\theta_f)},
    \end{equation}
    $\varepsilon_\perp=\sqrt{1+2B/B_q}$, $\kappa=\omega_f^2\sin^2\theta_f/(2B/B_q)$, $\zeta=\omega_i\omega_f(1-\cos\theta_f)$, and
    \begin{equation}
        \Lambda_s = (2\varepsilon_\perp+s){\omega_f^2\mathcal{T}-[\omega_i-\omega_f]} - s\varepsilon_\perp^2[\omega_i-\omega_f],
    \end{equation}
    \begin{equation}
        \omega_f^2\mathcal{T} = 2(\omega_i-\omega_f)-2\omega_i\omega_f(1+\omega_i){1-\cos\theta_f}+2\omega_i^2,
    \end{equation}
    \begin{equation}
        \Gamma_s = \left(1-\frac{s}{\varepsilon_\perp}\right)\Gamma, \ \Gamma=\alpha_f\frac{B}{B_q}I_1(B),
    \end{equation}
    \begin{align}
        I_1(B) &= \int_0^\Phi \frac{d\phi e^{-\phi}}{\sqrt{(\Phi-\phi)(1/\Phi-\phi)}} \left[1-\frac{\phi}{2}(\Phi+1/\Phi)\right], \nonumber\\
        \Phi &= \frac{\sqrt{1+2B/B_q}-1}{\sqrt{1+2B/B_q}+1}.
    \end{align}
    The reason why the mean free path of electron would exceed that of photons in resonant ICS mode is ascribed to the offset between the energy of the resonant incident photons $\epsilon({\rm res})$ (which provide the largest cross section) and the energy of the peak of blackbody radiation (which provide the largest number of photons).

    When magnetic field exceeds the critical magnetic field $B_q$, the resonant condition cannot be satisfied since the resonant photons are in the Klein-Nishina regime. The ICS cross section declines, so that the electron mean free path $l_e$ becomes much larger. In such a case, no solution can be found for Eq. (\ref{h_equal}) for resonant ICS mode. In any case, one may still have an ICS mode, even though the photon energies are not at the resonance. We then apply the minimum $l_e(\gamma_e)$ to define the gap height.

    For the resonant ICS mode, considering that the energy loss of particles in the acceleration process cannot exceed the maximum electric potential energy in the gap, we need to ensure that the voltage  along the gap is enough to accelerate particles to the Lorentz factor required. This condition can be expressed as \citep{Zhang1996}
    \begin{equation}\label{P_e}
        P_e \frac{h}{c}<e \Delta V(h).
    \end{equation}
    We adopt an approximation of the energy-loss rate at resonance \citep{Dermer1990}:
    \begin{equation}
        P_e=\frac{c\sigma_{\rm eff}}{\mu_+-\mu_-}(\frac{\epsilon_B^2}{\gamma_e}) \times\int_{\epsilon_B/\gamma_e(1-\beta_e\mu_-)}^{\epsilon_B/\gamma_e(1-\beta_e\mu_-)} d\epsilon \epsilon^{-2} n_{\rm ph}(\epsilon)(1-\frac{\epsilon}{\gamma_e\epsilon_B}),
    \end{equation}
    where $\sigma_{\rm eff}\approx 3\pi\sigma_T\alpha_f$ is the effective cross section, $\mu_+$, $\mu_-$ are cosines of the maximum and minimum incident angles, respectively, which we adopt as $\mu_\pm=\pm 1$ in the calculations.
    In most cases, such a requirement is satisfied.

\subsubsection{Two photon pair production mode}

    As for the $2\gamma$ process, we consider interaction between the photons created by high-energy particles (through CR or ICS) and thermal photons. The mean free path of a photon, $l_{2\gamma}$, is determined by \citep{Zhang1998}
    \begin{equation}
        \int_0^{l_{2\gamma}} 2.1\times10^{-6}T_6^3 {\rm Max}[f(\nu, x)]dx=1,
    \end{equation}
    \begin{equation}
        {\rm Max}[f(\nu, x)]=0.270-0.507\mu_c+0.237\mu_c^2,
    \end{equation}
where $T_6$ is the surface temperature of pulsar in units of $10^6\,{\rm K}$, $\mu_c$ is the cosine of the angle between the two photons. Since gap height is usually much shorter than the size of polar caps, most of the two-photon interactions occur at about $90^{\rm o}$ angle (one photon streaming out, while the other comes perpendicularly from other parts of the surface), so we simply adopt $\mu_c \sim 0$. A small change of $\mu_c$ would not noticeably affect the results. It is worth pointing out that the height of gap due to $2\gamma$ processes is only relative to the surface temperature and has nothing to do with the expression of $V(h)$; in other words, oscillations do not have an essential effect on $2\gamma$ processes. The $2\gamma$ process is more important in SGRs/AXPs thanks to their high surface temperatures \citep{Zhang2001}.

\section{EFFECT OF OSCILLATIONS ON INNER GAP PROPERTIES}
When an oscillation gets its maximum amplitude with the same direction of rotation, it has the maximum effect on the properties of the gap. If pair production condition is satisfied at this most optimistic condition, radio emission can be produced. Therefore in the following we mainly focus on this extreme condition and ignore the periodical variation of the oscillations. In other words, in Eq. (\ref{rho_GJ_osc}) we set $e^{-i\omega t}=1$. In principle the gap properties would vary with time during one oscillation, but it is difficult to have an observational effect due to the high oscillation frequency. We may estimate the frequency of oscillation in a pulsar based on its analogy with the earth. The pulsar oscilation frequency is $P_{\rm osc} \propto \frac{L}{v}$, where $L$ is the typical length of the star which is or the order of pulsar radius $\sim 10^6\,{\rm cm}$, $v$ is the speed of sound, which might be close to speed of light for both neutron stars and solid quark stars \citep[e.g.][]{Haensel2007}. From the observations of earth oscillations, we know that the period of its fundamental-frequency oscillation is $P_{\rm earth}\sim 1\,{\rm h}$. The radius of the earth is $R_{\rm earth}\sim 6.4\times10^8\,{\rm cm}$, and the speed of sound (e.g. that earthquake wave) is $v_{\rm earth} \sim 5\times10^5\,{\rm cm/s}$.
As a result, the period of pulsar oscillation may be estimated as
\begin{equation}
    P_{\rm osc} \sim \frac{R}{R_{\rm earth}} \frac{v_{\rm earth}}{v} P_{\rm earth} \sim 3\times10^{-4}\,{\rm s}.
\end{equation}
The frequency is
\begin{equation}
    \nu_{\rm osc} = \frac{1}{P_{\rm osc}} \sim 3\,{\rm kHz}.
\end{equation}
This is much higher than the rotational frequency, so that the periodical variation of $\rho_{\rm GJ,osc}$ would be smoothed out and cannot cause an observable, periodical signal.
In principle an integrated average effect may be arguably more relevant, but this involves much more complicated calculations. The treatment here can provide an estimate to correct order of magnitude. Moreover, since we will discuss the threshold condition for pair production (and hence, radio emission), the maximum potential would be more relevant to define such a condition.

    We show the general effects of oscillation for a typical pulsar period $P=1\,{\rm s}$ in Fig. \ref{P=1_T6=1_l=2_m=0}.
The temperature $T=10^6\,{\rm K}$, and magnetic field strengths $B=10^{12},10^{13},10^{14}\,{\rm G}$, are adopted, respectively. Both dipole and multipole magnetic configurations are considered. The point lines represent the gap heights, while the solid lines represent the voltages along the gap. The blue lines with star points, green lines with circle points and red lines with cross points represent CR, ICS, and $2\gamma$ modes, respectively. The gap properties are plotted against the dimensionless amplitude of oscillation $K$, which approximately equals to the ratio of oscillation velocity to rotational velocity at the surface of star. Our calculations start from $K=10^{-3}$, at which the differences of gap properties are small enough with respect to the cases without oscillation.  Note that all the cases we calculate satisfy the requirement in Eq. (\ref{P_e}), so we do not present the condition in the figures.

   Keeping other parameters the same, we calculate a faster pulsar with $P=0.1$ s in Fig. \ref{P=0.1_B12=1_T6=1_l=2_m=0}. A typical SGR/AXP with $P=10$ s, $T=5\times 10^6$ K, and $B = 10^{14}$ G is calculated in Fig. \ref{P=10_B12=100_T6=5_l=2_m=0}.

    As mentioned above, oscillations have the effect to enhance the voltage across the polar cap when we have the same gap height, which means that $\Delta V(h)$ increases. For the $2\gamma$ mode, the height of gap is only determined by temperature, therefore it remains unchanged as the oscillation amplitude and $\Delta V$ vary.
The CR and resonant ICS modes, on the other hand, are senstive to these variations. The gap height may decrease when oscillations occur. Notice that a larger voltage across the polar cap would accelerate particles to higher Lorentz factors, and photons emitted by high energy particles would have a higher typical energy, hence, a shorter mean free path, so that the gap would break down earlier and has a lower height. In this case, $\Delta V$ still increases but with a smaller slope than the $2\gamma$ mode case.

    In some cases the gap height in the CR and resonant ICS modes would also remain unchanged. These cases occur when $l_e > h_{\rm min}$ ($h_{\rm min}$ is the mean free path of photons when $\gamma_e=e\Delta V/(m_e c^2)$), so that the height of gap should be determined by Eq. (\ref{h_equal}). Notice that $\Delta V(h)$ does not appear in Eq. (\ref{h_equal}) or the expression of $l_e$, which means that the gap height does not react to oscillations when $l_e > h_{\rm min}$, so that the gap height cannot be shorter than a lower limit determined by Eq. (\ref{h_equal}) or a minimum of $l_e$. In the resonant ICS mode, $l_e> h_{\rm min}$ occurs more frequently (as discussed in \S\ref{sec:ICS}).

    Notice that in reality the gap properties are determined by the mode whose gap height is the lowest.
 As a result, mode changing can happen as the oscillation amplitude varies, or other parameters (e.g. surface temperature or magnetic field strength) change (see figures below). \cite{Zhang1997} first discussed such gap mode changing due to fluctuation of the pulsar surface temperature.

    The $2\gamma$ mode dominates only in high temperature and a relative small $V(h)$,  because a high temperature makes $2\gamma$ process more effective and a relative small $V(h)$ makes the CR and ICS modes less effective. Here a relative small $V(h)$ means a weak magnetic field, a large period without oscillations.
    When $V(h)$ is large, the CR and ICS modes would become more effective, therefore the $2\gamma$ process could not dominate, especially with oscillations. Comparing the CR mode with the resonant ICS mode, $l_\gamma$ in the resonant ICS mode is much shorter than that in the CR mode. However, when the magnetic field strength exceeds $B_q$, or is too weak with a dipole configuration, $l_e > h_{\rm min}$ often occurs in the resonant ICS mode, so that $l_e$ defines the gap height, which is higher.
In these cases, the CR mode would dominate the properties of gap in these cases. The ICS mode plays an important role only when the magnetic field does not exceed $B_q$, typically with a multipole configuration, but sometimes in a strong enough dipole field also.

\begin{figure*}
    \centering
    \includegraphics [width=15cm]{f1.eps}
    \caption{\label{P=1_T6=1_l=2_m=0} Gap properties as a function of $K$, the dimensionless amplitude of oscillations, which approximately equals to the ratio of oscillation velocity and rotational velocity in the surface of the pulsar. The dotted lines represent the gap heights, while the solid lines represent the voltages along the gap. The blue lines with star points, green lines with circle points, and red lines with cross points represent modes of curvature radiation (CR), inverse Compton scattering (ICS) and two-photon progress ($2\gamma$), respectively. Here $P=1\,{\rm s}$, $T=10^6\,{\rm K}$ and the $l=2,\ m=0$ oscillation mode are adopted. The top two. middle two, and bottom two panels are for $B=10^{12}\,{\rm G}$, $10^{13}\,{\rm G}$, $10^{14}\,{\rm G}$, respectively. The left three are for a dipole magnetic field configuration, whereas the right three are for a multipole configuration.}
\end{figure*}

The influence of different magnetic field configurations is reflected in the curvature radii of magnetic field lines. The dipole magnetic field lines have larger curvature radii, which make the gap heights in the CR and resonant ICS modes higher. It also influences the condition of $l_e > h_{\rm min}$.

\begin{figure*}
    \centering
    \includegraphics [width=15cm]{f2.eps}
    \caption{\label{P=0.1_B12=1_T6=1_l=2_m=0} Gap properties as a function of dimensionless oscillation amplitude $K$ for a typical radio pulsar with $P=0.1$ s, $T=10^6$ K, and $B=10^{12}$ G. The left and right panels are for the dipole and multipole configurations, respectively. Same line conventions as Fig.\ref{P=1_T6=1_l=2_m=0}.}
\end{figure*}

\begin{figure*}
    \centering
    \includegraphics [width=15cm]{f3.eps}
    \caption{\label{P=10_B12=100_T6=5_l=2_m=0} Gap properties as a function of dimensionless oscillation amplitude $K$ for a typical magnetar with $P=10$ s, $T=5\times 10^6$ K, and $B=10^{14}$ G. The left and right panels are for the dipole and multipole configurations, respectively. Same line conventions as Fig.\ref{P=1_T6=1_l=2_m=0}.}
\end{figure*}
After general considerations of the vacuum gap, we also analyze the oscillations effect on partially screened inner gaps. Following \cite{Gil2003,Gil2006}, we introduce a shielding factor defined as $\eta \equiv 1-\rho_i/\rho_{\rm GJ}$ where $\rho_i$ is the real electric charge density in the gap contributed by iron ions. We determine the shielding factor $\eta$ and surface temperature $T_s$ by \citep{Gil2006}
\begin{equation}
    \sigma_{\rm SB}T_s^4 = \gamma_e m_e c^3 \frac{\rho_{\rm GJ}}{e} \eta,
\end{equation}
\begin{equation}
    \eta \approx 1- \exp(30-\frac{\varepsilon_c}{k_B T_s}) \frac{\rho_{\rm GJ,rot}}{\rho_{\rm GJ}}, \
    \varepsilon_c \approx (0.18\,{\rm keV})(\frac{B}{10^{12}\,{\rm G}})^{0.7}
\end{equation}
where $\sigma_{\rm SB}$ is the Stefan-Boltzman constant and $\varepsilon_c$ is the binding energy in the neutron star surface.

In Fig. \ref{PSG_5B_q}, we calculate the properties of the partially screened gap as a function of the dimensionless oscillation amplitude $K$. Here we adopt
$P=1\,{\rm s}$,
the dipole magnetic field $B_{\rm di}=10^{12}\,{\rm G}$ and the multipole magnetic field $B_{\rm multi}=5B_q$, as is expected in the partially screened gap scenario. One can see that although oscillations increase the GJ charge density and are able to increase the voltage along the gap, they also enhance the heating by the back flowing particles and thus decrease the shielding factor. As a result, the voltage increase is not as obvious in partially screened gaps as in vacuum gaps.

\begin{figure*}
    \centering
    \epsscale{0.8}
    \plotone{f5.eps}

    \caption{\label{PSG_5B_q} The properties of a partially screened gap as a function of the dimensionless oscillation amplitude $K$. $P=1\,{\rm s}$,$B_{\rm di}=10^{12}\,{\rm G}$ and $B_{\rm multi}=5B_q$ are adopted. Same line conventions as Fig.\ref{P=1_T6=1_l=2_m=0}.}
\end{figure*}

\section{APPLICATIONS}

    \subsection{Normal Pulsars}
As seen in Fig. \ref{P=1_T6=1_l=2_m=0} and \ref{P=0.1_B12=1_T6=1_l=2_m=0}, the $2\gamma$ mode is not important due to the low temperature in normal pulsars. The properties of the gap was dominated by the CR mode or the resonant ICS mode, especially the CR mode when $K$ is large. More importantly, we find that the CR mode in normal pulsars usually does not meet the $l_e > h_{\rm min}$ criterion, so that $\Delta V$ only increases slowly with increasing $K$, due to decrease of the height. Since particle acceleration is closely related to $\Delta V$, one would expect that the magnetosphere activities would not change noticeably when a significant toroidal oscillation is excited. Observationally there is no noticeable emission property changes associated with glitches in most normal radio pulsars \citep[e.g. for Vela pulsar][]{Helfand2001}. This is understandable within the framework of our model.

Even though the gap properties do not change significantly, glitch-induced oscillations would enlarge the polar cap size, and modify the geometry of the radio beams. \cite{Weltevrede2011} discovered a glitch-induced pulse profile changes in PSR J1119-6127. Normally it has a single-peaked profile. After a large amplitude glitch, it revealed a double-peaked profile for some time, and finally returned to the single-peaked profile. The duration of the double-peaked profile is not well measured, but it is constrained to a range between minutes and months. We suggest that the change of pulse profile is due to the oscillation effect discussed in this paper. After the glitch, oscillations enlarge the polar cap angle and hence, also the size of the radiation cone. Imagine that the line of sight originally cuts across the edge of an emission cone, so that a single peak is observed. After the emission cone is enlarged due to the oscillation effect, the line of sight now cuts through a hollow cone, so that a double-peaked profile is observed. According to the observations of \cite{Weltevrede2011} (their Fig.5), the center of the single-peaked profile is not consistent to that of the double-peaked profile. This is possible within our model, since toroidal oscillations introduce another asymmetry with respect to the misalignment between the magnetic axis and the spin axis. More detailed simulations are needed to testify or falsify this picture. Future radio long-term observations following glitches will bring critical clues to test this model.

    \subsection{AXPs/SGRs}

    Pulsed radio emission is observed after flaring activities of some AXPs/SGRs, which are also accompanied by glitching activities \citep{Kaspi2003,Dib2014,Olausen2014}. This suggests that the condition of radio emission is satisfied after the flares/glitches. In the following, we show that the observations may be understood within the framework of our model.

    The physical nature of AXPs/SGRs is still an open question. The standard model is that they are neutron stars with superstrong magnetic fields, known as magnetars \citep{Thompson1995,Thompson1996}. Alternatively, a solid quark star model was suggested by \cite{Xu2003,Xu2013}. In the following, we discuss the radio activation mechanism within the framework of both models.

\subsubsection{Magnetar model}

    In Fig. \ref{P=10_B12=100_T6=5_l=2_m=0}, we calculate the gap properties for typical parameters of an AXP/SGR:  $P=10\,{\rm s}$, $T=5\times10^6\,{\rm K}$ and $B=10^{14}\,{\rm G}$. Because the magnetic field strength already exceeds the critical value $B_q$, the Klein-Nishina reduction effect makes cross section small, so that $l_e$ becomes very large. The ICS mode would not be important unless we consider the region far from the surface where magnetic field strength is below $B_q$. Since the inner gap height is much shorter than the radius of the pulsar, the resonant ICS mode is not important in any case. The $2\gamma$ mode is more dominant thanks to the high temperature. However, due to strong magnetic field, the CR mode would be more effective than $2\gamma$ mode and would determine the properties of the gap\footnote{It was suggested that photon splitting may suppress pair production in strong magnetic fields \citep{Baring1998,Baring2001}. This requires that both photon modes split. Later studies showed that photon-splitting are only valid in one polarization mode \citep{Usov2002}. The CR and ICS modes discussed in this paper therefore are also relevant in a magnetar environment.}.

    Radio activation of AXPs/SGRs after outburst/glitch may be understood within the following picture: With the strong magnetic field in a magnetar, the CR mode gap would be above the pulsar radio emission deathline even in the quiescent state. However, as slow rotators, the solid angle of the radio emission beam (if exist) would be very small, so that the chance for detecting radio emission from a magnetar during quiescent state is very low. After an outburst/glitch, toroidal oscillations would be excited. The strength of radio emission may be enhanced (due to the larger potential). More importantly, the effect polar cap angle of the pulsar (Eq. (\ref{polar_angle_m=0})) would be increased significantly when oscillations occur. The radio emission beam would be wider, so that it can be detected by an observer on earth. Later, as oscillations damp away, the radio emission decays due to the decrease of the voltage or due to a less favorable geometric configuration. It finally vanishes when the radio beam exits the line of sight. Within this picture, not all outburst/glitches in magnetars would excite radio emission, which seems to be consistent with the data.

    \subsubsection{Solid Quark Star Model}

In this sub-section, we discuss the solid quark star model for AXPs/SGRs \citep{Xu2003,Xu2013}.
Within this model, no strong magnetic field is needed to power outbursts. We therefore assume that AXPs/SGRs have a similar magnetic field as normal pulsars, i.e. $B \sim 10^{12}\,{\rm G}$\footnote{In this model, the spindown behavior of AXPs/SGRs is attributed to the existence of a fossil accretion disk \citep[e.g.][]{Alpar2001}.}. Due to their long periods, these objects are then usually below the radio emission ``death line'', so that they are radio quiet. When an outburst occurs (which attributes to star quakes in the solid quark star model), oscillations are excited. The oscillation-induced potential enhances voltage along the gap, so that one or more gap modes can be activated. The pulsar moves above to the death line, and becomes radio loud. Later, the oscillations are damped out. Radio emission fades away and eventually disappears when the gap state falls below the death line again.

It is worth reviewing here the physical picture of outbursts and toroidal oscillations within the solid quark model.
We consider bulk-variable starquakes \citep{Zhou2014} in solid quark stars. Various quark star equation-of-state defines a general positive $R-M$ relation, with a turn-over at the high $M$ range. There exists a maximum radius $R_{\rm max}$ in the $R-M$ diagram. Due to accretion from a fossil disk, the mass of the quark star increases and might exceed the mass corresponding to $R_{\rm max}$. As mass keeps increasing, the quark star would contract to seek a more stable configuration. Since the star is in the solid state, the elastic energy would be accumulated to resist radius contraction.
When the accumulated elastic energy finally exceeds a threshold, the solid would crack, and a bulk-variable starquake would happen. Strong oscillations are then excited. The gravitational and elastic energy released in a starquake may be estimated as
\begin{equation}\label{energy}
    \frac{GM^2}{R}\frac{\Delta R}{R} \sim 10^{53}\frac{\Delta R}{R}\,{\rm erg}
\end{equation}
for typical parameters: $M\sim 1.4M_\odot$, $R\sim10^6\,{\rm cm}$. Here $\Delta R$ is change of radius during configuration rearrangement. Such an energy is certainly large enough to power AXP/SGR outbursts and even giant flares. It has been suggested that oscillation-driven radiation could also act as the radiation mechanism of giant flares within this model \citep{Xu2006}. The starquakes induce oscillations, which convert the gravitational and elastic energy to the mechanical energy of oscillations. The oscillations then lead to acceleration of particles based on the mechanisms discussed in this paper, so that the mechanical energy of oscillations is converted to the kinetic energy of the particles, and is finally converted to high energy photon energy via CR or ICS mechanisms. An SGR/AXP outburst is then produced.
The kHz quasi-periodic oscillations (QPOs) observed after SGR giant flares \citep[e.g.][]{Israel2005} may hint towards global oscillations of the central compact star, which is consistent with this picture.

    Along the similar line, below we show that transient pulsed radio emission of several AXPs/SGRs detected after outbursts or giant flares is also consistent with such a picture.
    We calculate the gap properties of the four radio-loud AXPs/SGRs with different oscillation amplitudes. The magnetic field is assumed as $10^{12}\,{\rm G}$ in order to explain lack of radio emission in the quiescent state (below the death line). As an example, we show the result of 1E 1547.0-5408 in Fig. \ref{P=2.07_B12=1_T6=5_l=2_m=0}. The black solid line with inverted triangle represent the maximum voltage across the polar cap, i.e. $\Delta V$ less than this critical value is necessary for radio emission. Notice that the results of $K=10^{-3}$ can represent the gap properties without oscillation.
    We can see that with a dipole magnetic field consistent, the result is consistent with our expectation: In the quiescent state, there is no oscillation, $\Delta V$ of all three mode are larger than $\Delta V_{\rm max}$, so that the star is radio quiet. After an outburst, stellar oscillations are excited. As long as $K$ exceeds a critical value, there is at least one mode whose $\Delta V$ becomes less than $\Delta V_{\rm max}$, so that the gap is activated and the star would become radio loud. With oscillations damped away later, radio emission decays and finally vanishes.

    We note that a dipole magnetic field is not a necessary condition.
For a multipole field, gaps are more likely above the death line. However, if the magnetic field strength is even weaker (compared with the typical value $10^{12}\,{\rm G}$), the gap would be below the death line in the quiescent state. The mechanism discussed above then becomes valid.

    For other AXPs/SGRs with radio activation, we also carried out corresponding calculations. Similar to the result of 1E 1547.0-5408, we find that in a dipolar magnetic field with $B=10^{12}\,{\rm G}$, there always exists a critical $K$ value.
    When $K$ exceeds the critical value, the gaps would be activated above the death line. In Table \ref{T 1}, we show the critical values of $K$ of the four radio AXPs/SGRs.
    For solid quark stars, a relatively large $K$ would be possible. This may be justified by the following two arguments \citep[e.g.][]{Xu2006}. First, in solid quark stars, since the entire star is a rigid solid body, the large gravitational and elastic energy released in star quakes (Eq. \ref{energy}) are large enough to power the large-amplitude oscillations required by the model. For neutron stars, only the crust is believed to be solid. In order to attain the large oscillation energy, one must appeal to the magnetic energy reservoir of the magnetar. Second, since solid quark stars are self-bound via strong interaction, unlike the gravity-bound case for neutron stars, the surface velocity in solid quark stars can be close to or even exceed the Kepler velocity. This allows the star to reach a $K$ value greater than unity.
    We also note that here we only calculate the $l=2, m=0$ mode. For a larger $l$, the critical value of $K$ will decrease markedly, which would ease the radio reactivation condition. Our results suggest that all the radio-activating AXPs/SGRs can be well interpreted in a reasonable parameter space within this model.

\begin{figure*}
    \centering
    \includegraphics [width=15cm]{f4.eps}
    \caption{\label{P=2.07_B12=1_T6=5_l=2_m=0} Gap properties of 1E 1547.0-5408 as functions of dimensionless oscillations amplitude $K$. A dipole magnetic field $B=10^{12}\,{\rm G}$ is assumed. Other parameters are $P=2.07\,{\rm s}$, $T=5\times10^6\,{\rm K}$. The black solid line with inverted triangle markers represent the maximum voltage across the polar cap. If $\Delta V$ is less than this critical value, the gap becomes above the death line, and radio emission is reactivated. Other lines share the same meaning with Fig. \ref{P=1_T6=1_l=2_m=0}. The left and right panels are for the dipole and multipole magnetic field configurations, respectively.}
\end{figure*}

\begin{table}
    \begin{center}
        \begin{tabular}{cccc}
            \hline
            Radio AXPs/SGRs & $P/{\rm s}$ & $T/{\rm 10^6K}$ & critical $K$ \\
            \hline
            1E 1547.0-5408 & 2.07 & 5.0  & $2.54$ (CR) \\
            PSR J1622-4950 & 4.33 & 5.8  & $3.20$ ($2\gamma$) \\
            XTE J1810-197 & 5.54 & 2.1  & $8.90$ (CR) \\
            SGR J1745-2900 & 3.76 & 5.0\tablenotemark{1} & $5.53$ (CR) \\
            \hline
        \end{tabular}
        \tablenotetext{1}{Since the surface temperature of the SGR J174-2900 is unknown, we just adopt a typical value.}
        \caption{\label{T 1}The critical $K$ of the four AXPs/SGRs that show radio reactivation. When $K$ exceeds this value, $\Delta V$ will become less than $\Delta V_{\rm max}$, and radio emission would be activated. A dipole field with $B=10^{12}\,{\rm G}$ is assumed, and the $l$=2, $m$=0 oscillation mode is calculated.}
    \end{center}
\end{table}

\section{CONCLUSIONS AND DISCUSSIONS}
Along with rotation itself, toroidal stellar oscillations can provide additional voltage due to unipolar induction, change Goldreich-Julian charge density, and influence particle acceleration and properties of the pulsar inner gap. In this paper, we studied in detail the gap properties under toroidal oscillations for three different modes: CR, ICS, and $2\gamma$. We show that as the oscillation amplitude varies, inner gap mode changing could happen, including activating a radio pulsar below the radio emission death line. Oscillations can also enlarge the size of the effective polar cap. These interesting properties allow us to interpret the observed radio activation in several AXPs/SGRs within the frameworks of both magnetar model and solid quark star model. Within the magnetar model, the reactivation is interpreted as a geometric effect due to the enlargement of the effective polar cap size. Within the solid quark star model, on the other hand, the reactivation is interpreted as moving a low $B$ star from below the death line to above via oscillations.  The same model can also account for the post-glitch radio pulse profile change observed in PSR J1119-6127.

As discussed earlier in the paper, we only consider the maximum effect of stellar oscillations and ignore the periodical variations of the gap properties. The time scale for periodic variation is too short to give interesting observational effects, even though within a period, gap mode changing may still happen. Also when threshold condition of radio emission is considered (death line), the maximum effect would be the most relevant one to define the threshold condition.

Since the energy of oscillation is proportional to the square of amplitude, $K=\tilde{\eta}/\Omega B$ denotes the ratio of the oscillation velocity and the rotational velocity. The total energy of oscillation may be expressed as $E\propto K^2$. We may estimate it as $K^2$ times of the rotational energy, i.e.
\begin{equation}
    E \sim \frac{1}{2} I \Omega^2 K^2 = 1.58\times10^{46} M_1 R_6^2 P^{-2} K^2 \,{\rm erg},
\end{equation}
where $M_1$ and $R_6$ are the mass and radius of the pulsar normalized to $1 M_\odot$ and $10^6$ cm, respectively, and $P$ is the period of the pulsar in units of second.
Within this model, the oscillation energy should be larger than the observed energy release. This can set a rough lower limit to $K$.

In order to have an observational interest, a relatively large $K$ is needed. This is naturally expected within the solid quark star model, since oscillation occur in the entire star and the oscillation amplitude (associated with $K$) could be very large due to the solid rigidity. For neutron stars, on the other hand, oscillations may only occur in the crust. A large amplitude oscillation may break the crust. On the other hand, we note again that for oscillation modes with a higher $l$ value, a relatively smaller oscillation velocity would be needed in order have a same oscillation effect, as shown in  Eq. (\ref{rho_GJ_ratio}).
Furthermore, the oscillation velocities can vary largely at different locations on the neutron (quark) star's surface. For rotation, on the other hand, different points on the star's surface would have the same anglular velocity $\bm{\Omega}$, and $\rho_{\rm GJ,rot}\propto\bm{\Omega}\cdot\bm{B}$; but for oscillations, different points on the star's surface would have different anglular velocity component perpendicular to the direction of the magnetic field, denoted as $\Omega_{\rm eff\perp}$, and $\rho_{\rm GJ,osc}$ depends on $\Omega_{\rm eff\perp}$ in the polar cap region. For some oscillation modes, $\Omega_{\rm eff\perp}$ could be small in the equatorial region while large in the polar cap region. Therefore significant oscillation effects could be obtained without breaking the crust of a neutron star.

One may ask why not all AXPs/SGRs have radio emission after outbursts. Within the magnetar model, this may be interpreted as a viewing effect: not all activated radio emission can enter our line of sight. Within the solid quark star model, besides beaming effect, there is another possibility. Radio activation is related moving a pulsar from the grave yard, however, given certain parameters, even after oscillations the pulsar is still below the death line. So radio emission would not be observed in any case.

The solid quark star model requires a fossil disk to spin down the star. One related question would be whether the disk would absorb the radio emission from the pulsar magnetosphere. The fossil disks are believed to be very thin, so the probability of radio emission being absorbed should not be very large. In any case, when significant absorptions happen, this model would not produce radio emission in any case. These may correspond to those AXPs/SGRs that never activate.

\paragraph{Acknowledgments}

We thank an anonymous referee for a constructive report, Vicky Kaspi for drawing our attention to the observations of PSR J1119-6127, and Viktoriya Morozova for a helpful communication.
MXL and RXX would like to thank useful discussions at the pulsar group meetings at Peking University.
This work is supported by the National Basic Research Program of China (973 program) No. 2012CB821801,
the National Natural Science Foundation of China (Grant Nos. 11225314, Grant Nos. 11133002, NSFC-11103020),
National Found for Fostering Talents of Basic Science (No. J0630311)
and the Undergraduate Research Fund of Education Foundation of Peking University.


\section{Introduction}
The Whitham equations and other semi-Hamiltonian systems of hydrodynamic type represent an important subclass of diagonal systems of first-order
quasi-linear partial differential equations
\begin{equation}\label{eq:1}
\frac{\partial \beta_i}{\partial t_{2}}=\lambda _{i}(\beta _1,\ldots,\beta_N)\frac{\partial \beta _i}{\partial t_{1}},\qquad i=1,\ldots,N
\end{equation}
which exhibits rich mathematical structure and has a variety of applications in physics (see,
e.g., \cite{Wh, RY}).  The theory of Hamiltonian systems of hydrodynamic type has been developed by Dubrovin and Novikov \cite{DN}. Intimate connections with the theory of conjugate and orthogonal nets of classical differential geometry have been established in \cite{Per70,Gru84,T1,T2}. The construction of Whitham-type equations, that is, the determination of the associated characteristic speeds, usually relies heavily on the sophisticated theory of Riemann surfaces \cite{FFM, FL, Kr1, Kr2}. By contrast, the approach presented in this paper is based on the interpretation of the system
\eqref{eq:1} as a system of partial differential equations describing deformations of critical points and the classical theory of Darboux-type transformations for conjugate nets.

For any function $\Theta(x_1,\ldots,x_N;t_1,\ldots,t_n)$, the critical points
$\bbeta=(\beta_1,\ldots,\beta_N)$ of $\Theta$ are defined implicitly by
\begin{equation}\label{E1}
\left.\frac{\partial \Theta}{\partial x_i}\right|_{\x=\bbeta}=0,\qquad i=1,\ldots,N.
\end{equation}
In the current context, the function $\Theta(\x;\t)$ is assumed to be a solution of the system of linear equations
\begin{equation}\label{eq:2}
\frac{\partial^2\Theta}{\partial x_{i}\partial x_{k}}=A_{ik}\frac{\partial\Theta}{\partial x_{i}}+A_{ki}\frac{\partial \Theta}{\partial x_{k}},\qquad i\neq k
\end{equation
for some functions $A_{ik}(\x)$ and any $\t$.   The set of equations \eqref{eq:2} is well-known in the classical theory of conjugate nets \cite{Darboux}. Particular choices
of solutions $\Theta $ of the system \eqref{eq:2} give rise to different
semi-Hamiltonian systems of the type \eqref{eq:1}. The latter turn out to be invariant under the classical Combescure transformation for conjugate nets acting on $\Theta$. A privileged subclass of systems \eqref{eq:2} is given by the choice
\begin{equation}
A_{ik}=\frac{\varepsilon_k}{x_i-x_k},
\end{equation}
leading to the Euler-Poisson-Darboux (EPD) equations
\begin{equation}\label{eq:3}
\frac{\partial f}{\partial x_{i}\partial x_{k}}=\frac{1}{x_{i}-x_{k}}\left(\varepsilon _{k}\frac{\partial f}{\partial x_{i}}-\varepsilon
_{i}\frac{\partial f}{\partial x_{k}}\right),\qquad i\neq k
\end{equation
with constants $\varepsilon _{1},\ldots,\varepsilon _{N}$.
The latter system of equations is seen to play a distinguished role in our approach. The classical Fundamental, Levy and other transformations (see, e.g., \cite{Eisenh}) acting on the system \eqref{eq:3} allow us to construct large classes of systems \eqref{eq:2} and their solutions. In particular, it is shown that the characteristic speeds $\lambda _{i}$ associated with the standard $g$-phase Whitham equations for the Korteweg-de Vries (KdV) and nonlinear Schr\"odinger (NLS) equations may be obtained via iterated Darboux-type
transformations generated by Lauricella-type solutions (specific complete
hyperelliptic integrals) \cite{Lau} of an extended EPD system \eqref{eq:3} with all $
\varepsilon _{i}=\frac{1}{2}$. It is noted that, in the construction presented here, the sophisticated theory of Riemann surfaces need not be employed.

The general idea that (classes of) nonlinear partial differential equations may be regarded as the stationary (critical) points of functionals (functions) is classical and has been elaborated upon in a variety of contexts. In this paper, it is demonstrated that the notion of critical points is natural and may be used very effectively in the specific context of hydrodynamic-type systems \eqref{eq:1}. It is also emphasised that even though the significance of the EPD equations in the theory of Whitham equations has been established
in \cite{KS1, KS2, GKE, Tian}, their role is quite different in the approach presented here.

\section{Semi-Hamiltonian hydrodynamic type systems}

In the seminal paper \cite{T1}, Tsarev proved that, at least locally, all
solutions $\bbeta=\bbeta(t_1,t_2)$ of a semi-Hamiltonian system \eqref{eq:1} are implicitly given by the
system
\begin{equation}\label{eq:4}
t_{1}+\lambda _{i}(\bbeta )t_{2}-\omega _{i}(\bbeta )=0,\qquad
i=1,\ldots,N,
\end{equation
where the set of functions $\left\{ \omega_{i}\right\} $ is the solution of the
linear equations
\begin{equation}\label{eq:5}
\frac{\partial \omega _{i}}{\partial \beta _{k}} = B_{ik}(\omega _{k}-\omega_{i}),\qquad i\neq k
\end{equation
with the coefficients $B_{ik}$ being defined by
\begin{equation}\label{E2}
\frac{\partial \lambda _{i}}{\partial \beta _{k}} = B_{ik}(\lambda_k - \lambda_i).
\end{equation}
The condition for the existence of the functions $\omega_i$ (in the sense of a natural Cauchy problem) is equivalent to the assumption that the system (\ref{eq:1}) be semi-Hamiltonian and may be expressed as
\begin{equation}\label{E3}
\frac{\partial B_{ik}}{\partial\beta_l} = \frac{\partial B_{il}}{\partial\beta_k},\qquad i\neq k\neq l\neq i.
\end{equation}
The latter together with the compatibility conditions $\frac{\partial}{\partial\beta_l} (\frac{\partial \lambda_i}{\partial\beta_k})=\frac{\partial}{\partial\beta_k} (\frac{\partial \lambda_i}{\partial\beta_l})$
give rise to the first-order system
\begin{equation}\label{E4}
\frac{\partial B_{ik}}{\partial \beta_l} = B_{ik}B_{kl} - B_{ik}B_{il} + B_{il}B_{lk}.
\end{equation}

Semi-Hamiltonian systems admit an infinite set of
commuting symmetries given by
\begin{equation}\label{eq:6}
\frac{\partial \beta _i}{\partial t_{\alpha }}=\lambda _{i}^{\alpha }(\bbeta )\frac{\partial \beta _i}{\partial t_{1}},\qquad\alpha =2,3,\ldots,
\end{equation
where each set $\{\lambda_i^\alpha\}$ necessarily constitutes another solution of the linear system (\ref{E2}) due to the compatibility condition associated with (\ref{eq:6}). Here, we have made the identification $\lambda_i^2 = \lambda_i$.
The symmetries, in turn, give rise to an infinite family of conservation laws
\begin{equation}\label{E4b}
\frac{\partial P}{\partial t_{\alpha }}=\frac{\partial Q^\alpha}{\partial t_1},
\end{equation}
where the density $P$ and the fluxes $Q^\alpha$ are related by
\begin{equation}\label{eq:8}
\frac{\partial Q^{\alpha}}{\partial \beta _{i}}=\lambda _{i}^{\alpha }(\bbeta )\frac{\partial P}{\partial\beta _{i}}.
\end{equation
The existence of the fluxes is guaranteed if the density obeys the system of linear partial differential equations \cite{T1}
\begin{equation}\label{eq:9}
\frac{\partial P}{\partial\beta _{i}\partial \beta _{k}}=B_{ik}\frac{\partial P}{\partial\beta _{i}}+B_{ki}\frac{\partial P}{\partial \beta _{k}},\qquad i\neq k.
\end{equation
These are indeed compatible modulo the first-order system (\ref{E4}). In fact, the above linear system represents the basic equations of the theory of $N$-dimensional conjugate nets developed more than a century ago \cite{Dar}. In this context, the connection between any quantity $Q^\alpha$ and $P$ is known as a Combescure transformation (see, e.g., \cite{RogersSchief2002}). Moreover, the parametrisation
\begin{equation}
B_{ik}=\frac{\partial}{\partial \beta_k}\ln B_{i}
\end{equation}
implied by the ``conservation laws'' (\ref{E3}) leads to the integrable nonlinear Darboux system (see, e.g., \cite{Kon93}) encapsulated in (\ref{E4}). The relevance of conjugate nets, not necessarily of orthogonal (such as Egoroff) type, in the theory of semi-Hamiltonian systems has been noted in \cite{T2}.

A natural extension of Tsarev's theorem for commuting systems of hydrodynamic type is readily available. Thus, all (local) solutions $\bbeta = \bbeta(\t)$ of $n-1$ commuting flows (\ref{eq:6}) are given implicitly by the system
\begin{equation}\label{eq:10}
\sum_{\alpha=0}^n t_{\alpha }\lambda _{i}^{\alpha }(\bbeta )=0,\qquad
i=1,\ldots,N,
\end{equation}
where
\begin{equation}
 t_0 = 1,\qquad\lambda_i^1=1
\end{equation}
and $\{\lambda_i^0\}$ is the ``general solution'' of the linear system (\ref{E2}). This may be proven in the same manner as Tsarev's theorem for $n=2$ \cite{T1}.

The system of equations (\ref{eq:10}) possesses a distinguished property. Indeed, by virtue of the relations \eqref{eq:8}, it may be reformulated as
\begin{equation}\label{eq:11}
\sum_{\alpha=0}^nt_{\alpha }\frac{\partial Q^\alpha}{\partial \beta _{i}}=0,
\end{equation}
where $Q^0$ represents the ``flux'' associated with the set of functions $\{\lambda_i^0\}$ and $Q^1 = P$. Accordingly, if we introduce the parameter-dependent function
\begin{equation}\label{eq:12}
\Theta(\x;\t) =\sum_{\alpha=0}^nt_{\alpha }Q^{\alpha }(\x)
\end{equation}
then the criticality conditions
\begin{equation}
\left.\frac{\partial \Theta}{\partial x_i}\right |_{\x=\bbeta}=0,\qquad i=1,\ldots,N
\end{equation}
coincide with (\ref{eq:11}). Hence, the generalised Tsarev equations \eqref{eq:10} are descriptive of the critical points of the family of functions (\ref{eq:12}). We therefore conclude that  the theory of semi-Hamiltonian hydrodynamic-type systems of the form \eqref{eq:1} is essentially encoded in the theory of conjugate nets via the linear system \eqref{eq:9}, Combescure transformations \eqref{eq:8} and critical points of the family of functions \eqref{eq:12}. It is observed that Hamiltonian systems of the form \eqref{eq:1} may be viewed as the equations for critical (stationary) points of suitable functions. This fact may be traced back to the original derivation by Whitham \cite{Wh} of the equations for slow modulations (stationary phase) and has subsequently been discussed in different forms in \cite{Dubr1, Lo, KMM}.

In order to reveal the key structure behind the connection between semi-Hamil\-tonian hydrodynamic-type systems and critical points, we observe that elimination of the density $P$ in (\ref{eq:8}) via cross-differentiation shows that the fluxes $Q^\alpha$ are likewise solutions of conjugate net equations with coefficients depending on $\alpha$. However, on use of (\ref{eq:9}), differentiation of (\ref{eq:8}) produces
\begin{equation}\label{E4a}
 \frac{\partial Q^\alpha}{\partial\beta _{i}\partial \beta _{k}}=A_{ik}\frac{\partial Q^\alpha}{\partial\beta _{i}}+A_{ki}\frac{\partial Q^\alpha}{\partial \beta _{k}},\qquad i\neq k
\end{equation}
with coefficients
\begin{equation}
A_{ik} = B_{ki}\left.\frac{\partial P}{\partial \beta_k}\right/\frac{\partial P}{\partial \beta_i}
\end{equation}
which are independent of $\alpha$. Hence, $\Theta$ as defined by (\ref{eq:12}) may be regarded as a linear superposition of solutions of the {\em same} conjugate net equations (\ref{E4a}). It is noted that
\begin{equation}
\frac{\partial A_{ik}}{\partial\beta_l} = \frac{\partial A_{il}}{\partial\beta_k},\qquad i\neq k\neq l\neq i
\end{equation}
and, hence, the compatibility conditions for the linear system (\ref{E4a}) imply that the coefficients $A_{ik}$ constitute another solution of the first-order system (\ref{E4}).

\section{General scheme}

In order to formalise the observations made in the previous section, it is natural to consider functions $\Theta(\x)$ for which the associated Hessian matrix of second-order derivatives $\frac{\partial^2\Theta}{\partial x_i\partial x_k}$ is diagonal at any critical point $\x=\bbeta$. In the linear case, this condition may be met by demanding that $\Theta$ be a solution of a system of conjugate net equations
\begin{equation}\label{eq:13}
\frac{\partial^2\Theta}{\partial x_{i}\partial x_{k}}=A_{ik}\frac{\partial\Theta}{\partial x_{i}}+A_{ki}\frac{\partial\Theta}{\partial x_{k}},\qquad
i\neq k \in\{1,\ldots, N\}.
\end{equation
Indeed, if the criticality conditions are satisfied then the mixed derivatives of $\Theta$ vanish, that is,
\begin{equation}
\left.\frac{\partial \Theta}{\partial x_i}\right |_{\x=\bbeta}=0\qquad\Rightarrow\qquad \left.\frac{\partial^2 \Theta}{\partial x_i\partial x_k}\right |_{\x=\bbeta}=0,\qquad i\neq k.
\end{equation}
In this connection, it is convenient to introduce the notation
\begin{equation}
f_{\beta_i} = \left.\frac{\partial f}{\partial x_i}\right|_{\x=\bbeta}
\end{equation}
for any function $f = f(\x)$ so that the criticality conditions may be formulated as $\Theta_{\beta_i}=0$. As mentioned in the preceding, the system (\ref{eq:13}) is compatible if the coefficients $A_{ik}$ obey the first-order equations
\begin{equation}\label{darboux}
\frac{\partial A_{ik}}{\partial x_l} = A_{ik}A_{kl} - A_{ik}A_{il} + A_{il}A_{lk}.
\end{equation}
The latter imply that
\begin{equation}
A_{ik} = \frac{\partial}{\partial x_k}\ln A_i
\end{equation}
for some potentials $A_i$ so that the nonlinear Darboux system
\begin{equation}\label{eq:14}
\frac{\partial^2 A_i}{\partial x_{k}\partial x_{l}}=\frac{1}{A_k}\frac{\partial A_k}{\partial x_l}\frac{\partial A_i}{\partial x_{k}}+\frac{1}{A_l}\frac{\partial A_l}{\partial x_k}\frac{\partial A_i}{\partial x_{l}},\qquad i\neq k\neq l\neq i
\end{equation}
results.

In order to study critical points which depend on some parameters $\t = (t_1,\dots,t_n)$, we consider the linear combination
\begin{equation}\label{eq:16}
\Theta(\x;\t) =\sum_{\alpha =0}^nt_{\alpha }\Theta ^{\alpha }(\x),\qquad t_0=1
\end{equation
of functions $\Theta^{\alpha }$ which satisfy the same conjugate net system \eqref{eq:13}. The critical points $\bbeta$ of $\Theta$ are then implicitly but uniquely defined by
\begin{equation}\label{eq:17}
\Theta_{\beta_i} = \sum_{\alpha =0}^{n}t_{\alpha}\Theta^\alpha_{\beta _{i}} =0,\qquad i=1,\ldots,N,
\end{equation
wherever the Hessian determinant $\det{(\Theta _{\beta _{i}\beta _{k}})}_{i,k}$ does not vanish. The latter is equivalent to \mbox{$\Theta_{\beta_i\beta_i}\neq 0$} since $\Theta_{\beta_i\beta_k}=0$ for $i\neq k$. Hence, in the following, we confine ourselves to the regular sector for which these conditions hold. Now, differentiation of (\ref{eq:17}) with respect to $t_\gamma$ yields
\begin{equation}\label{E5}
\Theta^\gamma_{\beta_i} + \sum_{\alpha =0}^{n}\sum_{k=1}^N t_{\alpha}\Theta^\alpha_{\beta _{i}\beta_k}\frac{\partial\beta_k}{\partial t_\gamma} =0,
\end{equation}
while differentiation of (\ref{eq:16}) with respect to $x_i$ and $x_k$ and evaluation at the critical points $\bbeta$ produce
\begin{equation}
\Theta_{\beta_i\beta_k} = \sum_{\alpha=0}^n t_\alpha\Theta^\alpha_{\beta_i\beta_k}
\end{equation}
so that changing the order of summation in (\ref{E5}) leads to
\begin{equation}\label{eq:18}
\frac{\partial \beta _{i}}{\partial t_{\gamma }}=-\frac{\Theta^\gamma_{\beta _{i}}}{\Theta _{\beta _{i}\beta _{i}}},\qquad i=1,\ldots,N,\qquad \gamma =1,\ldots,n,
\end{equation
where we have used the fact that $\Theta_{\beta_i\beta_k}=0$ for $i\neq k$.

The system of first-order equations (\ref{eq:18}) determines the evolution of the critical points $\bbeta$ as a function of the ``times'' $t_\gamma$. Elimination of $\Theta_{\beta_i\beta_i}$ gives rise to the system of hydrodynamic type
\begin{equation}\label{eq:19}
\frac{\partial \beta _{i}}{\partial t_{\gamma }}=\lambda _{i}^{\gamma
}(\bbeta )\frac{\partial \beta _{i}}{\partial t_{1}},\qquad i=1,\ldots,N,\qquad\gamma
=2,\ldots,n
\end{equation
with characteristic speeds defined by
\begin{equation}\label{eq:20}
\lambda _{i}^{\gamma }(\x)=\left.\frac{\partial \Theta^{\gamma}}{\partial x_i}\right/\frac{\partial \Theta^{1}}{\partial x_i}
\end{equation
and Riemann invariants $\beta _{i}$. The corresponding generalised Tsarev equations are given by (\ref{eq:17}) divided by $\Theta^1_{\beta_i}$. Differentiation of (\ref{eq:20}) shows that
\begin{equation}\label{eq:21}
\frac{\partial \lambda _{i}^{\gamma}}{\partial x_{k}} = B_{ik}(\lambda
_{k}^{\gamma }-\lambda _{i}^{\gamma }),\qquad B_{ik} = A_{ki}\left.\frac{\partial \Theta^1}{\partial x_k}\right/\frac{\partial \Theta^{1}}{\partial x_i},\qquad i\neq k
\end{equation}
by virtue of the conjugate net equations (\ref{eq:13}). The latter also imply that
\begin{equation}
B_{ik}=\frac{\partial}{\partial x_k}\ln B_i,\qquad B_i = \frac{1}{A_i}\frac{\partial\Theta^1}{\partial x_i}
\end{equation
so that the definition of a semi-Hamiltonian hydrodynamic-type system
\begin{equation}\label{eq:22}
\frac{\partial B_{ik}}{\partial x_l} = \frac{\partial B_{il}}{\partial x_k},\qquad i\neq k\neq l\neq i
\end{equation}
is indeed met. Thus, any parametric family of solutions $\Theta(\x;\t)$ of the linear system \eqref{eq:13} of the form \eqref{eq:16} defines a commuting set of semi-Hamiltonian hydrodynamic-type systems \eqref{eq:19} governing the critical points $\bbeta$ of $\Theta$. Moreover, any Combescure transform
\begin{equation}
\tilde{\Theta}(\x;\t) = \sum_{\alpha=0}^nt_\alpha\tilde{\Theta}^\alpha(\x)
\end{equation}
of $\Theta(\x;\t)$ with the Combescure transforms $\tilde{\Theta}^\alpha$ of $\Theta^\alpha$ being defined by the compatible systems
\begin{equation}
  \frac{\partial \tilde{\Theta}^\alpha}{\partial x_i} = \sigma_i\frac{\partial \Theta^\alpha}{\partial x_i},\qquad \frac{\partial \sigma_i}{\partial x_k} = A_{ik}(\sigma_k - \sigma_i),\qquad i\neq k
\end{equation}
gives rise to the same criticality conditions (\ref{eq:17}) and, hence, the same critical points $\bbeta$. Accordingly, any (equivalence) class of Combescure-related functions $\Theta$ encodes a single set of commuting hydrodynamic-type systems (\ref{eq:19}), (\ref{eq:20}).

An immediate consequence of the hydrodynamic-type system \eqref{eq:19} is that
\begin{equation}\label{eq:23}
\frac{\partial \Theta^{\alpha }(\bbeta)}{\partial t_{\gamma }}=\frac{\partial
\Theta^{\gamma}(\bbeta)}{\partial t_{\alpha }},\qquad \alpha,\gamma\geq1.
\end{equation
In particular, for $\gamma=1$, the conservation laws (\ref{E4b}) are retrieved and (\ref{eq:20}) coincides with (\ref{eq:8}) (modulo $\beta_i\rightarrow x_i,\,\alpha\rightarrow\gamma$). In general, (\ref{eq:23}) implies that there exists a ``potential'' function $F(\t)$ such that
\begin{equation}\label{eq:24}
\Theta^{\alpha}(\bbeta)=\frac{\partial F}{\partial t_{\alpha }}.
\end{equation
In fact, it is easy to verify that
\begin{equation}
F(\t) = \Theta(\bbeta(\t);\t)
\end{equation}
up to an additive constant.

Large classes of solutions of the Darboux systems (\ref{eq:13}), (\ref{darboux}) have been constructed by, for instance, the $\partial$-bar-dressing method \cite{ZakMan85,Kon93} and  the algebro-geometric approach \cite{Kri97}. In the following, we treat the conjugate net equations (\ref{eq:13}) in a different manner.

\section{Lauricella functions and hydrodynamic-type systems}

Conjugate net equations of Euler-Poisson-Darboux-type represented by the choice
\begin{equation}\label{E6}
A_{i}\sim\prod_{k\neq
i}(x_{i}-x_{k})^{-\varepsilon _{k}}
\end{equation}
are of particular importance. The system of linear PDEs of the type \eqref{eq:3} was studied a long time ago not only in the context of classical differential geometry (see, e.g., \cite{Darboux}) but also in connection with multi-dimensional generalisations of Euler's hypergeometric function \cite{Lau}. The EPD system of equations \eqref{eq:3} has a number of remarkable properties. In
particular, its general solution may be written in the form
\begin{equation}\label{eq:26}
f(x_{1},\ldots,x_{N})=\sum_{k=1}^{N}\int_{{\mathfrak c}_{k}}f_{k}(z)\prod_{i=1}^{N}(z-x_{i})^{-\varepsilon _{i}}dz,
\end{equation
where $f_{k}(z)$ and ${\mathfrak c}_{k}$ are
arbitrary functions and contours of integration on the complex $z$-plane respectively (see, e.g.,
\cite{Lau, Loo, Sti}). A particular choice of functions $f_{k}(z)$ and contours ${\mathfrak c}_{k}$ gives rise to the Lauricella  function $F_{D}$ which appears in various studies. In what follows, we will refer to the functions \eqref{eq:26} as Lauricella-type functions.

A large variety of hydrodynamic-type systems \eqref{eq:19} may now be constructed by considering functions $\Theta$ which are linear combinations of Lauricella-type functions
\eqref{eq:26}. In the following, we briefly state the main results without going into details. Thus, for instance, if all functions $f_k$ vanish except for one which is of the form $ f_{k_0}=\frac{1}{2\pi i}\sum_{\alpha =1}^nt_{\alpha }z^{\alpha +\varepsilon
_{1}+\cdots+\varepsilon _{N}-1}$ and the corresponding contour ${\mathfrak c}_{k_0}$ constitutes a (counterclockwise oriented) circle of large radius then $\Theta$ may be taken to be
\begin{equation}
\Theta(\x;\t) = \Theta^0(\x) + \mbox{\rm Res}_{z=\infty
}\left(\sum_{\alpha =1}^nt_{\alpha }z^{\alpha -1}\prod _{i=1}^{N}\left(1-\frac{x_{i}}{z}\right)^{-\varepsilon _{i}}\right),
\end{equation}
where $\Theta^0$ is an arbitrary solution of the EPD system (\ref{eq:3}). Therefore, $\Theta^\alpha$ and, by virtue of (\ref{eq:20}), the characteristic speeds $\lambda_i^\alpha$ of the corresponding hydrodynamic type system (\ref{eq:19}) are polynomials in $\x$ of degree $\alpha$ and $\alpha-1$ respectively. In the case $\varepsilon _{i}=\frac{1}{2}$, $i=1,\ldots,N$, the system \eqref{eq:19} represents the dispersionless limit of the multi-component KdV equation or its higher-order counterparts \cite{KMM}. In particular, for $N=2$,
the dispersionless NLS equation or one-layer Benney system with $\lambda_{1}=\frac{1}{2}(3x_{1}+x_{2})$, $\lambda_{2}=\frac{1}{2}(x_{1}+3x_{2})$
is obtained \cite{Zak80,KMM}. On the other hand, if $\varepsilon _{1}=\varepsilon _{2}=-\frac{1}{2}$ then $f(x_{1},x_{2})=-\frac{1}{2}t_{1}(x_{1}+x_{2})-\frac{1}{8}t_{2}(x_{1}-x_{2})^{2}+\cdots$ \cite{KMM11} and the characteristic speeds are $\lambda _{1}=\frac{1}{2}(x_{1}-x_{2})$, $\lambda _{2}=\frac{1}{2}(x_{2}-x_{1})$. In terms of the variable $\phi =\exp[\frac{1}{4}(\beta_{1}-\beta _{2})^{2}]$, this system is equivalent to
\begin{equation}
\frac{\partial^2\phi}{\partial t_2^2}=\frac{\partial^2 \exp \phi}{\partial t_1^2}
\end{equation}
which is nothing but the dispersionless Toda equation \cite{Kod90} or the 1+1-dimensional Boyer-Finley equation \cite{BoyFin82}. More generally, for arbitrary $N\geq 2$ and all $\varepsilon _{i}=-\frac{1}{2}$, the $N$-component extensions of the dispersionless Toda equation are generated. In particular, for even $N$, the corresponding hydrodynamic-type systems govern deformations of the branch points for the special Seiberg-Witten spectral curves \cite{KMM13}.

The mixed cases with $\varepsilon _{i}=\pm \frac{1}{2}$ are likewise of interest. Thus, for $N=2$ and  $\varepsilon _{1}=\frac{1}{2}$, $\varepsilon _{2}=-\frac{1}{2}$, one obtains the hydrodynamic-type system
\begin{equation}
\frac{\partial \beta _{1}}{\partial t_{2}}=\frac{1}{2}(3\beta _{1}-\beta
_{2})\frac{\partial \beta _{1}}{\partial t_{1}},\qquad \frac{\partial \beta
_{2}}{\partial t_{2}}=\frac{1}{2}(\beta _{1}+\beta _{2})\frac{\partial \beta
_{2}}{\partial t_{1}}
\end{equation}
which, on applying the substitution $\beta _{2}\rightarrow -\beta _{2}$, may be regarded as a hybrid of the dispersionless NLS and Toda equations. The case $N=4$ and $\varepsilon _{1}=\varepsilon _{2}=\frac{1}{2},$ $\varepsilon_{3}=\varepsilon _{4}=-\frac{1}{2}$ leads to
\begin{equation}
\begin{array}{l}
f(x_{1},x_{2,}x_{3},x_{4})=\displaystyle t_{1}\frac{1}{2}(x_{1}+x_{2}-x_{3}-x_{4})\\[3mm]
\qquad\qquad\mbox{}+\displaystyle t_{2}\frac{1}{8}[3x_{1}^{2}+2x_{1}x_{2}+3x_{2}^{2}-2(x_{1}+x_{2})(x_{3}+x_{4})-(x_{3}-x_{4})^{2}]+\cdots
\end{array}
\end{equation}
 so that the corresponding hydrodynamic-type system with four Riemann invariants has the characteristic speeds
\begin{equation}
\begin{array}{rlrl}
\lambda _{1} =&\displaystyle\frac{1}{2}(3x_{1}+x_{2}-x_{3}-x_{4}), \qquad &
\lambda _{2} =&\displaystyle\frac{1}{2}(x_{1}+3x_{2}-x_{3}-x_{4})\\[4mm]
\lambda _{3} =&\displaystyle\frac{1}{2}(x_{1}+x_{2}+x_{3}-x_{4}), \qquad &
\lambda _{4} =&\displaystyle\frac{1}{2}(x_{1}+x_{2}+x_{4}-x_{3}).
\end{array}
\end{equation}
The latter give rise to a hydrodynamic-type system which contains the dispersionless NLS equation ($\beta_3=\beta_4=0$) and dispersionless Toda system ($\beta_1=\beta_2=0$) as particular reductions.

Characteristic speeds of an entirely different nature are obtained if one makes the choice
\begin{equation}\label{eq:27}
\Theta(\x;\t) = \Theta^0(\x) +
f(\x;\t)=\Theta^0(\x) + t_{1}\frac{1}{2}\sum_{i=1}^{N}x_{i}+t_{2
\int_{x_{k}}^{x_{k+1}}\prod_{i=1}^{N}(z-x_{i})^{-\frac{1}{2}}dz
\end{equation
for some $k<N$. Indeed, one is led to the system
\begin{equation}\label{eq:28}
\frac{\partial \beta _{i}}{\partial t_{2}}=\lambda _{i}(\bbeta )\frac
\partial \beta _{i}}{\partial t_{1}},
\end{equation
wherein the characteristic speeds $\lambda _{i}$ are derivatives of hyperelliptic integrals, that is,
 \begin{equation}
\lambda _{i}(\x)=2\frac{\partial}{\partial x_i}\left(\int_{x_{k}}^{x_{k+1}}\prod_{k=1}^{N}(z-x_{k})^{-\frac{1}{2}}dz\right).
\end{equation}
For $N=3,4$, such systems have been considered in \cite{Pav}. More general hydrodynamic-type systems may be constructed by choosing
\begin{equation}\label{eq:29}
\Theta = \Theta^0 + t_{1}\sum_{i=1}^{N}\varepsilon
_{i}x_{i}+t_{2}\int_{x_{k}}^{x_{k+1}}\prod_{i=1}^{N}(z-x_{i})^{
\varepsilon _{i}}dz.
\end{equation}

Another example of hydrodynamic-type systems with underlying Lauricella-type functions corresponds to
the choice
\begin{equation}\label{eq:30}
\Theta =\Theta^0 + t_{1}\sum_{i=1}^{N}\varepsilon _{i}x_{i}+t_{2}F_{D}(a,\varepsilon
_{1},\ldots,\varepsilon _{N},b;x_{1},\ldots,x_{N}),
\end{equation
where $F_{D}(a,\varepsilon _{1},\ldots,\varepsilon _{N},b;x_{1},\ldots,x_{N})$
is defined by
\begin{equation}\label{eq:Lau}
F_D:=\frac{\Gamma (b)}{\Gamma (a)\Gamma (b-a)}\int_{1}^{\infty }z^{\varepsilon
_{1}+\cdots+\varepsilon
_{N}-b-2}(z-1)^{b-a-1}\prod_{i=1}^{N}(z-x_{i})^{-\varepsilon _{i}}dz.
\end{equation}
The latter constitutes Lauricella's generalisation of the classical hypergeometric function
to the case of $N$ variables \cite{Lau}. Here, $a$ and $b$ are arbitrary
parameters and $\Gamma$ is the standard Gamma function.. The hydrodynamic-type system \eqref{eq:19}
is therefore of the form
\begin{equation}\label{eq:31}
\frac{\partial \beta _{i}}{\partial t_{2}}=\frac{1}{\varepsilon _{i}}\frac
\partial F_{D}(\bbeta )}{\partial \beta _{i}}\frac{\partial \beta _{i}}
\partial t_{1}}.
\end{equation
Different choices of the constants $a,\varepsilon _{1},\ldots,\varepsilon _{N},b$ give rise to various integrable hydrodynamic-type systems of interest.

\section{Whitham equations}

The relative simplicity of Lauricella-type functions make them very attractive as functions which may be employed in the classical theory of transformations of conjugate nets (see, e.g., \cite{Eisenh}). Thus, given particular  solutions of the EPD equations, one may construct large classes of more complicated systems of conjugate net equations. Specifically, gauge transformations are perhaps the simplest way of generating new systems of conjugate net equations from any given one. Indeed, if $f$ and $f^1$ are solutions of a given set of conjugate net equations (\ref{eq:2}) with coefficients $A_{ik}^0$ then one my directly verify that the gauge transformation
\begin{equation}
  f\rightarrow\Theta = \frac{f}{f^1}
\end{equation}
leads to a another system of conjugate net equations (\ref{eq:2}) with coefficients and potentials
\begin{equation}\label{E7}
A_{ik} = A_{ik}^0 - \frac{\partial}{\partial x_k}\ln f^1,\qquad A_i = \frac{A_i^0}{f^1}.
\end{equation}
In particular, if we choose the potentials
\begin{equation}
A_i^0 = \prod_{k\neq i}(x_i - x_k)^{-\varepsilon_{k}}
\end{equation}
corresponding to the EPD equations (\ref{eq:3}) then we are led to the new potentials \cite{KS}
\begin{equation}
A_i = \frac{1}{f^1}\prod_{k\neq i}(x_i - x_k)^{-\varepsilon_{k}}.
\end{equation}

In order to apply the theory developed in Section 3, we now consider the linear combination
\begin{equation}
f = \tilde{f} + t_1+ t_2 f^0,\qquad f^0 = \sum_{i=1}^{N}\varepsilon _{i}x_{i},
\end{equation}
where $\tilde{f}$ is an arbitrary solution of the EPD equations and $1$ and $f^0$ are constant and linear solutions respectively. Hence, if we superimpose the separable solutions of the EPD equations according to
\begin{equation}
f^1 = \oint_{\mathfrak c}
\prod_{i=1}^{N}(z -x_{i})^{-\varepsilon _{i}}dz
\end{equation}
for some contour $\mathfrak c$ on the complex $z$-plane then the solution $\Theta$ of the conjugate net equations with coefficients (\ref{E7}) is of the required form
\begin{equation}\label{E8}
\Theta = \Theta^0 + t_1\Theta^1 + t_2\Theta^2,\qquad \Theta^0 = \frac{\tilde{f}}{f^1},\qquad \Theta^1 = \frac{1}{f^1},\qquad \Theta^2 = \frac{f^0}{f^1}.
\end{equation}
Accordingly, the characteristic speeds $\lambda_i=\lambda_i^2$ of the associated semi-Hamiltonian hydrodynamic-type system (\ref{eq:19})$_{n=2}$ are given by
\begin{equation}\label{eq:32}
\lambda _{i} =\left.\frac{\partial \Theta^{2}}{\partial x_i}\right/\frac{\partial \Theta^{1}}{\partial x_i} = f^0 -f^1\left.\frac{\partial f^0}{\partial x_i}\right/\frac{\partial f^1}{\partial x_i}.
\end{equation}
For $N=3,4$, $\epsilon_i=\frac{1}{2}$ and an appropriate choice of the contour $\mathfrak c$, this hydrodynamic-type system coincides with the classical one-phase Whitham equations for the averaged KdV and NLS equations respectively (see, e.g., \cite{Wh,Pav2}). In the form \eqref{eq:32}, the
characteristic speeds for the one-phase Whitham equations have been given in \cite{GKE, Kud}.

In the classical differential-geometric context of conjugate nets, (\ref{eq:32}) regarded as a linear map $f^0\mapsto \lambda_i$ is known as a Levy transformation with respect to the independent variable $x_i$, generated by $f^1$ \cite{Eisenh}. This observation indicates that Levy transformations should play an important role in the theory of Whitham equations. However, it turns out that the identification of the underlying Levy transformations is not as straightforward in the case of the multi-phase Whitham equations. Thus, we  consider an $N+1$-dimensional extension of the EPD system
(\ref{eq:3}), namely
\begin{equation}\label{eq:3435}
\begin{array}{rl}
\displaystyle
\frac{\partial^2 f}{\partial x_{i}\partial x_{k}} &\displaystyle=\frac{1}{x_{i}-x_{k}}\left(\varepsilon
_{k}\frac{\partial f}{\partial x_{i}}-\varepsilon _{i}\frac{\partial f}{\partial x_{k}}\right),\\[6mm]
\displaystyle
\frac{\partial^2f}{\partial x_{i}\partial y} &\displaystyle=\frac{1}{x_{i}-y}\left(\varepsilon _{k}\frac{\partial f}{\partial x_{i}}-(1-g)\frac{\partial f}{\partial y}\right),
\end{array}
\qquad i\neq k\in\{1,\ldots,N\},
\end{equation}
 where $g$ is a positive integer and $y$ constitutes an auxiliary variable. It admits solutions of the form
\begin{equation}\label{eq:36}
f^{k}=\oint_{{\mathfrak c} _{k}}\frac{(z-y)^{g-1}}{\prod_{i=1}^{N}{(z-x_{i})}^{\varepsilon _{i}}}dz
\end{equation
for some contours ${\mathfrak c}_k$, $k=1,\ldots, N$. The $(g-1)$-fold Levy transformation of any solution $\varphi $ of \eqref{eq:3435} with respect to the auxiliary variable $y$ is given by (cf.\ \cite{MatSal91})
\begin{equation}\label{eq:37}
\varphi _{g-1}=\mathcal{L}_{g-1}[\varphi] =\frac{\left\vert
\begin{array}{cccc}
\varphi & \varphi _{y} & \cdots & \varphi _{(g-1)y} \\[1mm]
\hat{\fb} & \hat{\fb}_{y} & \cdots & \hat{\fb}_{(g-1)y
\end{array
\right\vert }{\left\vert
\begin{array}{ccc}
\hat{\fb}_{y} & \cdots & \hat{\fb}_{(g-1)y
\end{array
\right\vert },\qquad \hat{\fb} = \left(\begin{array}{c}f^2\\ \vdots\\ f^g\end{array}\right).
\end{equation
It constitutes a solution of the conjugate net equations \eqref{eq:2} with appropriate coefficients $A_{ik}$ which are independent of the choice of $\varphi $. For $g=1$, the above is to be interpreted as $\varphi_0 = \varphi$.

The Levy transforms $f^0_{g-1} = \mathcal{L}_{g-1}[f^0]$ and $f^1_{g-1} = \mathcal{L}_{g-1}[f^1]$ of the two particular solutions
\begin{equation}\label{eq:38}
f^{0}=\sum_{i=1}^{N}\varepsilon _{i}x_{i}+(1-g)y, \qquad f^{1}
\end{equation
of the extended EPD system (\ref{eq:3435})
therefore read
\begin{equation}\label{eq:39}
f_{g-1}^{0}=\frac{\sum_{i=1}^{N}\varepsilon _{i}x_{i}\left\vert
\begin{array}{ccc}
\hat{\fb}_{y} & \cdots & \hat{\fb}_{(g-1)y
\end{array
\right\vert +(g-1)\left\vert
\begin{array}{cccc}
\hat{\fb}-y\hat{\fb}_{y} & \hat{\fb}_{yy} & \cdots & \hat{\fb}_{(g-1)y
\end{array
\right\vert }{\left\vert
\begin{array}{ccc}
\hat{\fb}_{y} & \cdots& \hat{\fb}_{(g-1)y
\end{array
\right\vert }
\end{equation
and
\begin{equation}\label{eq:40}
f _{g-1}^{1}=\frac{\left\vert
\begin{array}{ccc}
\fb & \cdots & \fb_{(g-1)y
\end{array
\right\vert }{\left\vert
\begin{array}{ccc}
\hat{\fb}_{y} & \cdots & \hat{\fb}_{(g-1)y
\end{array
\right\vert },\qquad \fb = \left(\begin{array}{c} f^1\\ \vdots\\ f^g\end{array}\right).
\end{equation
Since the functions $f^{k}$ are polynomials in $y$, one concludes that
\begin{equation}\label{eq:41}
f_{g-1}^{0}=\frac{\sum_{i=1}^{N}\varepsilon _{i}x_{i}\left\vert
\begin{array}{ccc}
\hat{\mbox{\boldmath $I$}}_{g-2} & \cdots& \hat{\mbox{\boldmath $I$}}_{0
\end{array
\right\vert -\left\vert
\begin{array}{cccc}
\hat{\mbox{\boldmath $I$}}_{g-1} & \hat{\mbox{\boldmath $I$}}_{g-3} & \cdots & \hat
\mbox{\boldmath $I$}}_{0
\end{array
\right\vert }{\left\vert
\begin{array}{ccc}
\hat{\mbox{\boldmath $I$}}_{g-2} & \cdots & \hat{\mbox{\boldmath $I$}}_{0
\end{array
\right\vert }
\end{equation
and
\begin{equation}\label{eq:42}
f_{g-1}^{1}=\frac{\left\vert
\begin{array}{ccc}
{\mbox{\boldmath $I$}}_{g-1} & \cdots & {\mbox{\boldmath $I$}}_{0
\end{array
\right\vert }{\left\vert
\begin{array}{ccc}
\hat{\mbox{\boldmath $I$}}_{g-2} & \cdots & \hat{\mbox{\boldmath $I$}}_{0
\end{array
\right\vert },
\end{equation
where the components $I_l^k$ of $\mbox{\boldmath $I$}_{l}$ and $\hat{\mbox{\boldmath $I$}}_{l}$ are given by
\begin{equation}\label{eq:43}
I_{l}^k=\oint_{{\mathfrak c}_{k}}\frac{z
^{l}}{\prod_{i=1}^{N}{(z -x_{i})}^{\varepsilon _{i}}}dz.
\end{equation}
By construction, the functions $f_{g-1}^{0}$ and $f_{g-1}^{1}$ are solutions of the same
conjugate net equations \eqref{eq:2}. It is important to note that these solutions do not depend on
the auxiliary variable $y$. This is due to the fact that the particular solutions (\ref{eq:38}) are polynomials in $y$ of degree less than $g$.

Now, consistent with the interpretation of the $g-1$-fold Levy transformation  (\ref{eq:37}) for $g=1$ as the identity transformation, we observe that $f^0_0 = f^0$ and $f^1_0=f^1$. Hence, the canonical  generalisation of the function $\Theta$ given by (\ref{E8}) reads
\begin{equation}\label{eq:44}
\Theta = \Theta^0 + t_1\Theta^1 + t_2\Theta^2,\qquad \Theta^0 = \frac{\tilde{f}_{g-1}}{f^1_{g-1}},\qquad \Theta^1 = \frac{1}{f^1_{g-1}},\qquad \Theta^2 = \frac{f^0_{g-1}}{f^1_{g-1}},
\end{equation}
where $\tilde{f}_{g-1}$ is an arbitrary solution of the conjugate net equations generated by the iterated Levy transformation $\mathcal{L}_{g-1}$. As pointed out in the preceding, since $f_{g-1}^0, f_{g-1}^1,\tilde{f}_{g-1}$ and $1$ are all solutions of the same conjugate net equations, the ratios $\Theta^0,\Theta^1$ and $\Theta^2$ obey another system of conjugate net equations. Hence, once again, we may apply the theory developed in Section 3 and state that the functions
\begin{equation}\label{eq:45}
\lambda _{i} =\left.\frac{\partial \Theta^{2}}{\partial x_i}\right/\frac{\partial \Theta^{1}}{\partial x_i} = f^0_{g-1} -f^1_{g-1}\left.\frac{\partial f^0_{g-1}}{\partial x_i}\right/\frac{\partial f^1_{g-1}}{\partial x_i}
\end{equation}
constitute characteristic speeds of an associated semi-Hamiltonian hydrodynamic-type system (\ref{eq:1}). The nature of the latter depends on the choice of the parameters $\varepsilon_i$. The characteristic speeds $\lambda_i$ may be interpreted as the Levy transforms of $f^0_{g-1}$ with respect to the independent variables $x_i$. In a different context, the action of Levy transformations on hydrodynamic-type systems has been investigated in detail in \cite{Fer}.

Remarkably, for the particular choice $N=2g+1$, $\varepsilon _{i}=\frac{1}{2}$ and the contours
{\mathfrak c}_k$ being canonical cycles for the associated Riemann surfaces of
genus g, the system \eqref{eq:1} with characteristic speeds \eqref{eq:45} is nothing but the
classical $g$-phase Whitham system for the KdV equation. Indeed, if we set
\begin{equation}\label{eq:46}
H_{1}=\frac{\left\vert
\begin{array}{ccc}
\hat{\mbox{\boldmath $I$}}_{g-2} & \cdots & \hat{\mbox{\boldmath $I$}}_{0
\end{array
\right\vert }{\left\vert
\begin{array}{ccc}
{\mbox{\boldmath $I$}}_{g-1} & \cdots & {\mbox{\boldmath $I$}}_{0
\end{array
\right\vert },\qquad H_{2}=-\frac{\left\vert
\begin{array}{cccc}
\hat{\mbox{\boldmath $I$}}_{g-1} & \hat{\mbox{\boldmath $I$}}_{g-3} & \cdots & \hat
\mbox{\boldmath $I$}}_{0
\end{array
\right\vert }{\left\vert
\begin{array}{ccc}
{\mbox{\boldmath $I$}}_{g-1} & \cdots & {\mbox{\boldmath $I$}}_{0
\end{array
\right\vert }
\end{equation
and $\Gamma _{0}=1$, $\Gamma_{1}=\frac{1}{2}\sum_{i=1}^{2g+1}x_{i}$ then
\begin{equation}\label{eq:47}
\lambda _{i} =\left.\frac{\partial(H_{1}\Gamma _{1}+H_{2}\Gamma _{0})}{\partial x_i}\right/ \frac{\partial H_1}{\partial x_i},\qquad i=1,\ldots,2g+1
\end{equation
which coincides with the expressions for the characteristic speeds given in~\mbox{\cite{Tian, El}}. Other choices of the seed solution $f^{0}$ give rise to other members of Whitham hierarchies, all of which describe deformations of hyperelliptic curves \mbox{$p^{2}=\prod_{i=1}^{2g+1}(z-\beta _{i})$}. An extension of these results to the case $N=2g+2$ is straightforward. The corresponding equations \eqref{eq:1}, \eqref{eq:45} coincide with the $g$-phase Whitham equations for the NLS equation \cite{FL}. Hydrodynamic-type systems \eqref{eq:1}, \eqref{eq:45} with $\varepsilon _{i}=1/K$ govern the deformation of superelliptic  $(K,N)$ curves $p^{K}=\prod_{i=1}^{N}(z-\beta _{i})$. Finally, we note that our approach to Whitham type equations appears to be different from that used in \cite{OS} to construct hydrodynamic-type systems via generalised hypergeometric functions.

\section*{Acknowledgements}

The first author (YK) was partially supported by NSF grants DMS-1108813 and DMS-1410267.  The second author (BK) acknowledges support by the PRIN 2010/2011 grant 2010JJ4KBA\_003.