Self-similarity and recurrence in stability spectra of near-extreme Stokes waves
Bernard Deconinck Sergey A. Dyachenko Anastassiya Semenova


Deep-learning Optical Flow Outperforms PIV in
Obtaining Velocity Fields from Active Nematics
    Correspondence email addresses:
^∗hongpeng@brandeis.edu, ^hagan@brandeis.edu 

    April 29, 2024
============================================================================================




We consider steady surface waves in an infinitely deep two–dimensional ideal fluid with potential flow, focusing on high-amplitude waves near the steepest wave with a 120 corner at the crest. The stability of these solutions with respect to coperiodic and subharmonic perturbations is studied, using new matrix-free numerical methods. We provide evidence for a plethora of conjectures on the nature of the instabilities as the steepest wave is approached, especially with regards to the self-similar recurrence of the stability spectrum near the origin of the spectral plane.



Water waves, Stokes waves, Benjamin-Feir instability, High-frequency instability, superharmonic instability




§ INTRODUCTION


We consider the stability of spatially periodic waves that propagate with constant velocity without change of form, in potential flow of an ideal (incompressible and inviscid) two-dimensional fluid of infinite depth.
The study of such wave profiles has been the subject of much previous work, going back to <cit.> (also published in <cit.>). 
Stokes' work was followed by numerical computation of these waves by <cit.>, and existence was proved in the works of <cit.> and <cit.>, see also <cit.> for existence of the global branch of wave profiles.
The numerical study of these waves and the nature of their singularities was continued by <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.> and others.
 

In the context of water waves, such waves are usually referred to as Stokes waves.
It was suggested by <cit.> that there exists a progressive wave of maximum height, and that
the angle at the crest of this limiting wave should be 2π/3. Rigorous proofs of these Stokes conjectures came much later. 
<cit.> showed global existence of the limiting Stokes wave, but did not prove
that the angle at the crest is 2π/3.
This result was proved by <cit.> and <cit.> (Reported in English in <cit.>) independently. We refer to the Stokes wave of greatest height as the extreme wave and to waves of near-maximal amplitude as near-extreme waves. Even with the original Stokes conjectures resolved, the study of the graph of the wave profiles remains active, with a number of open problems, as detailed by <cit.>, see also below.  The works by  <cit.> and by <cit.> study the near-extreme waves using both asymptotic and numerical methods. The review by <cit.> discusses many currently active research directions. 


The investigation of the stability of Stokes waves was begun in the works of <cit.>,  <cit.>, <cit.> and <cit.>. Except for the influential experimental work by <cit.>, the focus of these works was on the dynamics of small disturbances of small-amplitude periodic Stokes waves. They unveiled the presence of the modulational or Benjamin-Feir instability with respect to long-wave disturbances in water of sufficient depth, kh>1.363…. Here h is the depth of the water and k=2π/L, with L the period of the Stokes wave. The first rigorous results on the Benjamin-Feir instability were established by <cit.>, followed up very recently by <cit.> and <cit.>. The numerical results of <cit.> reveal the presence of a figure-8 curve in the complex plane of the spectrum of the linear operator governing the linear evolution of the Stokes wave disturbances. Approximations to this figure-8 are obtained by <cit.> and by <cit.>, where the existence of the figure-8 was proven rigorously. <cit.> also examined the critical case kh=1.363…. 



 
<cit.> also brought to the fore the presence of the so-called high-frequency instabilities, existing for narrow ranges of the disturbance quasi-periods. These instabilities were further studied by <cit.> and by <cit.>, where their existence was proven rigorously. 

The instabilities mentioned above play a role in our study of the dynamics of 
large-amplitude Stokes waves, but we illustrate other instability mechanisms, not present for small-amplitude waves. Understandably, the study of large-amplitude Stokes waves, which cannot be thought of as perturbations of flat water, is harder, both from a computational and an analytical point of view. Nonetheless, some groundbreaking examinations have been done, for instance by  <cit.>, <cit.> and for near-extreme waves by <cit.>. These authors all consider perturbations of the Stokes waves with respect to co-periodic (or superharmonic) disturbances, i.e., the Stokes waves and the disturbance have the same minimal period. Their results are recapped in detail below, as they are instrumental to our own investigations. The results in this manuscript follow those of <cit.>, as we present a computational study of the instabilities of periodic Stokes waves, under the influence of disturbances parallel to the propagation direction of the wave. It should be emphasized that all figures presented below are 
quantitatively correct unless they are described as “schematic” in the caption. Similarly, all floating-point numbers given are approximate, of course, but all digits provided are believed to be correct.






§ ONE-DIMENSIONAL WAVES IN WATER OF INFINITE DEPTH



The equations of motion governing the dynamics of the one-dimensional free surface of a two-dimensional irrotational, inviscid fluid (see left panel of Fig. <ref>) are the Euler equations: 


    Φ_xx+Φ_yy=0,        x∈ℝ, y∈(-∞, η(x,t)), 
          η_t= .(-Φ_x η_x+Φ_y)|_y=η(x,t),        x∈ℝ,  y=η(x,t),
          .(Φ_t +1/2(Φ_x^2+Φ_y^2 ) ) 
    |_y = η(x,t)+gη= 0,        x∈ℝ,  y=η(x,t),
          lim_y→ -∞Φ_y=0,        x∈ℝ.


Here y=η(x,t) is the equation of the free surface and Φ(x,y,t) is the velocity potential (i.e., the velocity in the fluid is v=(Φ_x, Φ_y)), subscripts denote partial derivatives, x and y are the horizontal and vertical coordinate respectively, t denotes time, and g is the acceleration of gravity. We ignore the effects of surface tension. Although the Stokes waves are 2π-periodic, it is important to pose the problem above on x∈ℝ, since the perturbations we consider are not necessarily periodic. The first equation expresses the divergence-free property of the flow under the free surface determined by η(x,t). The second and third equation are nonlinear boundary conditions determining the free surface: the kinematic condition (<ref>) expresses that the free surface changes in the direction of the normal derivative to the surface (particles on the surface remain on the surface), whereas the dynamic condition (<ref>) states the continuity of pressure across the surface. Atmospheric pressure has been equated to zero, without loss of generality. 


Since the location of the surface y=η(x,t) is the main focus of the water wave problem, different reformulations have been developed that eliminate the velocity potential in the bulk of the fluid as an unknown.  

<cit.> shows how the problem (<ref>)–(<ref>) can be recast in terms of only the surface 
variables η(x,t) and φ(x,t)=Φ(x,η(x,t),t), 

and the dynamics of η(x,t) and φ(x,t) is governed by an infinite-dimensional Hamiltonian system with η(x,t) and φ(x,t) as canonical variables. The Hamiltonian is the total energy of the system (with potential energy renormalized to account for infinite depth), which depends on the velocity potential Φ(x,y,t) in the bulk of the fluid. 

To avoid the dependence on the bulk, Zakharov's formulation uses the Dirichlet-to-Neumann operator (DNO), producing the normal derivative of the velocity potential at the free surface (the right-hand side of (<ref>)) from the values of φ(x,t). For small-amplitude waves, the DNO is conveniently expressed as a series, as done by <cit.>. For large-amplitude waves, such an expansion is not readily available, and the DNO has to be approximated numerically. To avoid doing so, 
we use conformal variables, see Fig. <ref>: for a 2π-periodic wave, 
a time-dependent conformal transformation maps the half plane in the w=u+i v plane ((u,v) ∈ [-π,π]×(-∞,0])
into the area (x,y) ∈ [-π,π]×(-∞,η] in the physical z=x+iy plane occupied by the fluid. 
The horizontal line v = 0 is mapped into the fluid surface y=η(x,t). The implicit equations of motion in conformal variables are constructed in the works of <cit.>, see also  <cit.>, <cit.>,  <cit.>. 
We use this implicit formulation to study the stability of Stokes waves. 





From these works, the conformal map z(w,t) = x(w,t) + iy(w,t) is a complex-analytic function in ℂ^- that approaches the identity map z(w,t)→ w as w → -i∞, the image of a point at infinite depth. In the conformal variables, the Hamiltonian has the form


    ℋ = 1/2∫_-π^πψk̂ψ du + g/2∫_-π^π y^2 x_u  du,


where ψ(u,t)=φ(x,t) and the operator k̂ = -Ĥ∂_u. Here Ĥ is the periodic Hilbert 
transform defined by the principal-value integral


    Ĥf(u)
    = 1/2π^π_-π f(u') u'-u/2du'.


Equivalently, the Hilbert transform can be defined by its action on Fourier harmonics,
Ĥ e^iku = i (k)  e^iku.The equations of motion are derived by taking variational derivatives of the action 𝒮 = ∫ℒ dt with respect to x, y and ψ. The Lagrangian has the form


    ℒ=∫_-π^πψ(y_t x_u - y_u x_t)du -ℋ+∫_-π^π f(u)(y - Ĥ[x-u]) du,


where the Lagrange multiplier f(u) is chosen to enforce the relation x(u,t) = u-Ĥ[y(u,t)].
We refer to the work of <cit.> for the complete derivation of the equations of motion in conformal variables:


    y_t x_u - y_u x_t    = -Ĥψ_u, 
    
    x_t ψ_u - x_u ψ_t - Ĥ[y_t ψ_u -  y_uψ_t ]    =
    g(x_uy - 1/2Ĥ∂_u y^2).




 §.§ Traveling Waves


Using the conformal variables formulation (<ref>-<ref>), the Stokes waves are obtained by looking for a solution y=y(u-ct), ψ=ψ(u-ct), corresponding to stationary solutions in a frame of reference moving with constant speed c in physical variables, see <cit.>. This gives rise to the so-called <cit.> equation:


    (c^2 k̂ - g) y - g/2(k̂y^2 + 2yk̂ y) = 0.


Since we are interested in the stability of near-extreme Stokes waves, the accurate numerical solution of (<ref>) for near-limiting values of the speed is required. Details of such computations for the Babenko equation (<ref>) are given by <cit.>. In what follows, the ratio of the crest-to-trough height H to wavelength L is used as the definition of wave steepness s = H/L. 


The limiting Stokes wave has the steepness s_lim = 0.1410634839… and speed c_lim = 1.0922850485… as computed by <cit.>.

It is known that the speed c and the Hamiltonian H oscillate as a function of the wave steepness s, for values near the limiting value s_lim, see <cit.>. It is believed there is an infinite number of such oscillations. They are presented schematically in Fig. <ref>. The other details of this figure constitute some of the main results of this paper, discussed below.









 §.§ Linearization about a Stokes wave


In a reference frame traveling with the velocity of a Stokes wave, the Babenko equation describes a stationary solution of the equations (<ref>)–(<ref>). The linear stability of these Stokes waves is 
determined by the eigenvalue spectrum of the linearization in this traveling frame. To obtain the linearization, 
we transform (<ref>-<ref>) to a moving frame, using


    y(u,t)    → y(u-ct,t),      x(u,t) → u - Ĥ y(u-ct,t), 
    
    ψ(u,t)    →ϕ(u-ct, t) -  cĤ y(u-ct,t),


as in <cit.>. The nonlinear system (<ref>–<ref>) in the moving frame becomes


    x_u y_t - y_u x_t    =  - Ĥ(ϕ_u - c ), 
     
    x_u ϕ_t - x_tϕ_u - Ĥ[y_u ϕ_t - y_t ϕ_u]
    + 2cx_t   = c^2k̂ y - g(x_uy - Ĥ[yy_u]).


The linearization about a Stokes wave is found by substituting


    y(u,t)    → y(u) + δ y(u,t)+…    ϕ(u,t) → 0 + δϕ(u,t)+….


Here y(u) corresponds to the Stokes wave and δ y(u,t), δϕ(u,t) are small perturbations. 
Retaining only linear terms in δ y and δϕ leads to the following evolution equations for the perturbations δ y and δϕ:


    Ω̂_21^†δϕ_t - 2cĤδ y_t    = Ŝ_1 δ y, 
    Ω̂_21δ y_t    = k̂δϕ,


where the operator Ω̂_21 is defined via Ω̂_21 f = x_u f + y_u Ĥ f.  The operator Ω̂^†_21 is its adjoint: Ω̂^†_21 f = x_u f - Ĥ[y_u f] and 


    Ŝ_1(y)δ y = (c^2k̂ - g)δ y - 
    g[yk̂δ y + δ y k̂ y + k̂ (yδ y) ].
 

This is rewritten in matrix form as


    L̂∂_t
    [
    [  δϕ; δ y ]] =
    M̂[
    [  δϕ; δ y ]],


with

    L̂ =
    [       0   Ω̂_21; Ω̂_21^†    -2cĤ ],  M̂ =
    [  k̂   0;   0 Ŝ_1 ].


We examine the effect of quasi-periodic perturbations δ y, δϕ using the Fourier-Floquet-Hill (FFH) approach described in <cit.> and <cit.>.  The time dependence for δ y and δϕ is found using separation of variables. 
Moreover, in order to consider quasiperiodic perturbations we use a Floquet-Bloch decomposition in space, 


    [
    [ δ y(u,t);  δϕ(u,t) ]]=
    e^iλ t+iμ u[
    [ δ y(u);  δϕ(u) ]],


where  μ∈ (-1/2,1/2] is the Floquet parameter and λ(μ) ∈ℂ  is the eigenvalue. 
The resulting μ-dependent spectral problem is solved using a Krylov-based method and the shift-and-invert technique, see <cit.>. Details on Krylov methods are presented by <cit.>. We refer to the spectrum obtained this way as the stability spectrum of the Stokes wave. Note that the Floquet parameter is defined modulo 1, thus μ=1/2 is equivalent to μ=-1/2, see <cit.>. 




§ INSTABILITY




 §.§ The oscillating velocity and Hamiltonian


Both the velocity c and the Hamiltonian H depend on the Stokes wave. As the steepness s of the Stokes wave increases and approaches its limiting value s_lim, both quantities are not monotone, as observed by <cit.>. In fact, <cit.> produce an asymptotic result that implies the presence of an infinite number of oscillations for both quantities. These oscillations were studied more by <cit.>, <cit.> and <cit.>, and very recently by <cit.>. To our knowledge, no proof of an infinite number of oscillations in velocity c and Hamiltonian H exists.

We denote the steepness 
of a wave at the nth turning point of the speed by s_c,n, n=0, 1, 2, …, with s_c,0 = 0. Similarly, the nth extremizer of the Hamiltonian is denoted by s_H,n. These critical values of the velocity and the Hamiltonian are important for changes in the stability spectrum, as shown below. For the Hamiltonian, the importance of these values is known, due to the works of <cit.>, for instance. It appears that these extremizing values interlace, so that s_c,n<s_H,n+1<s_c,n+1, n=0, 1, ….  

In the recent work of <cit.>, 
a conjecture is made about an infinite number of secondary bifurcations associated with the Floquet multiplier μ=1/2, corresponding to perturbations whose period is twice that of the Stokes wave. It is unclear how these bifurcations are related to those in the works of <cit.>, <cit.> and <cit.>, since those works do not introduce a Floquet parameter. However, their importance to the stability results presented here is demonstrated below. We denote the steepness associated with the nth secondary bifurcation point by s_n^1/2, n=1,2, …. Further, we observe that s_c,n<s_n^1/2<s_H,n. These values are included in the schematic of Fig. <ref>. 


It is convenient to break up the range of steepness s from s=0 to s=s_lim in intervals from one extremizer of the wave speed to the next. 
For example, the first interval starts at the primary bifurcation s=0 and ends at the first maximizer of the wave speed  s=s_c,1; the second interval starts at s=s_c,1 and ends at the first minimizer of the wave speed s_c,2, and so on. The length of each interval shrinks as the extreme wave is approached, and following <cit.>
, we use a logarithmic scaling as illustrated in Fig. <ref>. Note that, because of the observed interlacing of the extremizers of the wave speed, those of the Hamiltonian, and the secondary bifurcations, each interval contains one extremizer of the Hamiltonian and one secondary bifurcation point. 





 §.§ A cyclus of changes in the spectrum


As the steepness increases and each interval is traversed, an instability emerges from λ = 0 in the spectral plane, giving rise to a sequence of λ(μ)-curves with changing topology, 
see Fig. <ref>. These changes for s∈ [s_c,0=0, s_c,1) are described below.




  * Initially, at s_c,0, a figure-8 emerges from the origin, expanding in size as steepness increases, Panels (a) and (b). 


  * At an isolated value of the steepness s=s_h,1, both tangents of the figure-8 at the origin become vertical, resulting in an hourglass shape, Panel (c). 


  * Next, the lobes of the figure-8 detach from the origin, forming two disjoint isles qualitatively reminiscent of the high-frequency instabilities of <cit.>. The band of Floquet values parameterizing the isles shrinks away from μ=0 as the steepness increases, Panels (d)-(h). 


  * At s=s_1^1/2, eigenvalues with Floquet parameter μ=1/2 bifurcate away from the origin onto the real line, creating an oval of eigenvalues with center at the origin, parameterized by Floquet values centered about μ=1/2, see Panels (f)-(h). 


  * As the steepness increases, the oval around the origin deforms to a bean shape, eventually re-absorbing the remnants of the figure-8, Panel (j). More detail on the changes in these remnants and their re-absorption is presented in Figs. <ref> and <ref>. 


  * At s=s_H,1, the bean shape pinches to form a figure-∞. The double point of the figure-∞ is at the origin and has a Floquet parameter μ=0. Thus it corresponds to perturbations with the same period as the Stokes wave, see <cit.>. In Panel (k) this co-periodic (or superharmonic) eigenvalue is marked in green. The unstable eigenvalue with μ=1/2 is marked in red and gives rise to the most unstable mode for the wave with steepness s=s_H,1. 


  * As s increases beyond s_H,1, the figure-∞ splits off from the origin into a pair of symmetric lobes, one moving to the right, the other to the left, Panel (l). Further interesting changes in the shape of these lobes are observed as the steepness increases and they move away from the origin, see <cit.> and Fig. <ref>. Importantly, we observe that the most unstable mode for this range of steepness s is either co-periodic with the Stokes wave (μ=0, green dot in Fig. <ref>) or has twice its period, i.e., it is subharmonic with μ=1/2 (red dot in Fig. <ref>). Figure <ref> illustrates two interchanges between these modes. We conjecture that such interchanges recur an infinite number of times as s→ s_lim. Note that the difference between s_lim=0.1410634839 and the steepness in the final panel of Fig. <ref> is about 0.000425 or 0.3%. 



Below we focus on what happens near the origin of the spectral plane as the steepness continues to increase, ever getting closer to its extreme value. 
  















 §.§ The Benjamin-Feir Instability




We observe that a figure-8 shape in the stability spectrum emerges from the origin when the steepness s=s_c,n, an extremizer of the velocity. 
The first three extrema of the velocity appear at the following values:

    s_c,0   = 0,
    
        s_c,1   = 0.138753,
    
        s_c,2   = 0.140920.


For small-amplitude waves (i.e., waves with steepness s near s_c,0=0), the figure-8 corresponds to the well-studied classical Benjamin-Feir, or the modulational instability. In what follows we refer to instabilities manifested through a figure-8 in the spectral plane as Benjamin-Feir instabilities. 
We refer to the Benjamin-Feir instability branches starting at steepness s_c,n as the (n+1)-th Benjamin-Feir branch, denoted BFI, BFII, BFIII, and so on. Below, we show that eigenvalues on the figure-8 near the origin (for BFII and BFII) give rise to modulational instabilities, as they do for small-amplitude waves, see <cit.>.

All the Benjamin-Feir branches that we compute experience the sequence of changes for increasing steepness  described above: they grow in size, their tangents at the origin become vertical followed by pinching off of the figure-8, resulting in the formation of isole on the positive and negative imaginary axis. For
each branch BFI, BFII and BFIII, we determine the figure-8 that gives rise to the eigenvalue with the largest real part, i.e., the maximal growth rate, see Fig. <ref>. Table <ref> displays these values of steepness, the corresponding eigenvalue with maximal real part and its Floquet exponent, for BFI, BFII and BFIII.
These computations illustrate that the widest figure-8 (green curves in Fig. <ref>) settles down to a universal shape as the extreme wave is approached, since the second and third panel appear indistinguishable). The values in Table  <ref> confirm this visual inspection. Further, for BFI, BFII and BFIII, we compute the hourglass shapes resulting from the figure-8's with vertical tangents at the origin, see Fig. <ref>. We overlay these shapes in Fig. <ref>, plotting the real part of the spectrum as a function of the Floquet parameter. This figure illustrates convergence to a universal hourglass shape as the steepest wave is approached. We conjecture that an infinite number of Benjamin-Feir instability branches exist as the steepest wave is approached and that all of them experience a universal sequence of transitions.

Finally, using the points marked by triangles in Fig. <ref>, we examine the eigenfunctions of (<ref>). 
The eigenfunctions related to BFI, BFII and BFIII are visibly different, while their spectra in Figs. <ref> and <ref> are, to the eye, indistinguishable. A second observation is that these eigenfunctions are indeed modulational in nature: Fig. <ref> displays 10 (=1/μ) Stokes wave periods of the eigenfunctions. Although their effect is increasingly localized at the crest in each wave period, there is a more global modulational effect when many periods are considered. Thus even the BFII and BFIII instabilities, for μ close to zero, deserve the modulational instability moniker.  























 §.§ The localized instability




The nth oval emerges from the origin of the spectral plane as the steepness increases past s^1/2_n. We observe that s^1/2_n>s_h,n, the value of the steepness for which the n-th Benjamin-Feir figure-8 separates from the origin. Thus prior (i.e., for s<s^1/2_n) to these ovals emerging from the origin, the spectrum near the origin is confined to the imaginary axis. The primary, secondary, and tertiary ovals form at the steepnesses


    s_1^1/2   = 0.128903,
    
        s_2^1/2   = 0.140487,
    
        s_3^1/2   = 0.141032049,

which correspond to steepnesses at which 4π-periodic Stokes waves bifurcate from the primary, 2π-periodic wave branch, see <cit.>.



The changing topology of the primary oval for s>s^1/2_1 is shown in Figs. <ref> and <ref>.
More detail is presented in Fig. <ref>. The oval develops for s_1^1/2≤ s < s_H,1. The oval stage is followed by a symmetric bean shape with a narrowing neck as steepness approaches s_H,1. The maximal growth rate associated with the localized instability quickly overtakes the maximal growth rate associated with the Benjamin-Feir isole higher on the imaginary axis, see <cit.>. Shortly before the steepness s  reaches s_H,1, the first extremizer of the Hamiltonian, the remnant of the Benjamin-Feir instability isola merges with the localized instability branch bean, as shown in the right panel of Fig. <ref>.

 We observe the recurrence of the process described above two more times, for the secondary and tertiary ovals that form at s=s^1/2_2 and s=s^1/2_3, respectively. This leads to the conjecture of an infinite number of such recurrences, the n-th one born at s=s^1/2_n, leading to the formation of the oval, gradually deforming to a bean shape, which pinches off at s=s_H,n, after which the resulting lobes move away from the origin along the real axis, ever decreasing in diameter. For s>s_H,n, the lobes are parameterized by the full range of Floquet exponents μ∈ [-1/2, 1/2). Further, for s∈ (s_H,n,s_c,n+1) there is no component of the spectrum other than the imaginary axis. 





The first two figure-∞'s are shows in Fig. <ref>. Like the figure-8's, they settle down to a universal shape as s→ s_lim. The difference between the real and imaginary parts of these first two figure-∞'s as a function of the Floquet exponent never exceeds 10^-3. 


Some eigenfunctions associated with the figure-∞ are displayed in Figs. <ref> and <ref>. For Floquet exponents close to zero (green and gold graphs), the modulational effect of the perturbation is clear from the polar plots. For other Floquet exponents (e.g. red and blue), the perturbation does not have a distinct modulational character. As for other high-amplitude Stokes waves, the localization of the eigenfunction (and thus the perturbation) near the crest of the waves is increasingly pronounced as s→ s_lim. 


When the oval forms at s=s_n^1/2, its eigenvalue with largest real part is real and has Floquet exponent μ=1/2, leading to eigenfunctions that have double the period of the Stokes wave. After pinch off, s>s_H,n, the left-most eigenvalue of the right lobe has μ=0 (co-periodic eigenfunctions). As for the primary lobe, we conjecture that the most unstable mode on the right lobe is either the μ=0 or the μ=1/2 mode, which interchange an infinite number of times as s→ s_lim, see <cit.>. As remarked above, the profile of the eigenfunction is strongly localized at the crests of the Stokes wave.  Modes with μ close to 0 have an envelope containing roughly 1/μ periods of the Stokes wave and could be called modulational. However, in contrast to the Benjamin-Feir instabilities, for these instabilities the μ=0 mode itself is unstable. For this reason, we refer to the instabilities emerging from the figure-∞'s as localized instabilities. 






 §.§ The maximal growth rate
 

We track the maximum growth rate γ (i.e., we track the eigenvalues with the largest real part) of the Benjamin-Feir and localized instabilities (plotted in black dotted and solid lines) as a function of the steepness of the Stokes wave in Fig.  <ref>. The maximum growth rate for BFI, BFII, and BFII are presented by green, blue, and purple curves respectively. 
Steepnesses at which the dominant instability switches from the Benjamin-Feir to the corresponding localized branch are marked by circles. These switches are presented in the corresponding insets.
The steepness values where the maximal growth rates for BFI and BFII vanish, correspond to the case where the Benjamin-Feir remants are absorbed into the localized instabilities.



 §.§ The high-frequency instabilities
 

Since the work presented here focuses on the evolution of the spectrum for increasing steepness in the vicinity of the origin in the spectral plane, we discuss the high-frequency instabilities only briefly. As shown by <cit.> and <cit.>, the high-frequency instabilities emanate from purely imaginary double eigenvalues for steepness s=0, giving rise to an isola of unstable eigenvalues centered on the imaginary axis, away from the origin. As steepness is increased, these isole may collapse back on the imaginary axis, and new ones may form, see <cit.>. Unlike the Benjamin-Feir (figure-8's) and localized instability branches (figure-∞'s), the high-frequency isole are highly localized in the space of Floquet exponents: indeed, <cit.> show that often a range of Floquet exponents of width no more than 10^-4 parameterizes an isola. This complicates their numerical detection. Figure <ref> presents plots of a few high-frequency isole for near-extreme increasing steepness, showing the collapse of one into the imaginary axis. For the top isola plotted, 
μ∈ [0.00092,0.00095], for the one below μ∈ [0.00057,0.00059]. For the two isole on bottom, μ∈ [0.000251, 0.000294] (outer), μ∈ [0.000263,0.000282] (inner). This demonstrates the isole can be captured using our method. A detailed study of the evolution of the high-frequency instabilities as steepness increases is kept for future work. 








§ CONCLUSIONS






We have presented a numerical exploration of the stability spectrum of Stokes waves near the origin of the spectral plane, focusing on the topological changes in the stability spectrum as the wave steepness grows. The main challenge of the study is due to the non-smooth nature of the extreme Stokes wave, which has a 120 corner at its crest. Thus for waves whose steepness approaches that of this extreme wave, our Fourier-based method requires the use of millions of Fourier modes. Indeed, for the computation of waves with steepness near the third extremizer of the Hamiltonian (see Fig. <ref>) nearly ten million Fourier modes are used. To examine the stability of these waves, we linearize about them, resulting in a generalized operator eigenvalue problem. The numerical approximation of this problem results in a generalized matrix eigenvalue problem with matrices of dimension equal to the square of twice the number of Fourier modes used for approximating the underlying Stokes wave, since each component of the perturbation requires a comparable number of Fourier modes to reach the same accuracy. Storing and manipulating such matrices is prohibitive, and our investigations are only possible because of the matrix-free approaches to the conformal mapping formulation, introduced by <cit.>. 

In <cit.> we used this same method to investigate the largest growth rate of perturbations of near-extreme Stokes waves, as a function of their steepness. Among other outcomes, this led to the conclusion that long-lived ocean swell is confined to moderate amplitudes. In this paper, we focus instead on the behavior of the stability spectrum near the origin of the spectral plane, as the recurring, self-similar behavior may provide an indication of how the stability of near-extreme Stokes waves may be approached analytically. Specifically, previous work and our numerical explorations lead to the following conjectures. 




  * The Hamiltonian H and the velocity c have an infinite number of local extrema (s_H,n, n=1,2,… and s_c,n, n=0,1,… respectively) as the steepness s increases to s_lim, the steepness of the extreme wave. This conjecture is not new, and the asymptotics of <cit.> provides a strong indication as to its validity. We include this conjecture here because all others below depend upon it. 


  * The maximal instability growth rate approaches infinity as the steepness increases to that of the steepest wave. Convincing evidence for this conjecture is presented by <cit.>. This would imply that the Euler water wave problem for the evolution of the extreme Stokes wave is ill posed. This is not a surprise as capillary effects need to be incorporated when the curvature at the crest is too large. This is discussed in more detail by <cit.>. 



The conjectures below are a direct outcome of the investigations presented in this paper. A graphical overview of which features occur at which steepness, according to these conjectures, is presented in Fig. <ref>.





  (iii) As the steepness s increases from 0 to that of the steepest wave, there exists an infinite number of Benjamin-Feir figure-8 instabilities. These emanate from the origin at each extremum s_c,n, n=0,1, … of the velocity. Upon formation, these figure-8's persist for a range of steepness. After their tangents at the origin become vertical (resulting in an hourglass shape) at steepness s_h,n, n=1,2,…, the figure-8 separates in two lobes on the imaginary axis. 


  (iv) After the Benjamin-Feir figure-8 separates from the origin, the nth oval appears at the origin, for the steepness s^1/2_n, n=1,2,…, 
corresponding to the period-doubling bifurcation points from the primary branch of Stokes waves.


  (v) As the steepness s increases from 0 to that of the extreme wave, there are an infinite number of figure-∞ instabilities. These occur at the origin at each extremum of the Hamiltonian/energy. As the steepness is increased, the figure-∞'s detach instantaneously. In other words, the figure-∞ shapes occur only at isolated values of the steepness, for which the Hamiltonian has a local extremum. 


  (vi) These figure-8's and figure-∞'s alternate in their occurrence. Stated differently, the extremizers of the Hamiltonian interlace the extremizers of the velocity. 


  (vii) For Stokes waves with amplitude greater than that of the first maximizer of the Hamiltonian, the most unstable mode is either superharmonic (co-periodic) or subharmonic with twice the period of the Stokes wave (anti-periodic). Further, there exists an infinite number of interchanges between which of these two modes is dominant. Throughout these interchanges, no other mode is the most unstable.
These observations were already discussed  by <cit.>. 






In the context of the full water wave problem, the present work reveals two primary mechanisms for the breaking of ocean waves: (i) for Stokes waves with s < s_2=0.12894 (the steepness at which the dominant instability switches from Benjamin-Feir to the localized branch, the abscissa of the green circle in Fig. <ref>), when the unstable envelope of multiple periods of a train of Stokes waves enters the nonlinear stage, a complicated pattern of waves is observed on the free surface. The pattern tends to self focus leading to the formation of a large 
unsteady wave whose crest forms a plunging breaker <cit.>; 
(ii) for Stokes waves with s > s_2, the dynamics 
of the wave 
is dominated by the localized instability at the wave crest, see <cit.>. The localized instability immediately leads to wave breaking of either 
every other wave crest in the train (if μ = 0 is dominant), or every other
crest (if μ = 1/2 is dominant) as discussed by <cit.>. More work is needed to understand the fully nonlinear stage of the many different instabilities computed. 

A complete understanding of the stability of Stokes waves with respect to bounded perturbations (see <cit.>) requires further study of the spectrum of the operators associated with the linearization of the Euler equations governing the dynamics of these perturbed waves. Nonetheless, our study provides numerical evidence that the (quasi-) periodic eigenfunctions that we examine, are fundamental to this problem. 




Acknowledgements. The authors wish to thank Eleanor Byrnes, Diane M. Henderson, and Pavel M. Lushnikov for helpful discussions. Also, the authors thank <cit.>, the developers of FFTW and the whole GNU project for developing and supporting this important and free software. S.D. thanks the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, for support and hospitality during the programme “Dispersive hydrodynamics” where work on this paper was partially undertaken.  Partial support for A. S. is provided by a PIMS-Simons postdoctoral fellowship.




Declaration of Interests. The authors report no conflict of interest.


jfm


