Optical solitons in near $\mathcal{PT}-$symmetric Rosen-Morse potential
K. Hari K. Manikandan R. Sankaranarayanan





Department of Physics, National Institute of Technology, Tiruchirappalli - 620015, Tamil Nadu, India.


manikandan.cnld@gmail.com
Department of Physics, National Institute of Technology, Tiruchirappalli - 620015, Tamil Nadu, India.


 sankar@nitt.edu
Department of Physics, National Institute of Technology, Tiruchirappalli - 620015, Tamil Nadu, India.



We investigate the existence of stable soliton solution in a system described by complex Ginzburg-Landau (CGL) equation with near parity reflection - time reversal (𝒫𝒯)  symmetric Rosen-Morse potential. In this study, the stability of solution is examined by numerical analysis to show that solitons are stable for some parameter ranges. The dynamical properties such as evolution and transverse energy flow for both self-focusing and self-defocusing nonlinear mode are also analyzed.  The obtained results are useful for experimental designs and applications in related fields. 

42.25.Bs, 42.65.-k, 42.65.Tg, 11.30.Er

Optical solitons in near 𝒫𝒯-symmetric Rosen-Morse potential
    R. Sankaranarayanan
    December 30, 2023
===========================================================





§ INTRODUCTION

The introduction of 𝒫𝒯 operation has paved a new way into realm of undiscovered physical systems.  In recent years, nonlinear 𝒫𝒯-symmetric systems have become a fascinating area in research <cit.>.  Both theoretical and experimental investigations are developing the subject into vast regimes.  The idea of 𝒫𝒯-symmetry and the possibility for investigation of real energy spectrum using this symmetry, despite the presence of complex terms in potential, was originated from Bender and Boettcher in 1998 <cit.>. In the 𝒫𝒯-symmetry operation, the parity reflection (𝒫) is defined by p̂→-p̂, x̂→-x̂, i→ i and time-reversal (𝒯) is defined as p̂→ -p̂, x̂→x̂, i→-i <cit.>. The experimental realization of such 𝒫𝒯-symmetric potential has led to discover new kind of optical lattices, lasers, metamaterials, etc <cit.>. In nonlinear optical devices the complex refractive index constitutes the 𝒫𝒯-symmetric potential which balances the loss and gain of optical pulse in the medium.  𝒫𝒯-symmetry breaking is also an important physical phenomenon because some of the desired properties are achieved at the point of symmetry breaking <cit.> and in the symmetry broken phase <cit.>.  𝒫𝒯-symmetry breaking exploited in many experiments <cit.>.

Many theoretical studies have been done on nonlinear Schrödinger equation (NLS) with 𝒫𝒯-symmetric potentials such as harmonic potential <cit.>, sextic anharmonic double-well potential <cit.>, Scarf 2 potential <cit.>, Rosen-Morse potential <cit.> and Gaussian potential <cit.>. More interestingly, in Ref. [16] the authors found that the 𝒫𝒯-symmetric Rosen-Morse potential with NLS equation does not accommodate stable soliton solutions for any potential parameters. This is due to the nonvanishing terms in the imaginary part of the potential. In this study, we extend the analysis of Rosen-Morse potential with CGL equation to attain stable soliton solutions. The CGL is one the important equation describing a nonlinear optical system <cit.> and investigations on such systems are hardly done due to the non-invariant 𝒫𝒯-symmetry of the system. Additional parameters in CGL equation give us the freedom to create new stable soliton solution in this system. The potential is no more 𝒫𝒯-symmetric but near 𝒫𝒯-symmetric <cit.> due to the presence of asymmetric terms. 

Through this work we reflect the dynamical properties of a near 𝒫𝒯-symmetric potential by playing with the system parameters. We present some novel results in which the near 𝒫𝒯-symmetric Rosen-Morse potential accommodates stable soliton solutions. We have extensively studied the stability of soliton solutions along with the dynamical properties such as evolution dynamics and energy flow. Our work is also extended to both self-focusing and self-defocusing nonlinear modes.



§ A PHYSICAL MODEL OF THE SYSTEM

In a physical model of 𝒫𝒯-symmetric system, the loss and gain is balanced. But the physical system we consider has some additional physical parameters which lead to imbalance in gain and loss in the medium, thus leading to 𝒫𝒯-symmetric breaking. The transmission of such an optical lattice is given by CGL equation <cit.> with complex constants and potentials as

    i∂Ψ/∂ z+(α_1+i α_2)∂^2Ψ/∂ x^2+[V(x)+i W(x)] Ψ +
    σ(β_1+i β_2)|Ψ|^2Ψ = 0,
 
where Ψ(x,z) is the amplitude of complex electric field of the optical pulse, z is the propagation distance, x is the spatial coordinate, α_1 is the diffraction coefficient, β_1 is the Kerr-nonlinearity coefficient, α_2 spectral filtering, β_2 gives the nonlinear gain/loss in the system, σ=± 1 gives the self-focusing and self-defocusing nonlinear mode, V(x) is the real part of potential and W(x) is the imaginary part of potential.  For invariant 𝒫𝒯-symmetry the potential should be of the form, V(-x)=V(x) and W(-x)=-W(x). It is noted that the CGL equation is non-𝒫𝒯 invariant due to the presence of complex coefficients. 

Our analysis starts by assuming stationary solution in the form Ψ(x,z)=ϕ(x)e^iμ z, where μ is the real valued propagation constant and ϕ(x) is the complex field.

Substituting this solution into Eq. (<ref>), we obtain the following second-order ordinary differential equation of the form

    (α_1+ iα_2) d^2ϕ/dx^2 +[V(x)+iW(x)]ϕ
     +σ(β_1+iβ_2)|ϕ|^2ϕ-μϕ=0.

We use the condition μ=α_1 for the above equation. 

In order to investigate the optical beam propagation of  (<ref>), we consider the near 𝒫𝒯-symmetric Rosen-Morse potential which admits stationary soliton solution is of the form


    V(x)=     -a(a+1)^2(x)-2(α_2/α_1)btanh(x)+b^2/α_1, 
     W(x)=    2btanh(x)+W_1^2(x)+α_2(b^2/α_1^2-1),


where W_1=2α_2-(a(a+1)+2α_1)( β_2/β_1), a and b are positive real values for describing the strength of the potential. The original Rosen-Morse potential is given by


    V(x)=-c(c+1)^2(x),


    W(x)=2d tanh (x),


where c and d describe the strength of real and imaginary parts of the potential respectively. Comparing Eqs. (<ref>) with Eqs. (<ref>), it is evident that the existence of additional terms in the potential leads to near 𝒫𝒯-symmetry. Here  the physical model (<ref>) is described by many system parameters and for our convenience, we fix the values a=0.1 and α_1=β_1=1. The existence of system parameters α_2 and β_2 breaks the 𝒫𝒯-symmetry of the system. If the physical effects described by α_2 and β_2 is very negligible then the system leads to 𝒫𝒯-symmetry.   In other words if the values of these parameters are zero then the potential is exactly 𝒫𝒯-symmetric. The qualitative nature of near 𝒫𝒯-symmetric Rosen-Morse potential for some parameter values are given in Fig. <ref>. Unlike the 𝒫𝒯-symmetric Rosen-Morse potential, here both V(x) and W(x) are non-vanishing asymptotically.

We have taken the values for parameters in such a way that we can illustrate the dependence of parameter in both potentials.  Initially we fix the value for b in Figs. <ref>(a)-<ref>(b) and varied α_2 and β_2. There is a appreciable shift in W(x) and the depth of the potential W(x) increases.  Then we change the value of b for the same variations of α_2 and β_2 as seen in Figs. <ref>(c)-<ref>(d).  The increase in value of b is also contributing to the shape of the potential.  In addition, 
we also observe that the asymmetry of both the potentials increase with b.  These are the major factors that actually control the potential. In a physical system the values of α_2 and β_2 are important because these values are externally controlled to make slight modifications to the potentials. In other way we can say that these parameters can be used to tune the system to reach a particular regime where we can achieve stable propagation of light pulses.



§ STATIONARY SOLITON SOLUTION AND LINEAR STABILITY ANALYSIS



 §.§ Self-focusing nonlinear mode

Our investigation of stationary solution starts with self-focusing nonlinearity, σ=1. Now the system admits an exact solution for Eq. (<ref>) in the form

    ϕ(x)=√(a(a+1)+2α_1/β_1)(x) e^ibx/α_1

For this particular solution, we impose the condition μ=α_1 for which the above is true.

The profile of stationary soliton solution of Eq. (<ref>) is shown in Fig. <ref>. When we increase the parameter b from 0.1 to 0.4, there is no change in total intensity of stationary solution.  But we can observe an increase in amplitude of imaginary part and decrease in the width of real part of stationary solution. 

The stability of this solution is evaluated by adding infinitesimal perturbation with the stationary solution. If such a perturbation leads to deviation from the original solution, then the solution is unstable. The perturbed solution <cit.> is given by

    Ψ(x,z) = (ϕ(x)+ϵ [f(x)e^λ z+g^*(x)e^λ^* z])e^iμ z,

where ϵ≪ 1, f(x) and g(x) are very small perturbation functions and λ is the stability parameter. Substituting the perturbed solution in Eq.  (<ref>) and linearizing with respect to ϵ, one can obtain a set of equations for f(x) and g(x) which can be solved by forming a matrix of the system of equations. 

The linear eigenvalue problem is described by 

    [    L_0    L_1; -L_1^* -L_0^* ][ f(x); g(x) ]=-iλ[ f(x); g(x) ]

where L_0=(α_1+iα_2)∂_xx+V(x)+iW(x)+2(β_1+iβ_2)|ϕ|^2-μ and L_1=(β_1+iβ_2)ϕ^2.
This eigenvalue equation can be solved using Fourier collocation method <cit.>.  Stable solutions are only present if the eigenvalues have a negative real part or purely imaginary. This means that when the system is perturbed from a singular point, the perturbation will eventually decay for negative real part of the eigenvalues. The stationary solution with purely imaginary eigenvalues oscillates around the singular point for small perturbation.  In case of positive eigenvalues, the terms in perturbation diverge and stability cannot be achieved even for infinitesimal perturbation. Figure <ref> illustrates the eigenvalues for different parameter values.




Thus we have obtained the parameter values for stable and unstable modes using linear stability analysis.  The stable regions are small ranges in different parameters.   All soliton solutions for our considered system are stable only for α_2<0.  Values for stable solutions are obtained and eigenvalues for the same are given in Figs. <ref>(a)-<ref>(f).  Figures <ref>(g) and <ref>(h) show eigenvalues for unstable solutions.  Here, the eigenvalues have positive real part which will not converge when perturbed.

We follow the convention that the soliton propagates in z - direction.  Figure <ref> represents intensity plot of solitons for the same parameter values as in Fig. <ref>. It is interesting to note from Figs. <ref>(d) and <ref>(e) where the soliton profile is not smooth as other stable solutions. During propagation, the amplitude oscillates rather than maintaining a constant value.  Thus we infer that there exists some cases where the amplitude of solitons oscillates during propagation without affecting their stability. These oscillations are unknown from stability analysis. For stable soliton solutions, the maximum intensity is at x=0 plane and reduces to zero when x→±5. In Fig. <ref>(g) the soliton is unstable, deviates from its path and dissipates. But in Fig. <ref>(h) as α_2 increases, the soliton is not able to propagate even a little distance, where it scatters due to α_2 > 0. Thus, we find that due to the 𝒫𝒯-symmetry breaking, the near 𝒫𝒯-symmetric Rosen-Morse potential accommodates stable soliton solutions. 



 §.§ Self-defocusing nonlinear mode

Extending our insight into self-defocusing nonlinearity, for which σ=-1, the exact soliton solution for Eq. (<ref>) is given by 	

    ϕ(x)=√(-a(a+1)+2α_1/β_1)sech(x) e^ibx/α_1,

where all the terms are already discussed in Sec. 2.  The difference between the solutions of focusing and defocusing mode is the only change in the sign of the term inside square root of the amplitude of soliton. Here also the condition for obtaining this solution is μ=α_1.


The solution, Eq. (<ref>) is given by Fig. <ref> with the real part, imaginary part and amplitude of the exact soliton solution.  In Fig. <ref>(a), we can see that the imaginary part of solution is more dominant than the real part due to the negative term inside the square root (see Eq.  (<ref>)).  When we increase the parameter b from 0.1 to 0.4, there is no change in total intensity of stationary solution as shown in Fig. <ref>(b).  In addition, we notice the increase in amplitude of real part and decrease in the width of imaginary part of stationary solution.  The linear stability analysis for the above solution is studied as mentioned earlier and our results are plotted in Fig. <ref>  for α_2=-1.  

From Fig. <ref> the stability analysis for self-defocusing nonlinear mode also shows the same sign pattern of α_2, but with less number of combinations. The range of values are very short compared with self-focusing nonlinearity. The evolution of stationary solution for the parameters obtained by stability analysis confirms the smoothness of soliton. 




The evolution of soliton is given in Fig. <ref>.  Even the solutions are stable for the given values of parameters, the soliton profile does not guarantee propagation without loss is intensity. Figs. <ref>(a) and <ref>(c) show the evolution of stable soliton solution with a small variation in amplitude in the beginning of propagation. When we reduce the value of β_2, the soliton dissipates after propagating a small distance as in Figs. <ref>(b) and <ref>(d). Comparing with the self-focusing nonlinear mode, for the stable soliton solutions, self-defocusing nonlinear mode has narrow range for parameters. Keeping other parameters constant and increasing the value of b the intensity of soliton increases which means we can amplify the intensity of light pulse with modulating the potential strength, that is the refractive index of wave guide.



§ ENERGY FLOW FOR EXACT SOLITON SOLUTION



 §.§ Self-focusing nonlinear mode

The equation of continuity for energy flow (where the energy is not conserved) of CGL Eq. (<ref>) gives us the relation between the energy density (ρ = |Ψ|^2) and the energy flux (j) which is given by <cit.> 

    E = ∂ρ/∂ z+∂ j/∂ x,

where E determines the gain or loss distribution of energy and the energy flux is described by j=i/2(ϕϕ_x^*-ϕ_xϕ^*).  If the system is conservative then E=0 which means the loss and gain in the system is balanced. When the loss and gain of a system is not balanced, energy will flow from one region to another region.  For the system of our interest with σ=1, we find

    E=    -2b/α_1β_1(a+a^2+2α_1)sech^2(x)tanh(x), 
    
    j=    b(a(a+1)+2α_1)/α_1β_1^2(x).

Figure <ref> shows the variation of energy flux with spatial coordinate. 

The plot is similar to the shape of the stationary soliton, that is the value of j peaks at x=0, which implies that maximum intensity is transferred through x=0. The energy flux depends on the value of b since a, α_1  and β_1 are fixed. The direction and quantity of energy flow are illustrated in Fig. <ref>.   Here we observe that the light pulse gains energy in x<0 region

and looses energy in x>0 region.  In other words, the energy flow is effected from left to right.  An important observation from Eq. (<ref>) is that the energy flow due to unbalanced loss and gain will not depend on α_2 and β_2. Thus, the strength of the potential controls the energy flow in near 𝒫𝒯-symmetric potentials.


 §.§ Self-defocusing nonlinear mode

Similarly, for the self-defocusing case (σ=-1), the gain or loss distribution of energy and the energy flux are given by 

    E=   2b/α_1β_1(a+a^2+2α_1)sech^2(x)tanh(x),
    
    j=     -b(a(a+1)+2α_1)/α_1β_1^2(x).

The energy flux for the self-defocusing nonlinear mode is shown in Fig. <ref>.

Further, maximum energy flux is at x=0  and decreases on either side as earlier.  Figure <ref> gives the gain or loss distribution of energy in near 𝒫𝒯-symmetric unbalance in potential. It is clear that the gain or loss distribution of energy in negative x-axis corresponds to loss, and positive in x-axis corresponds to gain.  Altogether, for self-defocusing nonlinearity the distribution of energy flows from positive x direction to negative x direction. 

Here also the energy flow is independent of α_2 and β_2 but dependent on the strength (b) of the potential. Therefore, when the sign of nonlinearity in the CGL equation changes the direction of energy flow also changes.




§ CONCLUSION

In our study, we modified the 𝒫𝒯-symmetric Rosen-Morse potential with NLS equation to a near 𝒫𝒯-symmetric Rosen-Morse potential with CGL equation.  Exact 𝒫𝒯-symmetric soliton solutions for this potential along with stability analysis and evolution of soliton solution for different parameter ranges are analyzed for both self-focusing and self-defocusing modes.  For some parameter ranges, the 𝒫𝒯-symmetric soliton in near 𝒫𝒯-symmetric potential shows excellent behavior compared to the 𝒫𝒯-symmetric potential where such solutions are unstable.  The stable soliton solutions are apparently possible for combination of α_2<0 and β_2≥ 0.  We have also investigated the energy flow in the potential and observed that the direction of energy flow in self-focusing and self-defocusing nonlinear modes are opposite.  𝒫𝒯-symmetry breaking is an important phenomenon to create near 𝒫𝒯-symmetric potentials, and analyzing the possibilities of stable soliton solutions can be further developed into experimental realizations in future.




§ ACKNOWLEDGMENTS

KM wishes to thank the Department of Science and Technology - Science and Engineering Research Board, Government of India, for providing National Post Doctoral fellowship under the Grant No. PDF/2016/001620/PMS.

50

1
V. V. Konotop, J. Yang and D. A. Zezyulin, Rev. Mod. Phys. 88, 035002 (2016).

2
C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).

3
C. M. Bender, Contemp. Phys. 46, 277 (2005).

4
C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).

5
A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides and U. Peschel, Nature 488, 167, (2012).

6
H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides and M. Khajavikhan, Science 346, 975 (2014).

7
D. R. Smith, J. B. Pendry and M. C. Wiltshire, Science 305, 788 (2004).

8
N. Lazarides and G. P. Tsironis, Phys. Rev. Lett. 110, 053901 (2013).

9
V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis and R. Carretero-González, Phys. Rev. A 86, 013808 (2012).

10
J. Yang, Opt. Lett. 39, 5547 (2014).

10a
H. Ramezani, T. Kottos, R. El-Ganainy and D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).

10b
T. Kottos, Nat. Phy. 6, 166 (2010).

10c 
S. Karthiga, V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, Phys. Rev. A 94, 023829 (2016).

10d
S. Karthiga, V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, Phys. Rev. A 95, 033829 (2017).

11
M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang and S. Zhang,  Phys. Rev. Lett. 113, 093901 (2014).

12
L. Feng, Z. J. Wong, R. M. Ma, Y. Wang and X. Zhang, Science 346, 972 (2014).

12a
C.E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev and D. Kip, Nat. Phy. 6, 192 (2010).

12b
A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides and U. Peschel, Nature 488, 167 (2012).

13
D. A. Zezyulin and V. V. Konotop, Phys. Rev. A 85, 043840 (2012).

14
Z. C. Wen and Z. Yan, Phys. Lett. A 379, 2025 (2015).

15
Z. H. Musslimani, K. G. Makris, R. El-Ganainy and D. N. Christodoulides, Phys Rev. Lett. 100, 030402 (2008).

16
B. Midya and R. Roychoudhury, Phys. Rev. A 87, 045803 (2013).

17
S. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng and W. Hu, Phys. Rev. A 84, 043818 (2011).

18
P. A. Bélanger, L. Gagnon and C. Paré, Opt. Lett. 14, 943 (1989).

19
Y. Chen, Z. Yan and W. Liu, Optics Express, 26, 33022 (2018).

20
Y. He and D. Mihalache, JOSA B, 29, 2554 (2012).

21
Y. He and D. Mihalache, Phys. Rev. A 87, 013812 (2013).

22
E. A. Kuznetsov, A. M. Rubenchik and V. E. Zakharov, Phys. Rep. 142, 103 (1986).

23
J. Yang, Nonlinear Waves in Integrable and Non-Integrable Systems (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2010).


