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Soliton Management for ultrashort pulse: dark and anti-dark solitons of Fokas-Lenells equation with a damping like perturbation and a gauge equivalent spin system

Riki Dutta, Gautam K Saharia, Sagardeep Talukdar, Sudipta Nandy

Abstract

We investigate the propagation of an ultrashort optical pulse using Fokas-Lenells equation (FLE) under varying dispersion, nonlinear effects and perturbation. Such a system can be said to be under soliton management (SM) scheme. At first, under a gauge transformation, followed by shifting of variables, we transform FLE under SM into a simplified form, which is similar to an equation given by Davydova and Lashkin for plasma waves, we refer to this form as DLFLE. Then, we propose a bilinearization for DLFLE in a non-vanishing background by introducing an auxiliary function which transforms DLFLE into three bilinear equations. We solve these equations and obtain dark and anti-dark one-soliton solution (1SS) of DLFLE. From here, by reverse transformation of the solution, we obtain the 1SS of FLE and explore the soliton behavior under different SM schemes. Thereafter, we obtain dark and anti-dark two-soliton solution (2SS) of DLFLE and determine the shift in phase of the individual solitons on interaction through asymptotic analysis. We then, obtain the 2SS of FLE and represent the soliton graph for different SM scheme. Thereafter, we present the procedure to determine N-soliton solution (NSS) of DLFLE and FLE. Later, we introduce a Lax pair for DLFLE and through a gauge transformation we convert the spectral problem of our system into that of an equivalent spin system which is termed as Landau-Lifshitz (LL) system. LL equation (LLE) holds the potential to provide information about various nonlinear structures and properties of the system.

Soliton Management for ultrashort pulse: dark and anti-dark solitons of Fokas-Lenells equation with a damping like perturbation and a gauge equivalent spin system

Abstract

We investigate the propagation of an ultrashort optical pulse using Fokas-Lenells equation (FLE) under varying dispersion, nonlinear effects and perturbation. Such a system can be said to be under soliton management (SM) scheme. At first, under a gauge transformation, followed by shifting of variables, we transform FLE under SM into a simplified form, which is similar to an equation given by Davydova and Lashkin for plasma waves, we refer to this form as DLFLE. Then, we propose a bilinearization for DLFLE in a non-vanishing background by introducing an auxiliary function which transforms DLFLE into three bilinear equations. We solve these equations and obtain dark and anti-dark one-soliton solution (1SS) of DLFLE. From here, by reverse transformation of the solution, we obtain the 1SS of FLE and explore the soliton behavior under different SM schemes. Thereafter, we obtain dark and anti-dark two-soliton solution (2SS) of DLFLE and determine the shift in phase of the individual solitons on interaction through asymptotic analysis. We then, obtain the 2SS of FLE and represent the soliton graph for different SM scheme. Thereafter, we present the procedure to determine N-soliton solution (NSS) of DLFLE and FLE. Later, we introduce a Lax pair for DLFLE and through a gauge transformation we convert the spectral problem of our system into that of an equivalent spin system which is termed as Landau-Lifshitz (LL) system. LL equation (LLE) holds the potential to provide information about various nonlinear structures and properties of the system.
Paper Structure (13 sections, 39 equations, 8 figures)

This paper contains 13 sections, 39 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Bilinearization of FLE with non-vanishing background
  3. Dark and Anti-Dark 1SS
  4. $D(t) = R(t) = 1$ (or some other constants) and $\Gamma(t)$ vanishes
  5. $D(t) = 1$, $R(t) = 1 + \sigma cos(kt)$ and $\Gamma(t)$ is non-vanishing
  6. $D(t) = 1$, $R(t) = 1 + \sigma e^{(-k t^2)}$ and $\Gamma(t)$ is non-vanishing
  7. $D(t) = R(t) = e^{(\sigma t)}\ cos(k t)$ and $\Gamma(\tau)$ vanishes
  8. $D(t) = e^{(\sigma t)}\ cos(k t)$ and $R(t) = cos(k t)$ and $\Gamma(t)$ is non-vanishing
  9. Dark and Anti-Dark 2SS
  10. Asymptotic analysis of 2SS
  11. Scheme for obtaining NSS
  12. Gauge Connected Landau-Lifshitz equation
  13. Conclusion

Figures (8)

  • Figure 1: 3D plot representation of 1SS under SM scheme: $D(t) = R(t) = 1$. (a) represents anti-dark soliton and (b) represents dark soliton.
  • Figure 2: 3D plot representation of 1SS under SM scheme: $D(t) = 1$, $R(t) = 1 + \sigma cos(kt)$ with $\sigma = 0.1$ and $k = 10 \pi$. (a) represents anti-dark soliton and (b) represents dark soliton. (c) represents anti-dark soliton but under SM scheme $D(t) = 1 + \sigma cos(kt)$, $R(t) = 1$.
  • Figure 3: 3D plot representation of 1SS under SM scheme: $D(t) = 1$, $R(t) = 1 + \sigma e^{(-k t^2)}$ with $\sigma = 0.25$ and $k = 100$. (a) represents anti-dark soliton and (b) represents dark soliton. (c) represents anti-dark soliton but under SM scheme: $D(t) = 1 + \sigma e^{(-k t^2)}$, $R(t) = 1$.
  • Figure 4: 3D plot representation of 1SS under SM scheme: $D(t) = R(t) = e^{(\sigma t)}\ cos(k t)$ with $\sigma = 0.25$ and $k = \pi$. (a) represents anti-dark soliton and (b) represents dark soliton.
  • Figure 5: 3D plot representation of 1SS under SM scheme: $D(t) = e^{(\sigma t)}\ cos(k t)$ and $R(t) = cos(k t)$ with $\sigma = 0.25$ and $k = \pi$. (a) represents anti-dark soliton and (b) represents dark soliton.
  • ...and 3 more figures