Anti-dark and Mexican-hat solitons in the Sasa-Satsuma equation on the continuous wave background

Abstract

In this letter, via the Darboux transformation method we construct new analytic soliton solutions for the Sasa-Satsuma equation which describes the femtosecond pulses propagation in a monomode fiber. We reveal that two different types of femtosecond solitons, i.e., the anti-dark (AD) and Mexican-hat (MH) solitons, can form on a continuous wave (CW) background, and numerically study their stability under small initial perturbations. Different from the common bright and dark solitons, the AD and MH solitons can exhibit both the resonant and elastic interactions, as well as various partially/completely inelastic interactions which are composed of such two fundamental interactions. In addition, we find that the energy exchange between some interacting soliton and the CW background may lead to one AD soliton changing into an MH one, or one MH soliton into an AD one.

The algebraic representation for high order solution of Sasa-Satsuma equation

“…What should be pointed out that, most of the single localized wave solutions have been obtain in the previous literature. For the focusing case, the breather solution [1,30], rogue wave solution [4], rational W-shape soliton [34] and degenerate resonant soliton on NVBC [32] have been obtained in the previous literature. For the defocusing case, the dark soliton and W-shape dark soliton (dark double-hump soliton) have been been obtain through bilinear method and symbolic computation [12].…”

“…1975 1976 LIMING LING then equation (2) could be rewritten asSSE (4) is one of a limited number of integrable models and has been a field of active research for the past two decades. Thanks to the integrability, the sophisticated soliton construct underlying this wave equation can, therefore, be achieved using an array of mathematical tools such as inverse scattering transform [22,14,17,33], Darboux transformation (DT) [27,30,1,4,32,25], Hirota bilinear method [6,5,26], and others. Although the DT for the SSE was given in the previous study, the reduction of DT is not complete.…”

“…Recently, there are some works focus on the solutions of SSE with the non-vanishing background (NVBC). For instance: the breather solution [30, 1], W-shape soliton [32], rational W-shape soliton [34] and the twist rogue wave solution [4] are obtained for the focusing SSE in the recent literature. In this work, we construct the solitonic solution for both the focusing and defocusing SSE on the NVBC systematically.…”

The N‐coupled higher‐order nonlinear Schrödinger equation: Riemann‐Hilbert problem and multi‐soliton solutions

“…To our knowledge, the RH problem for the N ‐CHNLS equations and its multi‐soliton solutions have not been studied so far, which is the purpose of this work. Here, there are two main reasons for taking the N ‐CHNLS equations as a model: firstly, due to its important applications in physics, it is well known that Equation degenerates into the Sasa‐Satusma higher‐order nonlinear Schrödinger equation describing the dynamics of the ultrashort pulses under the effects of the TOD, the SS, and the SRS with N =1 . Besides, when N =2, Equation can be reduced to the coupled Sasa‐Satsuma equations to investigate the effects of birefringence on pulse propagation in the femtosecond regime .…”

“…Here, there are two main reasons for taking the N-CHNLS equations as a model: firstly, due to its important applications in physics, it is well known that Equation (2) degenerates into the Sasa-Satusma higher-order nonlinear Schrödinger equation describing the dynamics of the ultrashort pulses under the effects of the TOD, the SS, and the SRS with N = 1. 25,26 Besides, when N = 2, Equation (2) can be reduced to the coupled Sasa-Satsuma equations to investigate the effects of birefringence on pulse propagation in the femtosecond regime. [27][28][29] Like the earlier works, the Sasa-Satsuma equations can be generalized to the integrable form of the N-CHNLS equations, which can enrich the properties of the equation.…”