Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Soliton solutions to the coupled Sasa-Satsuma-mKdV equation

Changyan Shi, Bao-Feng Feng

Abstract

We consider the soliton solutions of a recently proposed coupled Sasa-Satsuma-mKdV equation using the Kadomtsev-Petviashvili reduction method. The system consists of a complex-valued component coupled with a real-valued one. Under zero or nonzero boundary conditions, we derive four distinct classes of soliton solutions: bright-bright, dark-dark, bright-dark, and dark-bright. These solutions are derived from the vector Hirota equation, for which the bright, dark, and bright-dark soliton solutions are provided in the Appendix. We perform asymptotic analysis of soliton collisions for each class of solutions, in which inelastic collisions are observed between bright-bright solitons. In the dark-dark case, we identify soliton profiles similar to the Sasa-Satsuma equation, including double-hole, Mexican hat, and anti-Mexican hat solutions; this study further explores the collisions between these structures and hyperbolic tangent shaped kink solitons. Regarding the bright-dark case, beyond the expected soliton-kink interactions, we report and analyze a notable collision occurring between kink solitons.

Soliton solutions to the coupled Sasa-Satsuma-mKdV equation

Abstract

We consider the soliton solutions of a recently proposed coupled Sasa-Satsuma-mKdV equation using the Kadomtsev-Petviashvili reduction method. The system consists of a complex-valued component coupled with a real-valued one. Under zero or nonzero boundary conditions, we derive four distinct classes of soliton solutions: bright-bright, dark-dark, bright-dark, and dark-bright. These solutions are derived from the vector Hirota equation, for which the bright, dark, and bright-dark soliton solutions are provided in the Appendix. We perform asymptotic analysis of soliton collisions for each class of solutions, in which inelastic collisions are observed between bright-bright solitons. In the dark-dark case, we identify soliton profiles similar to the Sasa-Satsuma equation, including double-hole, Mexican hat, and anti-Mexican hat solutions; this study further explores the collisions between these structures and hyperbolic tangent shaped kink solitons. Regarding the bright-dark case, beyond the expected soliton-kink interactions, we report and analyze a notable collision occurring between kink solitons.
Paper Structure (9 sections, 10 theorems, 76 equations, 26 figures)

This paper contains 9 sections, 10 theorems, 76 equations, 26 figures.

Table of Contents

  1. Introduction
  2. Bilinearization of the coupled Sasa-Satsuma-mKdV equation
  3. Soliton solutions to the coupled Sasa-Satsuma-mKdV equation
  4. Dynamics of bright-bright solitons
  5. Dynamics of dark-dark solitons
  6. Dynamics of bright-dark solitons
  7. Dynamics of dark-bright solitons
  8. Results on vector Hirota equation
  9. Corresponding bilinear equations and $\tau$-functions from KP-Toda hierarchy

Key Result

Theorem 3.1

Equation ss_mkdv_1-ss_mkdv_2 admits the bright soliton solutions given by $u=g_1/f,\ v=g_2/f$ with $f,\ g_1, \ g_2$ defined as where $M$ is an $N\times N$ matrix, $\Phi$, $\bar{\Psi}$, are $N$-component row vectors whose elements are defined respectively as Here, $p_i$, $\xi_{i0}$, $C_i$, $D_i$ are complex parameters which satisfy the following restrictions

Figures (26)

  • Figure 1: One-bright-one-anti-bright and one-bright-one-bright soliton solution to Eq. \ref{['ss_mkdv_1']}-\ref{['ss_mkdv_2']} with parameters (a-b) $p_1= 1, C_1 = 1 + \mathrm{i}, D_1 = -3, \xi_{1,0} = 0$, (c) $p_1= 1, C_1 = 1 + \mathrm{i}, D_1 = 3, \xi_{1,0} = 0$.
  • Figure 2: One-oscillated soliton solution to Eq. \ref{['ss_mkdv_1']}-\ref{['ss_mkdv_2']} with parameters $p_1= 1+\mathrm{i}, C_1 = 1 + 1\mathrm{i}, C_2 = 1 - 2\mathrm{i}, D_1 = 1+2\mathrm{i}, \xi_{1,0} = 0$.
  • Figure 3: Soliton solution to Eq. \ref{['ss_mkdv_1']}-\ref{['ss_mkdv_2']} with collision between oscillated soliton and traveling soliton solution under parameters $p_1= 1+\mathrm{i}, p_2 = 2, C_1 = 1 + \mathrm{i}, C_2 = 1 - 2\mathrm{i}, C_3 = 2 + 2\mathrm{i}, D_1 = 1+2\mathrm{i}, D_2 = 2, \xi_{1,0} = \xi_{2,0} = 0$.
  • Figure 4: Solution to Eq. \ref{['ss_mkdv_1']}-\ref{['ss_mkdv_2']} with collision between traveling solitons under parameters $p_1= \frac{2}{3}, p_2 = 1, C_1 = C_2 = C_3 = 2, D_1 = 1 + \mathrm{i}, D_2 = 1, \xi_{1,0} = \xi_{2,0} = 0$.
  • Figure 5: Y-shaped Solution to Eq. \ref{['ss_mkdv_1']}-\ref{['ss_mkdv_2']} under parameters $p_1= \frac{2}{3}+\mathrm{i}, p_2 = 1, C_1 = 1+2\mathrm{i}, C_2 = 2+2\mathrm{i}, C_3 = 3-\mathrm{i}, D_1 = 1 + 2\mathrm{i}, D_2 = 0, \xi_{1,0} = \xi_{2,0} = 0$.
  • ...and 21 more figures

Theorems & Definitions (10)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem A.1: Bright soliton solution to Eq. \ref{['m-Hirota']}
  • Theorem A.2: Dark soliton solution to Eq. \ref{['m-Hirota']}
  • Theorem A.3: Bright-dark soliton solution to Eq. \ref{['m-Hirota']}
  • Lemma B.1
  • Lemma B.2
  • Lemma B.3