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Exact solutions of the reverse space-time higher-order modified self-steepening nonlinear Schrödinger equation

Yanan Wang, Xi-hu Wu

TL;DR

This work studies the reverse space-time higher-order modified self-steepening NLS equation, a nonlocal PT-symmetric extension, and proves its integrability via a Lax pair and infinite conservation laws. Using a Darboux transformation, the authors construct a broad class of exact solutions on zero and nonzero backgrounds, including solitons, kinks, breathers, rogue waves, and their interactions. The reverse-space-time symmetry yields novel dynamics absent in local models, such as kink waves and asymmetric rogue waves. The results provide a systematic framework for exploring nonlinear wave dynamics in nonlocal systems and expand the catalog of exact solutions for high-order nonlocal NLS-type equations.

Abstract

This paper investigates a reverse space-time higher-order modified self-steepening nonlinear Schrödinger equation, which distinguishes its standard local counterparts through the reverse space-time symmetry. The integrability of this nonlocal equation is rigorously verified by presenting its associated Lax pair and infinitely many conservation laws. Utilizing the Darboux transformation, we systematically construct a diverse range of localized wave solutions on both zero and nonzero backgrounds. These patterns, such as kinks, exponentially decaying solitons, asymmetric rogue waves and their interaction solutions, exhibit novel dynamical behaviors that are not found in the local counterparts. This work not only enriches the family of solutions for the equation, but also highlights the effectiveness of the Darboux transformation in exploring nonlinear wave dynamics in nonlocal systems.

Exact solutions of the reverse space-time higher-order modified self-steepening nonlinear Schrödinger equation

TL;DR

This work studies the reverse space-time higher-order modified self-steepening NLS equation, a nonlocal PT-symmetric extension, and proves its integrability via a Lax pair and infinite conservation laws. Using a Darboux transformation, the authors construct a broad class of exact solutions on zero and nonzero backgrounds, including solitons, kinks, breathers, rogue waves, and their interactions. The reverse-space-time symmetry yields novel dynamics absent in local models, such as kink waves and asymmetric rogue waves. The results provide a systematic framework for exploring nonlinear wave dynamics in nonlocal systems and expand the catalog of exact solutions for high-order nonlocal NLS-type equations.

Abstract

This paper investigates a reverse space-time higher-order modified self-steepening nonlinear Schrödinger equation, which distinguishes its standard local counterparts through the reverse space-time symmetry. The integrability of this nonlocal equation is rigorously verified by presenting its associated Lax pair and infinitely many conservation laws. Utilizing the Darboux transformation, we systematically construct a diverse range of localized wave solutions on both zero and nonzero backgrounds. These patterns, such as kinks, exponentially decaying solitons, asymmetric rogue waves and their interaction solutions, exhibit novel dynamical behaviors that are not found in the local counterparts. This work not only enriches the family of solutions for the equation, but also highlights the effectiveness of the Darboux transformation in exploring nonlinear wave dynamics in nonlocal systems.

Paper Structure

This paper contains 7 sections, 1 theorem, 35 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Lax pair and conservation laws
  3. Darboux transformation
  4. Exact solutions on different backgrounds
  5. zero background
  6. nonzero background
  7. Conclusions and discussions

Key Result

Proposition 1

Assume that $\Phi_{k}=(\phi_{1,k},\phi_{2,k})^T$ are the eigenfunctions of eq:app2 corresponding to the spectral parameters $\lambda=\lambda_i$. Then the $N$-fold DT of the coupled system is given by where $A_{2,N-1}=\frac{|B_{11}|}{|B_{12}|},A_{3,N-1}=\frac{|B_{21}|}{|B_{22}|}$, with $B_{11}$ is obtained by replacing the second column of $B_{12}$ with and $B_{21}$ is derived by replacing the f

Figures (10)

  • Figure 1: (a) The bell-shaped soliton with $\rho=0,\lambda_1=\lambda_2^*=1+\mathrm{i}$; (b) The exponentially decaying soliton with $\rho=0,\lambda_1=1+\mathrm{i}, \lambda_2=2-2\mathrm{i}$; (c) The cross-sectional view of (b) at $t=-8, t=0, t=3$.
  • Figure 2: (a) The periodic wave solution with $\rho=0, \lambda_1=\frac{3}{2}, \lambda_2=1$; (b) The kink solution with $\rho=1, \lambda_1=2\mathrm{i}, \lambda_2=\frac{1}{2}-2\mathrm{i}$; (c) The cross-sectional view of (b) at $t=-3, t=0, t=3$.
  • Figure 3: The interaction solutions with $\rho=0, \lambda_1=1+\mathrm{i}, \lambda_2=2-2\mathrm{i}$. (a) Two classical solitons with $\lambda_3=1-\mathrm{i}, \lambda_4=2+2\mathrm{i}$; (b) The classical soliton and the exponentially decaying soliton with $\lambda_3=1-\mathrm{i}, \lambda_4=2+\mathrm{i}$; (c) Two exponentially decaying solitons with $\lambda_3=1-1.01\mathrm{i}, \lambda_4=2+1.25\mathrm{i}$.
  • Figure 4: The interaction solutions. (a) The soliton and the periodic wave with $\rho=0, \lambda_1=\lambda_2^*=1+\mathrm{i}, \lambda_3=2, \lambda_4=3$; (b) The kink and the periodic wave with $\rho=1, \lambda_1=2\mathrm{i}, \lambda_2=3\mathrm{i}, \lambda_3=\frac{1}{2}-2\mathrm{i}, \lambda_4=\frac{1}{3}-3\mathrm{i}$; (c) The kink and the breather-like solution with $\rho=1, \lambda_1=2\mathrm{i},\lambda_2=\frac{1}{2}-2\mathrm{i}, \lambda_3=\lambda_4^*=1-\mathrm{i}$.
  • Figure 5: (a) The interaction solutions with $\rho=0, a=1, b=\frac{11}{2}, c=1, c_1=c_2=1$. (a) Two bright solitons with $\lambda_1=-\mathrm{i}, \lambda_2=2\mathrm{i}$; (b) The bright soliton and the dark soliton with $\lambda_1=\mathrm{i}, \lambda_2=2\mathrm{i}$; (c) Two dark solitons with $\lambda_1=\mathrm{i}, \lambda_2=-2\mathrm{i}$.
  • ...and 5 more figures

Theorems & Definitions (1)

  • Proposition 1