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Exact solitary wave solutions for a coupled gKdV-Schrodinger system by a new ODE reduction method

Stephen C. Anco, James Hornick, Sicheng Zhao, Thomas Wolf

TL;DR

The paper addresses the challenge of finding exact solitary-wave solutions for a coupled generalized Korteweg-de Vries and linear Schrödinger (gKdV-LS) system. It introduces a novel, fully systematic ODE reduction method based on symmetry multi-reduction and a hodograph transformation, reducing the travelling-wave ODE system to a single nonlinear base ODE for which all polynomial solutions can be found by symbolic computation. This approach yields 22 explicit phase-modulated solitary-wave families for nonlinearity powers $p=1,2,3,4$, revealing features such as bright/dark peaks, symmetric side peaks, and kink-shaped LS phase; many of these are new, especially for $p>1$. The work also derives conservation laws and a variational principle, enabling a pathway toward stability analysis, and demonstrates that the method extends to other coupled nonlinear dispersive systems and to generalized Hénon–Heiles type ODEs. Overall, the results provide a broad catalogue of exact travelling-wave solutions and introduce a powerful framework for analyzing complex coupled dispersive waves with multiple nonlinearities.

Abstract

A new method is developed for finding exact solitary wave solutions of a generalized Korteweg-de Vries equation with p-power nonlinearity coupled to a linear Schrödinger equation arising in many different physical applications. This method yields 22 solution families, with p=1,2,3,4. No solutions for p>1 were known previously in the literature. For p=1, four of the solution families contain bright/dark Davydov solitons of the 1st and 2nd kind, obtained in recent work by basic ansatze applied to the ODE system for travelling waves. All of the new solution families have interesting features, including bright/dark peaks with (up to) p symmetric pairs of side peaks in the amplitude and a kink profile for the nonlinear part in the phase. The present method is fully systematic and involves several novel steps which reduce the travelling wave ODE system to a single nonlinear base ODE for which all polynomial solutions are found by symbolic computation. It is applicable more generally to other coupled nonlinear dispersive wave equations as well as to nonlinear ODE systems of generalized Hénon-Heiles form.

Exact solitary wave solutions for a coupled gKdV-Schrodinger system by a new ODE reduction method

TL;DR

The paper addresses the challenge of finding exact solitary-wave solutions for a coupled generalized Korteweg-de Vries and linear Schrödinger (gKdV-LS) system. It introduces a novel, fully systematic ODE reduction method based on symmetry multi-reduction and a hodograph transformation, reducing the travelling-wave ODE system to a single nonlinear base ODE for which all polynomial solutions can be found by symbolic computation. This approach yields 22 explicit phase-modulated solitary-wave families for nonlinearity powers , revealing features such as bright/dark peaks, symmetric side peaks, and kink-shaped LS phase; many of these are new, especially for . The work also derives conservation laws and a variational principle, enabling a pathway toward stability analysis, and demonstrates that the method extends to other coupled nonlinear dispersive systems and to generalized Hénon–Heiles type ODEs. Overall, the results provide a broad catalogue of exact travelling-wave solutions and introduce a powerful framework for analyzing complex coupled dispersive waves with multiple nonlinearities.

Abstract

A new method is developed for finding exact solitary wave solutions of a generalized Korteweg-de Vries equation with p-power nonlinearity coupled to a linear Schrödinger equation arising in many different physical applications. This method yields 22 solution families, with p=1,2,3,4. No solutions for p>1 were known previously in the literature. For p=1, four of the solution families contain bright/dark Davydov solitons of the 1st and 2nd kind, obtained in recent work by basic ansatze applied to the ODE system for travelling waves. All of the new solution families have interesting features, including bright/dark peaks with (up to) p symmetric pairs of side peaks in the amplitude and a kink profile for the nonlinear part in the phase. The present method is fully systematic and involves several novel steps which reduce the travelling wave ODE system to a single nonlinear base ODE for which all polynomial solutions are found by symbolic computation. It is applicable more generally to other coupled nonlinear dispersive wave equations as well as to nonlinear ODE systems of generalized Hénon-Heiles form.
Paper Structure (24 sections, 9 theorems, 279 equations, 3 figures, 9 tables)

This paper contains 24 sections, 9 theorems, 279 equations, 3 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Hamiltonian structure and dimensionless form
  3. Dimensionless variables
  4. Symmetries and conservation laws
  5. Conservation laws
  6. ODE system for phase-modulated travelling waves
  7. Variational formulation and symmetries
  8. Reduction to a decoupled second-order ODE
  9. First integrals from symmetry multi-reduction
  10. Hodograph transformation to triangular form
  11. Abundant solitary wave solutions for nonlinearity powers $p=1,2,3,4$
  12. Polynomial solutions of the decoupled ODE
  13. Classification of solitary wave solutions
  14. Solutions for $\boldsymbol{p=1}$
  15. Solutions for $\boldsymbol{p=2}$
  16. ...and 9 more sections

Key Result

Proposition 1

The dimensionless form of the gKdV-LS system u.eqn.phys--psi.eqn.phys is given by where and The transformation to this form is given by

Figures (3)

  • Figure 1: Shape of potential for bright/dark solitary wave solutions
  • Figure 2: Bright ($s_3=1$) and dark ($s_3=-1$) peaks for $U(\xi)$ on zero background
  • Figure 3: Bright/dark peaks and tails for $A(\xi)$

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Proposition 5
  • Proposition 6