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Soliton solutions associated with a class of third-order ordinary linear differential operators

Tuncay Aktosun, Abdon E. Choque-Rivero, Ivan Toledo, Mehmet Unlu

TL;DR

This work develops an inverse scattering framework for a third-order linear operator with Schwartz potentials, focusing on reflectionless data to construct explicit soliton solutions of associated nonlinear equations. By formulating a Riemann–Hilbert problem tied to a Lax pair $(L,A)$ and analyzing bound-state data, the authors derive $\mathbf N$-soliton solutions for the Sawada-Kotera equation and a modified bad Boussinesq equation, providing a physical interpretation of Hirota’s solitons via bound-state parameters. The core contribution is a determinant-based, IST-driven method that yields closed-form $Q(x)$ and $P(x)$, and under suitable reality constraints, real-valued solitons, with explicit 1– and multi-soliton examples and connections to Hirota’s construction. This framework clarifies how bound-state poles and their time-evolved constants encode multi-soliton interactions and can be extended to other third-order operator–evolution pairs.

Abstract

Explicit solutions to the related integrable nonlinear evolution equations are constructed by solving the inverse scattering problem in the reflectionless case for the third-order differential equation $d^3ψ/dx^3+Q\,dψ/dx+Pψ=k^3ψ,$ where $Q$ and $P$ are the potentials in the Schwartz class and $k^3$ is the spectral parameter. The input data set used to solve the relevant inverse problem consists of the bound-state poles of a transmission coefficient and the corresponding bound-state dependency constants. Using the time-evolved dependency constants, explicit solutions to the related integrable evolution equations are obtained. In the special cases of the Sawada--Kotera equation and the modified bad Boussinesq equation, the method presented here explains the physical origin of the constants appearing in the relevant $\mathbf N$-soliton solutions algebraically constructed, but without any physical insight, by the bilinear method of Hirota.

Soliton solutions associated with a class of third-order ordinary linear differential operators

TL;DR

This work develops an inverse scattering framework for a third-order linear operator with Schwartz potentials, focusing on reflectionless data to construct explicit soliton solutions of associated nonlinear equations. By formulating a Riemann–Hilbert problem tied to a Lax pair and analyzing bound-state data, the authors derive -soliton solutions for the Sawada-Kotera equation and a modified bad Boussinesq equation, providing a physical interpretation of Hirota’s solitons via bound-state parameters. The core contribution is a determinant-based, IST-driven method that yields closed-form and , and under suitable reality constraints, real-valued solitons, with explicit 1– and multi-soliton examples and connections to Hirota’s construction. This framework clarifies how bound-state poles and their time-evolved constants encode multi-soliton interactions and can be extended to other third-order operator–evolution pairs.

Abstract

Explicit solutions to the related integrable nonlinear evolution equations are constructed by solving the inverse scattering problem in the reflectionless case for the third-order differential equation where and are the potentials in the Schwartz class and is the spectral parameter. The input data set used to solve the relevant inverse problem consists of the bound-state poles of a transmission coefficient and the corresponding bound-state dependency constants. Using the time-evolved dependency constants, explicit solutions to the related integrable evolution equations are obtained. In the special cases of the Sawada--Kotera equation and the modified bad Boussinesq equation, the method presented here explains the physical origin of the constants appearing in the relevant -soliton solutions algebraically constructed, but without any physical insight, by the bilinear method of Hirota.

Paper Structure

This paper contains 6 sections, 1 theorem, 271 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The direct scattering problem
  3. The bound states and bound-state dependency constants
  4. The Riemann--Hilbert problem in the reflectionless case
  5. Soliton solutions
  6. A modified bad Boussinesq equation

Key Result

Theorem 2.1

Assume that the potentials $Q(x)$ and $P(x)$ in 1.1 belong to the Schwartz class $\mathcal{S(\mathbb R)}.$ Let $\Omega_1$ and $\Omega_3$ be the sectors defined in 2.6 and 2.8, respectively, and let $\overline{\Omega_1}$ and $\overline{\Omega_3}$ denote the corresponding closures, respectively. We ha

Figures (8)

  • Figure 2.1: The directed half lines $\mathcal{L}_1,$$\mathcal{L}_2,$$\mathcal{L}_3,$$\mathcal{L}_4$ and the open sectors $\Omega_1,$$\Omega_2,$$\Omega_3,$$\Omega_4$ in the complex $k$-plane.
  • Figure 2.2: The complex $k$-plane is divided into the six sectors $\Omega^\text{\rm{up}}_1,$$\Omega^\text{\rm{down}}_1,$$\Omega_2,$$\Omega^\text{\rm{down}}_3,$$\Omega^\text{\rm{up}}_3,$ and $\Omega_4$ as shown on the left plot, and the $k$-domains of three basic solutions to \ref{['1.1']} in each of the six regions, respectively, are shown on the right plot.
  • Figure 4.1: The $k$-domains of $T_{\text{\rm{l}}}(k) f(k,x),$$m(k,x),$$g(k,x),$ and $n(k,x),$ respectively, are shown on the left plot. The right plot shows the plus and minus regions in the complex $k$-plane separated by the directed full line $\mathcal{L},$ as well as the plus and minus functions in their respective $k$-domains.
  • Figure 5.1: The snapshots for the $1$-soliton solution $Q(x,t)$ to \ref{['1.10']} at $t=0,$$t=0.2,$$t=0.4,$ and $t=0.6,$ respectively.
  • Figure 5.2: The snapshots for the $2$-soliton solution $Q(x,t)$ to \ref{['1.10']} at $t=-0.1,$$t=-0.04,$$t=0,$ and $t=0.06,$ respectively.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Theorem 2.1
  • Example 5.1
  • Example 5.2
  • Example 5.3
  • Example 5.4
  • Example 6.1
  • Example 6.2