A uniqueness result for 2-soliton solutions of the KdV equation

TL;DR

This paper proves that 2-soliton solutions of the fourth-order stationary KdV equation are the only nonsingular solutions that vanish at infinity, using integrability theory of finite-dimensional Hamiltonian systems.

Contribution

It establishes the uniqueness of 2-soliton solutions among nonsingular solutions vanishing at infinity for a specific stationary KdV equation.

Findings

2-soliton solutions are unique among nonsingular solutions vanishing at infinity

The proof uses integrability of the associated Hamiltonian system

Results contribute to understanding solution structure of stationary KdV equations

Abstract

Multisoliton solutions of the KdV equation satisfy nonlinear ordinary differential equations which are known as stationary equations for the KdV hierarchy, or sometimes as Lax-Novikov equations. An interesting feature of these equations, known since the 1970's, is that they can be explicitly integrated, by virtue of being finite-dimensional completely integrable Hamiltonian systems. Here we use the integration theory to investigate the question of whether the multisoliton solutions are the only nonsingular solutions of these ordinary differential equations which vanish at infinity. In particular we prove that this is indeed the case for $2$-soliton solutions of the fourth-order stationary equation.