A note on the uniqueness properties of solutions for the Schr\"odinger-Korteweg de Vries system

TL;DR

This paper proves the uniqueness of solutions for the coupled Schrödinger-Korteweg de Vries system under certain decay conditions at two different times, extending understanding of solution behavior in weighted Sobolev spaces.

Contribution

It establishes a uniqueness result for solutions with exponential decay at two times, a novel contribution to the analysis of the Schrödinger-Korteweg de Vries system.

Findings

Solutions are unique if they agree at two times with exponential decay.

The result applies to smooth solutions with decay in weighted Sobolev spaces.

Provides conditions under which solutions must coincide.

Abstract

In this work we prove that if $(u_i,v_i)$, $i=1,2$, are smooth enough solutions of the coupled Schr\"odinger-Korteweg-de Vries system \begin{align*} \left. \begin{array}{rl} i u_t+\partial_x^2 u &\hspace{-2mm}=\beta uv - |u|^2 u,\\ \partial_t v + \partial_x^3 v &\hspace{-2mm}=\gamma \partial_x |u|^2-\frac12\partial_x (v^2) \end{array} \right\} \end{align*} with appropriate decay at infinity such that at two different times $t_0=0$ and $t_1=1$ satisfy that $$u_1(0)-u_2(0),u_1(1)-u_2(1),v_1(0)-v_2(0),v_1(1)-v_2(1)\in H^1(e^{ax^{2}}dx),$$ for $a>0$ big enough, then $u_1=u_2$ and $v_1=v_2$. (Let us recall that $f\in H^1(e^{ax^{2}} dx)$ iff $f\in L^2(e^{ax^{2}}dx)$ and $\partial_x f\in L^2(e^{ax^{2}}dx)$).