On uniqueness and decay of solution for Hirota equation

TL;DR

This paper proves that a decay property of the difference between two solutions at two different times guarantees their uniqueness for the Hirota equation, a nonlinear dispersive PDE.

Contribution

It establishes a new uniqueness criterion for solutions of the Hirota equation based on decay properties at two time points.

Findings

Decay property implies solution uniqueness.

Uniqueness holds under specific decay conditions.

Results contribute to understanding solution behavior of the Hirota equation.

Abstract

We address the question of the uniqueness of solution to the initial value problem associated to the equation \partial_{t}u+i\alpha \partial^{2}_{x}u+\beta \partial^{3}_{x}u+i\gamma|u|^{2}u+\delta |u|^{2}\partial_{x}u+\epsilon u^{2}\partial_{x}\bar{u} = 0, \quad x,t \in \R, and prove that a certain decay property of the difference $u_1-u_2$ of two solutions $u_1$ and $u_2$ at two different instants of times $t=0$ and $t=1$, is sufficient to ensure that $u_1=u_2$ for all the time.