Weak stability of the sum of two solitary waves for Half-wave equation

TL;DR

This paper investigates the weak stability of the sum of two solitary waves in the subcritical one-dimensional half-wave equation, highlighting the effects of strong interactions and employing energy and monotonicity methods.

Contribution

It demonstrates weak stability for two solitary waves with large separation in the half-wave equation, introducing novel analysis of non-local effects and interaction strength.

Findings

Weak stability when the solitary waves are sufficiently separated

Use of energy method and local mass monotonicity in analysis

Analysis of non-local effects via Calderón estimate and integral formulas

Abstract

In this paper, we consider the subcritical half-wave equation in one dimension. Let $R_k(t,x)$, $k=1,2$, represent two-solitary wave solutions of the half-wave equation, each with different translations $x_1,x_2$. We prove that if the relative distance $x_2-x_1$ between the two solitary waves is large enough, then the sum of $R_k(t)$ is weakly stable. Our proof relies on an energy method and the local mass monotonicity property. Unlike the single-solitary wave or NLS cases, the interactions between different waves are significantly stronger here. To establish the local mass monotonicity property, as well as to analyze non-local effects on localization functions and non-local operator $D$, we utilize the Carlder\'on estimate and the integral representation formula of the half-wave operator.