# Asymptotic Linear Stability of the Benney-Luke equation in 2D

## TL;DR

This paper analyzes the linear stability of line solitary waves in the 2D Benney-Luke equation, revealing resonant eigenvalues and describing their evolution, with implications for understanding water wave dynamics.

## Contribution

It provides a detailed spectral analysis of the transverse linear stability of solitary waves in the 2D Benney-Luke equation, highlighting resonant eigenvalues and their effects.

## Key findings

- Resonant continuous eigenvalues near zero in weak surface tension case
- Linear evolution of eigenmodes described by a 1D damped wave equation
- Exponential decay of solutions in weighted space over time

## Abstract

In this paper, we study transverse linear stability of line solitary waves to the $2$-dimensional Benney-Luke equation which arises in the study of small amplitude long water waves in $3$D. In the case where the surface tension is weak or negligible, we find a curve of resonant continuous eigenvalues near $0$. Time evolution of these resonant continuous eigenmodes is described by a $1$D damped wave equation in the transverse variable and it gives a linear approximation of the local phase shifts of modulating line solitary waves. In exponentially weighted space whose weight function increases in the direction of the motion of the line solitary wave, the other part of solutions to the linearized equation decays exponentially as $t\to\infty$.