Investigation of the maximum amplitude increase from the Benjamin-Feir instability

TL;DR

This paper investigates the Benjamin-Feir instability in surface waves modeled by the NLS equation, revealing that the maximum amplitude increase during instability is finite and limited to three times the initial amplitude.

Contribution

The study links linearized NLS solutions to exact nonlinear soliton solutions, providing a finite bound on amplitude growth during Benjamin-Feir instability.

Findings

Maximum amplitude increase is at most three times the initial amplitude.

The linear growth predicted by the linearized NLS is bounded in the nonlinear regime.

The analysis connects linear instability to nonlinear exact solutions.

Abstract

The Nonlinear Schr\"odinger (NLS) equation is used to model surface waves in wave tanks of hydrodynamic laboratories. Analysis of the linearized NLS equation shows that its harmonic solutions with a small amplitude modulation have a tendency to grow exponentially due to the so-called Benjamin-Feir instability. To investigate this growth in detail, we relate the linearized solution of the NLS equation to a fully nonlinear, exact solution, called soliton on finite background. As a result, we find that in the range of instability the maximum amplitude increase is finite and can be at most three times the initial amplitude.