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

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.