Stability of the NLS equation with viscosity effect

TL;DR

This paper derives a viscous nonlinear Schrödinger equation from a viscous KdV equation and investigates how viscosity influences the modulational instability of plane-wave solutions, showing that viscosity acts as a stabilizing factor.

Contribution

It introduces a NLS equation with viscosity derived via multiple time-scales and analyzes its modulational instability, highlighting viscosity's stabilizing effect.

Findings

Viscosity stabilizes modulational instability in the NLS equation.

The modulational dispersion relation is quadratic with complex coefficients.

Viscous effects derived from KdV influence wave stability.

Abstract

A nonlinear Schr\"{o}dinger (NLS) equation with an effect of viscosity is derived from a Korteweg-de Vries (KdV) equation modified with viscosity using the method of multiple time-scales. It is well known that the plane-wave solution of the NLS equation exhibits modulational instability phenomenon. In this paper, the modulational instability of the plane-wave solution of the NLS equation modified with viscosity is investigated. The corresponding modulational dispersion relation is expressed as a quadratic equation with complex-valued coefficients. By restricting the modulational wavenumber into the case of narrow-banded spectra, it is observed that a type of dissipation, in this case, the effect of viscosity, stabilizes the modulational instability, as confirmed by earlier findings.