Stability of the NLS Equation with Viscosity Effect

Abstract EN

A nonlinear Schrödinger (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 scale.

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.