Modulational instability of small amplitude periodic traveling waves in the $b$-family of Novikov equation

TL;DR

This paper analyzes the modulational instability of small-amplitude periodic traveling waves in the $b$-family of Novikov equations, deriving an index that predicts spectral instability for long-wavelength perturbations.

Contribution

It introduces a spectral perturbation approach to derive a modulational instability index specific to the $b$-family of Novikov equations, confirming Benjamin-Feir instability.

Findings

Negative instability index indicates spectral instability.

Confirms Benjamin-Feir instability in the $b$-family.

Provides a criterion for long-wavelength perturbation instability.

Abstract

We study the modulational instability of smooth, small-amplitude periodic traveling wave solutions to the $b$-family of Novikov equation with cubic nonlinearity with an arbitrary coefficient $b>0$. Our approach is based on applying spectral perturbation theory to the corresponding linearization process. We derive a modulation instability index dependent on the nonlinear parameter $b$ and the fundamental wave number, and prove that when this index is negative, sufficiently small periodic traveling waves in the Novikov equation $b$-family exhibit spectral instability to long-wavelength perturbations. This confirms the well-known Benjamin-Feir instability in the $b$-family of Novikov equation.