Modulational instability in the b-family equation

Abstract

Consideration in this paper is the modulational instability of periodic traveling waves in the vicinity of the origin in the spectral plane of the $b$-family equation admitting quadratic nonlinearity with an arbitrary coefficient $b\in R$. We derive modulational instability index as functions of both the nonlinear parameter $b$ and the wave number of the underlying wave, and demonstrate that a sufficiently small periodic traveling wave of the $b$-family equation is spectrally unstable to long wavelength perturbations when the modulational instability index is negative. Based on this, the effects of the nonlinear terms on the instability mechanism are discussed and a phenomenon so-called the unstable “island” is observed.