A Note on Modulational Instability of a Nonlinear Klein-Gordon Equation

Abstract

The linear stability of uniform solutions of a complex nonlinear Klein-Gordon equation A_tt−A_xx−GA−N|A|²A=0, where G and N are real coefficients, is investigated. Two kinds of uniform solutions are treated: one is a wave train solution a₀ exp (iσt) where a₀ is a constant real amplitude; the other is a temporally periodic solution in terms of Jacobian elliptic function. The former is found to be unstable in the subcritical state (i.e. G<0, N>0); the latter is found to be unstable in both subcritical and supercritical (i.e. G>0, N<0) states. The stability digrams are given in both cases.