Modulational instability in the dynamics of interacting wave packets: the extended Korteweg-de Vries equation

This paper is concerned with interacting wave packet dynamics for long waves. The Kortweg-de Vries equation is the most well-known model for weakly nonlinear long waves. Although the dynamics of a single wave packet in this model is governed by the defocusing nonlinear Schrödinger equation, implying that a plane wave is modulationally stable, the dynamics of two interacting wave packets is governed by a coupled system of nonlinear Schrödinger equations; although each component is defocusing by itself, the coupling may lead nevertheless to modulational instability. In this paper we consider the analogous problem for the extended Korteweg–de Vries equation, which governs the evolution of weakly nonlinear long waves in those situations where cubic nonlinearity is comparable in magnitude with quadratic nonlinearity. In contrast to the Korteweg-de Vries equation, this extended long wave model can support modulational instability in the dynamics of a single wave packet for certain parameter régimes. Thus, for the case of two interacting wave packets, the governing system is again a coupled system of nonlinear Schrödinger equations, but now each component can be either focusing or defocusing. In this case, we find new classes of modulation instabilities arising from the coupling. In particular, cross-phase modulation-induced instabilities will depend critically on the sign of the cubic nonlinearity.