Instabilities of multiphase wave trains in coupled nonlinear Schrödinger equations: A bisymplectic framework

Hamiltonian systems, with bisymplectic structure, are known to model a wide range of interesting phenomena occurring in optics, oceanography, biochemistry, geology, and materials science. Examples of such systems are nonlinear Schrödinger (NLS) equations and Klein-Gordon (KG) equations. The paper focuses on a general class of the former and presents a linear stability theory for the interaction of a basic class of periodic traveling wave solutions, which exploits the geometric structure of the system. A criterion for linear instability is derived. Additionally, for the qualitatively tractable cases, criteria for linear instability are given explicitly in terms of: the amplitudes of the modes; the parameters of the system that characterize the medium as well as the interaction between component modes; and, when the solutions of the system are both time-and space dependent, the wave numbers. An extension to the coupled NLS equations case study is introduced, namely the consideration of a related class of coupled KG equations, which has the potential to lead to further development for the underlying bisymplectic systems theory.