HAMILTONIAN BIFURCATIONS OF THE SPATIAL STRUCTURE FOR COUPLED NONLINEAR SCHRODINGER-EQUATIONS

A coupled pair of nonlinear Schrodinger equations is used as a model to study bifurcations in space of translation-invariant infinite-dimensional Hamiltonian systems. It is usual in translation-invariant systems to fix the spatial structure (for example: spatially periodic functions or functions that decay at infinity). In this paper the temporal structure (relative equilibria or time-periodic functions for example) is fixed and an evolution equation in space is constructed for the unknown spatial structure. The spatial evolution equation is shown to be, Hamiltonian and its basic properties are related to the fluxes of energy and momentum of the time-dependent problem. The gauge fluxes are used to reduce the spatial system to a Poisson system on a reduced phase space with two Casimir functions. Hamiltonian bifurcation theory is used to analyze the spatial Hamiltonian system. Particular attention is paid to the role of invariants in spatial bifurcations and temporal stability. For spatially periodic travelling waves a global result relating the momentum flux to the position of spatial Floquet multipliers is proved.