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Bifurcations of relative periodic orbits in NLS/GP with a triple-well potential. (English) Zbl 1378.35277

Summary: The nonlinear Schrödinger/Gross-Pitaevskii (NLS/GP) equation is considered in the presence of three equally-spaced potentials. The problem is reduced to a finite-dimensional Hamiltonian system by a Galerkin truncation. Families of oscillatory orbits are sought in the neighborhoods of the system’s nine branches of standing wave solutions. Normal forms are computed in the neighborhood of these branches’ various Hamiltonian Hopf and saddle-node bifurcations, showing how the oscillatory orbits change as a parameter is increased. Numerical experiments show agreement between normal form theory and numerical solutions to the reduced system and NLS/GP near the Hamiltonian Hopf bifurcations and some subtle disagreements near the saddle-node bifurcations due to exponentially small terms in the asymptotics.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
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