Bifurcations of relative periodic orbits in NLS/GP with a triple-well potential

TL;DR

This paper investigates bifurcations of relative periodic orbits in the nonlinear Schrödinger equation with a triple-well potential, combining normal form theory and numerical analysis to understand orbit dynamics near bifurcations.

Contribution

It introduces a finite-dimensional Hamiltonian system model for NLS/GP with triple-well potential and analyzes bifurcations of oscillatory orbits using normal form computations.

Findings

Normal form theory accurately predicts orbit changes near Hamiltonian Hopf bifurcations.

Numerical experiments confirm theoretical predictions with some discrepancies near saddle-node bifurcations.

Agreement between theory and numerics is observed in key bifurcation regimes.

Abstract

The nonlinear Schr\"odinger/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.