Existence, Stability and Dynamics of Harmonically Trapped One-Dimensional Multi-Component Solitary Waves: The Near-Linear Limit

TL;DR

This paper investigates the existence, stability, and dynamics of two-component one-dimensional solitary waves in a trapped nonlinear Schrödinger setting, providing explicit eigenvalue formulas and analyzing instability-induced vibrational behaviors.

Contribution

It develops a systematic existence and stability theory for multi-component solitary waves in a parabolic trap, including explicit eigenvalue expressions and dynamic instability analysis.

Findings

Explicit eigenvalue formulas for various states

Identification of stability and instability regions

Dynamics of unstable states showing vibrational evolution

Abstract

In the present work, we consider a variety of two-component, one-dimensional states in nonlinear Schrodinger equations in the presence of a parabolic trap, inspired by the atomic physics context of Bose-Einstein condensates. The use of Lyapunov-Schmidt reduction methods allows us to identify persistence criteria for the different families of solutions which we classify as (m,n), in accordance with the number of nodes in each component. Upon developing the existence theory, we turn to a stability analysis of the different configurations, using the Krein signature and the Hamiltonian-Krein index as topological tools identifying the number of potentially unstable eigendirections for each branch. A systematic expansion of suitably reduced eigenvalue problems when perturbing off of the linear limit permits us to obtain explicit expressions for the eigenvalues of each of the states considered. Finally, when the states are found to be unstable, typically by virtue of Hamiltonian Hopf bifurcations, their dynamics is studied in order to identify the nature of the respective instability. The dynamics is generally found to lead to a vibrational evolution over long time scales.