Kink-Antikink Interaction Forces and Bound States in a nonlinear Schr{\"o}dinger Model with Quadratic and Quartic dispersion

TL;DR

This paper investigates the complex interactions and bound states of kinks in a nonlinear Schr{"o}dinger model with mixed quadratic and quartic dispersion, revealing rich bifurcation structures and stability properties through analytical and numerical methods.

Contribution

It introduces the first six families of multikink solutions in this model and develops an effective particle ODE theory to analyze their stability and bifurcations.

Findings

Six families of multikink solutions identified.

Rich bifurcation structure connecting different kink states.

Effective ODE theory accurately captures kink equilibria and modes.

Abstract

In the present work we explore the competition of quadratic and quartic dispersion in producing kink-like solitary waves in a model of the nonlinear Schr{\"o}dinger type bearing cubic nonlinearity. We present the first 6 families of multikink solutions and explore their bifurcations as the strength of the quadratic dispersion is varied. We reveal a rich bifurcation structure for the system, connecting two-kink states with states involving 4-, as well as 6-kinks. The stability of all of these states is explored. For each family, we discuss a ``lower branch'' adhering to the energy landscape of the 2-kink states. We also, however, study in detail the ``upper branches'' bearing higher numbers of kinks. In addition to computing the stationary states and analyzing their stability within the partial differential equation model, we develop an effective particle ordinary differential equation theory that is shown to be surprisingly efficient in capturing the kink equilibria and normal (as well as unstable) modes. Finally, the results of the bifurcation analysis are corroborated by means of direct numerical simulations involving the excitation of the states in a targeted way in order to explore their instability-induced dynamics.