Stable NLS solitons in a cubic-quintic medium with a delta-function potential

TL;DR

This paper analyzes one-dimensional cubic-quintic nonlinear Schrödinger equations with a delta-function potential, explicitly characterizing bound states, their stability, and coexistence regimes using bifurcation and orbital stability theories.

Contribution

It provides explicit formulas for bound states and demonstrates their stability and coexistence in a trapping potential, advancing understanding of soliton behavior in nonlinear media.

Findings

Explicit formulas for bound states derived

Identification of stable and coexisting solitons

Demonstration of orbital stability using bifurcation theory

Abstract

We study the one-dimensional nonlinear Schr\"odinger equation with the cubic-quintic combination of attractive and repulsive nonlinearities, and a trapping potential represented by a delta-function. We determine all bound states with a positive soliton profile through explicit formulas and, using bifurcation theory, we describe their behavior with respect to the propagation constant. This information is used to prove their stability by means of the rigorous theory of orbital stability of Hamiltonian systems. The presence of the trapping potential gives rise to a regime where two stable bound states coexist, with different powers and same propagation constant.