# Orbital stability of standing waves for the nonlinear Schr\"odinger equation with attractive delta potential and double power repulsive nonlinearity

## TL;DR

This paper investigates the existence and orbital stability of standing wave solutions for a one-dimensional nonlinear Schrödinger equation with an attractive delta potential and a double power repulsive nonlinearity, providing explicit constructions and stability proofs.

## Contribution

It introduces explicit constructions of standing waves and proves their orbital stability in a novel setting combining attractive delta potential with double power repulsive nonlinearity.

## Key findings

- Existence of standing wave solutions for certain parameters
- Orbital stability of these solutions under the flow
- Explicit construction methods for solutions

## Abstract

In this paper, a nonlinear Schr\"odinger equation with an attractive (focusing) delta potential and a repulsive (defocusing) double power nonlinearity in one spatial dimension is considered. It is shown, via explicit construction, that both standing wave and equilibrium solutions do exist for certain parameter regimes. In addition, it is proved that both types of wave solutions are orbitally stable under the flow of the equation by minimizing the charge/energy functional.