# Ground states and energy asymptotics of the nonlinear Schr\"{o}dinger equation

## TL;DR

This paper analytically investigates the existence, uniqueness, and asymptotic behavior of ground states for the nonlinear Schrödinger equation with various nonlinearities and potentials, supported by numerical simulations.

## Contribution

It provides explicit approximations of ground states and energies in different regimes and explores the bifurcation phenomenon as nonlinearity becomes very large.

## Key findings

- Explicit ground state approximations in weak and strong regimes
- Identification of bifurcation in ground states at large nonlinearity
- Numerical validation in 1D and 2D cases

## Abstract

We study analytically the existence and uniqueness of the ground state of the nonlinear Schr\"{o}dinger equation (NLSE) with a general power nonlinearity described by the power index $\sigma\ge0$. For the NLSE under a box or a harmonic potential, we can derive explicitly the approximations of the ground states and their corresponding energy and chemical potential in weak or strong interaction regimes with a fixed nonlinearity $\sigma$. Besides, we study the case where the nonlinearity $\sigma\to\infty$ with a fixed interaction strength. In particular, a bifurcation in the ground states is observed. Numerical results in 1D and 2D will be reported to support our asymptotic results.