On a parameter-stability for normalized ground states of two-dimensional cubic–quintic nonlinear Schrödinger equations

Abstract

In this paper we consider the 2D cubic–quintic nonlinear Schrödinger equation with harmonic potential:

$$\begin{aligned} i\partial _t \psi =-\Delta \psi +|x|^2\psi -|\psi |^2\psi +\epsilon |\psi |^{4}\psi ,\ \ \ (t,x)\in {\mathbb {R}}\times {\mathbb {R}}^2. \end{aligned}$$

where $\epsilon >0$. We study the $\epsilon$-stability and $\epsilon$-instability (blow-up) of the ground state solitary waves of the form $\psi (t,x)=e^{-i\mu t}u_{\epsilon }(x)$ as $\epsilon \rightarrow 0^+$. The $\epsilon$-stability occurs if $\Vert u_{\epsilon }\Vert ^2_{L^2}<\rho _c$ (where $2/\rho _c$ is the best constant of Gagliardo–Nirenberg inequality), while $\epsilon$-blow-up occurs if $\Vert u_{\epsilon }\Vert ^2_{L^2}\ge \rho _c$. Two optimal blow-up rates will be computed. We show that as $\epsilon \rightarrow 0^+$, $\mathop \int \limits _{{\mathbb {R}}^2}|\nabla u_\epsilon |^2\,\mathrm{d}x\sim \epsilon ^{-\frac{1}{3}}$ for $\Vert u_{\epsilon }\Vert ^2_{L^2}=\rho _c$, and $\int _{{\mathbb {R}}^2}|\nabla u_\epsilon |^2\,\mathrm{d}x\sim \epsilon ^{-1}$ for $\Vert u_{\epsilon }\Vert ^2_{L^2}>\rho _c$. Moreover, two different blow-up profiles will be obtained.