Orbitally Stable Standing Waves of a Mixed Dispersion Nonlinear Schrödinger Equation

Abstract

We study the mixed dispersion fourth order nonlinear Schrödinger equation $i \partial_t \psi -\gamma \Delta^2 \psi +\beta \Delta \psi +|\psi|^{2\sigma} \psi =0\ \text{in}\ \mathbb{R} \times\mathbb{R}^N,$ where $\gamma,\sigma>0$ and $\beta \in \mathbb{R}$. We focus on standing wave solutions, namely, solutions of the form $\psi (x,t)=e^{i\alpha t}u(x)$ for some $\alpha \in \mathbb{R}$. This ansatz yields the fourth order elliptic equation $\gamma \Delta^2 u -\beta \Delta u +\alpha u =|u|^{2\sigma} u.$ We consider two associated constrained minimization problems: one with a constraint on the $L^2$-norm and the other on the $L^{2\sigma +2}$-norm. Under suitable conditions, we establish existence of minimizers and we investigate their qualitative properties, namely, their sign, symmetry, and decay at infinity as well as their uniqueness, nondegeneracy, and orbital stability.