Orbital Stability of Standing Waves for BNLS]{Orbital Stability of Standing Waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions

Tingjian Luo, Shijun Zheng, Shihui Zhu

Published 2019-04-04Version 1

In this paper, we study the standing wave solutions of the bi-harmonic nonlinear Schr\"{o}dinger equation with the Laplacian term (BNLS). By taking into account the role of second-order dispersion term in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity, we prove that in the mass-subcritical regime $p\in (1,1+\frac{8}{d})$, there exist orbitally stable standing waves for BNLS, when $\mu\geq 0$, or $-\lambda_0\le\mu<0$, for some $\lambda_0>0$. Moreover, we prove that in the mass-critical case $p=1+\frac{8}{d}$, the BNLS is orbital stable when $-\lambda_1\le\mu<0$, for some $\lambda_1>0$, and the initial data is below the ground state for the bihamonic operator. This shows that the sign of the second-order dispersion has crucial effect on the existence of orbitally stable standing waves for the BNLS with the mixed dispersions.