# 理学院：线上学术报告：Orbital stability of standing waves for the bi-harmonic nonlinear Schrödinger equation with mixed dispersions

In this paper, we study the standing wave solutions for the  bi-harmonic nonlinear Schrödinger equation with a Laplacian term (BNLS),  modelling the propagation of intense laser beams in a bulk medium with Kerr nonlinearity.   By taking into account the role of second-order dispersion term, 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 $\mu\in [-\lambda_0,0)$, for some $\lambda_0:=\lambda_0(p, \|Q_p\|_2)>0$. Moreover, in the mass-critical case $p=1+\frac{8}{d}$, we  prove that the standing waves for the  BNLS are orbital stable when given $\mu\in (-\dfrac{4\|\nabla Q^*\|_2^2}{\|Q^*\|_2^2}, 0)$, and $b\in (b_*,b^*)$, for some $b^*:=\|Q^*\|_2^{\frac{8}{d}}$, $b_*:=b^*(\mu, \|Q^*\|_{H^2})\geq 0$. 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 mixed dispersions. This work joint with Tingjian Luo(Guangzhou University), and Shijun Zheng(Georgia Southern University).