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Dynamics of solutions in the generalized Benjamin-Ono equation: A numerical study
Journal of Computational Physics ( IF 4.1 ) Pub Date : 2021-07-24 , DOI: 10.1016/j.jcp.2021.110570
Svetlana Roudenko , Zhongming Wang , Kai Yang

We consider the generalized Benjamin-Ono (gBO) equation on the real line, ut+x(Hux+1mum)=0,xR,m=2,3,4,5, and perform numerical study of its solutions. We first compute the ground state solution to QHQ+1mQm=0 via Petviashvili's iteration method. We then investigate the behavior of solutions in the Benjamin-Ono (m=2) equation for initial data with different decay rates and show decoupling of the solution into a soliton and radiation, thus, providing confirmation to the soliton resolution conjecture in that equation. In the mBO equation (m=3), which is L2-critical, we investigate solutions close to the ground state mass, and, in particular, we observe the formation of stable blow-up above it. Finally, we focus on the L2-supercritical gBO equation with m=4,5. In that case we investigate the global vs finite time existence of solutions, and give numerical confirmation for the dichotomy conjecture, in particular, exhibiting blow-up phenomena in the supercritical setting.



中文翻译:

广义 Benjamin-Ono 方程中解的动力学:数值研究

我们在实线上考虑广义 Benjamin-Ono (gBO) 方程, +X(-HX+1)=0,X电阻,=2,3,4,5,并对其解进行数值研究。我们首先计算基态解--H+1=0通过 Petviashvili 的迭代方法。然后我们研究了 Benjamin-Ono (=2) 具有不同衰减率的初始数据方程,并显示解耦合为孤子和辐射,因此,证实了该方程中的孤子分辨率猜想。在 mBO 方程中 (=3),即 2- 关键的是,我们研究了接近基态质量的解决方案,特别是,我们观察到在其上方形成稳定爆炸。最后,我们专注于2- 超临界 gBO 方程与 =4,5. 在这种情况下,我们研究解的全局与有限时间存在性,并对二分法猜想给出数值确认,特别是在超临界环境中表现出爆炸现象。

更新日期:2021-08-07
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