论文标题
概述的本杰明·诺克方程中解决方案的动力学：数值研究
Dynamics of solutions in the generalized Benjamin-Ono equation: a numerical study
论文作者
论文摘要
我们考虑了真实行上的概括的Benjamin-Ono（GBO）方程，$ u_t + \ \ partial_x（ - \ Mathcal H u_ {x} + \ tfrac1 {m} u^m）= 0，x \ in \ Mathbb r，m = 2,3,4,4,5 $，并执行了其解决方案的研究。我们首先将基态解决方案计算为$ -Q- \ MATHCAL H Q^\ PRIME +\ frac1 {M} Q^m = 0 $通过PETVIASHVILI的迭代方法。然后，我们研究了benjamin-ono（$ M = 2 $）方程中解决方案的行为，用于具有不同衰减速率的初始数据，并将溶液脱耦到孤子和辐射中，从而向该方程中的Soliton分辨率求解提供了确认。在MBO方程（$ M = 3 $）中，即$ l^2 $ - 批判性，我们调查靠近基块质量的解决方案，尤其是我们观察到其上方稳定的爆破的形成。最后，我们专注于$ l^2 $ -Superitical GBO方程，$ M = 4,5 $。在这种情况下，我们研究了解决方案的全球与有限时间的存在，并给出了二分法猜想的数值确认，特别是在超临界环境中表现出爆炸现象。
We consider the generalized Benjamin-Ono (gBO) equation on the real line, $ u_t + \partial_x (-\mathcal H u_{x} + \tfrac1{m} u^m) = 0, x \in \mathbb R, m = 2,3,4,5$, and perform numerical study of its solutions. We first compute the ground state solution to $-Q -\mathcal H Q^\prime +\frac1{m} Q^m = 0$ 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 $L^2$-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 $L^2$-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.
