论文标题
在修改的分数Korteweg-de Vries和相关方程式上
On the modified fractional Korteweg-de Vries and related equations
论文作者
论文摘要
我们在本文中考虑修改了分数Korteweg-de Vries和相关方程（修改后的Burgers-Hilbert和Whitham）。它们具有相对于通常的分数KDV方程的优势，可以具有不同动力学的散落案例。我们将区分弱分散的情况，在该情况下，相位速度对于低频而无限，并且在无穷大时倾向于零，并且相分散的情况下，相位速度在原点消失并且无穷大的无穷大。在前一种情况下，非线性双曲线效应主要是大数据的主导，导致可能形成冲击的可能性，尽管分散效应在可能的小初始数据中表现出可能的散射。在后一种情况下，在聚焦案例中可能会有有限的时间爆炸，而不是冲击形成。在散落的情况下，在能量亚临界情况下，预计全球存在和散射是预期的，而在能量超临界情况下，预计有限的时间爆炸。 我们严格地建立了冲击的存在，并在弱分散案例中明确计算了爆破的时间和位置，而强烈分散案例的大多数结果都是通过数值模拟得出的，用于大型解决方案。此外，可以将冲击形成结果扩展到弱分散方程，并具有一些广义的非线性。 我们还将简要评论这些方程式的BBM版本。
We consider in this paper modified fractional Korteweg-de Vries and related equations (modified Burgers-Hilbert and Whitham). They have the advantage with respect to the usual fractional KdV equation to have a defocusing case with a different dynamics. We will distinguish the weakly dispersive case where the phase velocity is unbounded for low frequencies and tends to zero at infinity and the strongly dispersive case where the phase velocity vanishes at the origin and goes to infinity at infinity. In the former case, the nonlinear hyperbolic effects dominate for large data, leading to the possibility of shock formation though the dispersive effects manifest for small initial data where scattering is possible. In the latter case, finite time blow-up is possible in the focusing case but not the shock formation. In the defocusing case global existence and scattering is expected in the energy subcritical case, while finite time blow-up is expected in the energy supercritical case. We establish rigorously the existence of shocks with blow-up time and location being explicitly computed in the weakly dispersive case, while most of the results on the strongly dispersive case are derived via numerical simulations, for large solutions. Moreover, the shock formation result can be extended to the weakly dispersive equation with some generalized nonlinearity. We will also comment briefly on the BBM versions of those equations.
