We study the Whitham equations for the defocusing complex modified KdV (mKdV) equation. These Whitham equations are quasilinear hyperbolic equations and they describe the averaged dynamics of the rapid oscillations which appear in the solution of the mKdV equation when the dispersive parameter is small. The oscillations are referred to as dispersive shocks. The Whitham equations for the mKdV equation are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the solutions of the Whitham equations when the initial values are given by a step function. We also compare the results with those of the defocusing nonlinear Schrodinger (NLS) equation. For the NLS equation, the Whitham equations are strictly hyperbolic and genuinely nonlinear. We show that the weak hyperbolicity of the mKdV-Whitham equations is responsible for an additional structure in the dispersive shocks which has not been found in the NLS case.
Modulation of Camassa--Holm equation and reciprocal transformations
本文导出了Camassa-Holm（CH）方程的调制方程或Whitham方程。我们证明了调制方程是双曲型的，并且具有双Hamilton结构。此外，它们通过倒数变换连接到Korteweg de Vries（KdV）方程的第一负流的调制方程。倒数变换由KdV平均流量的第二个Poisson括号的Casimir生成。我们表明，几何的双哈密顿结构的KdV和CH调制方程是完全不同的：事实上，KdV平均双哈密顿结构总是可以与一个半简单的Frobenius流形，而CH不能。
Riemann-Hilbert problem for the small dispersion limit of the KdV equation and linear overdetermined systems of Euler-Poisson-Darboux type
利用Riemann-Hilbert（RH）方法研究了具有小色散和单调增长初值的Korteweg de Vries（KdV）方程的Cauchy问题.使用振荡Riemann-Hilbert问题的最速下降法获得了零色散极限下柯西问题的解。渐近解完全由一个标量函数$\g$描述，该函数满足一个标量RH问题和一组由代数不等式约束的代数方程。标量函数$\g$等价于Lax-Levermore最大化问题的解。这组代数方程的解满足Whitham方程。我们证明了标量函数$\g$和Lax-Levermore极大化子可以表示为Euler-Poisson-Darboux型线性超定方程组的解。我们还表明，一组代数方程和代数不等式可以表示在一组不同的线性超定系统的Euler-Poisson-Darboux型方程的解决方案。此外，我们证明了代数方程组是等价的经典解的Whitham方程表示的速端变换。
Bifurcation diagram of a one-parameter family of dispersive waves
小色散的Korteweg de Vries（KdV）方程是一维色散激波形成和传播的模型。KdV中的色散激波的特征在于在具有光滑初始数据的柯西问题的解中出现快速调制振荡区。这些振荡及其相互作用的振幅、频率和波数在时间和空间上的调制近似地由$g$ -相位Whitham方程描述。研究了一类单参数单调递减初值族的Whitham方程的初值问题。我们得到了相互作用振荡区数$g$的分岔图。
Connection Between the Exact Moving Solutions of the Negative Korteweg-de Vries (nKdV) Equation and the Negative Modified Korteweg-de Vries (nmKdV) Equation and the Static Solutions of 1+1 Dimensional $ϕ^4$ Field Theory
Two-stage initial-value iterative physics-informed neural networks for simulating solitary waves of nonlinear wave equations
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Brown University（布朗大学）
