Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow 108840, Russia and Moscow Institute of Physics and Technology, Institutsky lane 9, Dolgoprudny, Moscow region 141700, Russia.
Phys Rev E. 2019 Jan;99(1-1):012203. doi: 10.1103/PhysRevE.99.012203.
We suggest a method for calculation of parameters of dispersive shock waves in the framework of Whitham modulation theory applied to nonintegrable wave equations with a wide class of initial conditions corresponding to propagation of a pulse into a medium at rest. The method is based on universal applicability of Whitham's "number of waves conservation law" as well as on the conjecture of applicability of its soliton counterpart to the above mentioned class of initial conditions which is substantiated by comparison with similar situations in the case of completely integrable wave equations. This allows one to calculate the limiting characteristic velocities of the Whitham modulation equations at the boundary with the smooth part of the pulse whose evolution obeys the dispersionless approximation equations. We show that explicit analytic expressions can be obtained for laws of motion of the edges. The validity of the method is confirmed by its application to similar situations described by the integrable Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations and by comparison with the results of numerical simulations for the generalized KdV and NLS equations.
我们提出了一种在 Whitham 调制理论框架下计算弥散激波参数的方法，该理论适用于具有广泛初始条件的不可积波动方程，这些初始条件对应于脉冲在静止介质中的传播。该方法基于 Whitham 的“波数守恒定律”的普遍适用性，以及其孤子对应物适用于上述初始条件类别的猜想，这一猜想通过与完全可积波动方程的类似情况进行比较得到了证实。这使得我们能够计算 Whitham 调制方程在与遵循无弥散近似方程演化的脉冲光滑部分的边界处的极限特征速度。我们表明，可以为边缘的运动定律获得显式解析表达式。该方法的有效性通过将其应用于由可积 Korteweg-de Vries（KdV）和非线性 Schrödinger（NLS）方程描述的类似情况以及与广义 KdV 和 NLS 方程的数值模拟结果进行比较得到了验证。
