Abstract

A method to derive formulas for the number of solitons into which a sufficiently intense initial pulse in the form of a localized simple wave evolves at asymptotically large time has been proposed on the basis of the Gurevich–Pitaevskii theorem on the number of oscillations entering the dispersive shock wave region. In application to integrable equations, this method reproduces Karpman-type formulas following from the semiclassical approximation for a linear spectral problem associated with the considered equation. It has been shown on particular examples that expressions previously obtained by means of the formal continuation of the solution of modulation Whitham equations to the dispersionless region of the wave can be derived in the case of nonintegrable equations.

中文翻译：

---

在渐近较大时间从强烈的初始脉冲生成的孤子数量

摘要

基于Gurevich-Pitaevskii定理，提出了一种进入孤子数量公式的方法，该孤子数量在渐进的大时间里以局部简单波的形式发展到足够强的初始脉冲，该定理基于进入色散的振荡数量冲击波区域。在可积分方程的应用中，此方法根据与考虑的方程相关的线性谱问题的半经典近似，重现了Karpman型公式。在特定的例子中已经表明，在不可积分方程的情况下，可以导出先前通过将调制Whitham方程的解形式形式地延续到波的无色散区域而获得的表达式。