标题： 半经典孤子系综用于三波共振相互作用方程 显示英文标题

标题： Semiclassical soliton ensembles for the three-wave resonant interaction equations

摘要： 三波共振相互作用方程是一个非色散的偏微分方程组，具有二次耦合，描述三个共振波模的复振幅的时间演化。 波包的碰撞通过泵浦和衰减在不同模式之间引起能量转移。 我们通过应用逆散射变换，在半经典极限下分析两个或三个波包的碰撞。 利用WKB分析，我们构建了一个相关的半经典孤子集合，这是一个通过其散射数据定义的无反射解族，旨在半经典极限下准确逼近初始数据。 从初始波包到孤子集合的映射是显式的，便于进行渐近和数值分析。 孤子集合的图示表明，时空平面被划分为包含静止、缓慢变化或快速振荡波的区域。 这种行为类似于Korteweg-de Vries方程和非线性薛定谔方程等方程中已知的色散冲击波的生成，尽管在没有色散的情况下，物理机制必须不同。 显示英文摘要

摘要： The three-wave resonant interaction equations are a non-dispersive system of partial differential equations with quadratic coupling describing the time evolution of the complex amplitudes of three resonant wave modes. Collisions of wave packets induce energy transfer between different modes via pumping and decay. We analyze the collision of two or three packets in the semiclassical limit by applying the inverse-scattering transform. Using WKB analysis, we construct an associated semiclassical soliton ensemble, a family of reflectionless solutions defined through their scattering data, intended to accurately approximate the initial data in the semiclassical limit. The map from the initial packets to the soliton ensemble is explicit and amenable to asymptotic and numerical analysis. Plots of the soliton ensembles indicate the space-time plane is partitioned into regions containing either quiescent, slowly varying, or rapidly oscillatory waves. This behavior resembles the well-known generation of dispersive shock waves in equations such as the Korteweg-de Vries and nonlinear Schrodinger equations, although the physical mechanism must be different in the absence of dispersion.