Whitham调制理论在色散方程间断初值问题中的应用

近年来，数学和物理对色散介质流体动力学的研究兴趣日益增长。Whitham调制理论是一种描述色散冲击波的有力数学工具，该理论是G.B.Whitham于1965年首次提出的，采用平均理论推导出Whitham调制方程(即黎曼不变量满足的方程)，并根据Whitham调制方程研究稀疏波和色散冲击波解性态。目前有四种方法可以推出Whitham调制方程，即多尺度扰动展开法、平均守恒律法、finite-gap积分法和Lagrange平均方法。本项目拟开展以下三个方面的研究：(1)采用基于Lax对的finite-gap积分法研究带有间断初值的Hirota方程色散冲击波解及稀疏波的判别条件、近似解析表达式和作用机制；(2)采用多尺度扰动展开法研究RLW-Burgers方程带有间断初值色散冲击波及稀疏波的判别条件、近似解析表达式和作用机制；(3)利用直接数值模拟分析验证解的正确性。

There is growing physical and mathematical interest in the hydrodynamics of dispersive media. Whitham modulation theory is a powerful mathematical tool for describing dispersive shock waves. This theory was first proposed by G.B.Whitham in 1965. The Whitham averaging method is used to derive the Whitham modulation equations (the equations satisfied by the Riemann invariant). On the basis of the Whitham modulation equations, we can analyze the properties of the dispersive shock waves. There are currently four methods that can be used to obtain the Whitham modulation equations, namely the multiple scales perturbation theory, the averaging conservation laws method, the finite-gap integration method, and the averaged Lagrange procedure. This project intends to carry out the following three aspects of research: (1)The finite-gap integration method based on Lax pairs is used to obtain the Whitham modulation equations of Hirota equation with discontinuous initial values. By using the Whitham modulation equations, we can derive analytic formulae of the rarefaction waves, dispersive shock waves and analyze their properties and behaviors; (2)The multiple scales perturbation theory is used to derive the Whitham modulation equations of the RLW-Burgers equations with discontinuous initial value. Then, we can derive analytic formulae of the rarefaction waves, dispersive shock waves and analyze their properties and behaviors; (3)Confirm the accuracy of analytic formulae by comparison with numerical solutions of the original system.