Lax pairs and a new spectral method for linear and integrable nonlinear PDEs
Lax pairs and a new spectral method for linear and integrable nonlinear PDEs
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Lax 对和线性和可积非线性偏微分方程的新谱方法
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发表时间：
1998
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通讯作者：
A. Fokas
中科院分区：
文献类型：
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作者：
A. Fokas
抽象的。文献[1]中提出了一种新的变换方法，用于求解两个独立变量的线性和可积非线性偏微分方程的初边值问题。对于线性偏微分方程，该方法包括：（a）将给定的偏微分方程表示为两个线性方程的相容条件，通过与非线性理论的类比，我们称之为Lax对;（B）通过对定义Lax对的两个方程进行同时谱分析来表示一个经典的数学问题，即所谓的Riemann-Hilbert问题;（c）导出给定偏微分方程解的边界值所满足的某些整体关系。本文用这种方法求解了热传导方程、线性化的Korteweg-deVries方程和拉普拉斯方程。一些算例表明，新方法可以有效地用于处理变类型边界条件和不可分边界条件等复杂边界条件问题。结果表明，对于简单的边界条件的整体关系（c）可以分析，仅使用代数操作，而对于复杂的边界条件，需要解决一个额外的Riemann-Hilbert问题。指出了该问题与经典Wiener-Hopf技巧的关系。最后讨论了上述结果在可积非线性方程中的推广。特别地，四分之一平面中的Korteweg-deVries方程被线性化。
Abstract. A new transform method for solving initial-boundary value problems for linear and integrable nonlinear PDEs in two independent variables has been recently introduced in [1]. For linear PDEs this method involves: (a) formulating the given PDE as the compatibility condition of two linear equations which, by analogy with the nonlinear theory, we call a Lax pair; (b) formulating a classical mathematical problem, the so-called Riemann-Hilbert problem, by performing a simultaneous spectral analysis of both equations defining the Lax pair; (c) deriving certain global relations satisfied by the boundary values of the solution of the given PDE. Here this method is used to solve certain problems for the heat equation, the linearized Korteweg-deVries equation and the Laplace equation. Some of these problems illustrate that the new method can be effectively used for problems with complicated boundary conditions such as changing type as well as nonseparable boundary conditions. It is shown that for simple boundary conditions the global relations (c) can be analyzed using only algebraic manipulations, while for complicated boundary conditions, one needs to solve an additional Riemann-Hilbert problem. The relationship of this problem with the classical Wiener-Hopf technique is pointed out. The extension of the above results to integrable nonlinear equations is also discussed. In particular, the Korteweg-deVries equation in the quarter plane is linearized.