The unified transform for linear, linearizable and integrable nonlinear partial differential equations
The unified transform for linear, linearizable and integrable nonlinear partial differential equations
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DOI：
10.1088/0031-8949/89/03/038004
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发表时间：
2014-02
期刊：
影响因子：
2.9
通讯作者：
A. Fokas;S. De Lillo
中科院分区：
文献类型：
--
作者：
A. Fokas;S. De Lillo
逆散射为一类可积的非线性发展偏微分方程的初值问题的分析提供了一种强有力的方法。90年代末，第一作者受逆散射的启发，提出了一种新的边值问题分析方法。该方法为线性、可线性化和可积的非线性偏微分方程提供了统一的处理方法。在这里，这种方法，这通常被称为统一变换，说明了以下具体情况：半线上的热方程;半线上的非线性薛定谔方程;半线上的伯格方程;和移动边界上的伯格方程。
So-called inverse scattering provides a powerful method for analyzing the initial value problem for a large class of nonlinear evolution partial differential equations which are called integrable. In the late 1990s, the first author, motivated by inverse scattering, introduced a new method for analyzing boundary value problems. This method provides a unified treatment for linear, linearizable and integrable nonlinear partial differential equations. Here, this method, which is often referred to as the unified transform, is illustrated for the following concrete cases: the heat equation on the half-line; the nonlinear Schrödinger equation on the half-line; Burger's equation on the half-line; and Burger's equation on a moving boundary.