偏微分方程的代数方法
Algebraic Approaches to Partial Differential Equations
摘要：
偏微分方程是数学、科学和工程中的基本工具。这本书主要是一个博览会的各种代数技术解决偏微分方程的精确解作者在最近几年开发的，重点是物理方程，如：一维量子多体系统的Calogero-Sutherland模型，麦克斯韦方程组，自由Dirac方程，广义声学系统，KdV方程，Kadomtsev和Petviashvili（KP）方程、跨音速气流方程、短波方程、非线性声学中的Khokhlov和Zabolotskaya方程、位势预报方程、光学中的非线性薛定谔方程和耦合非线性薛定谔方程、三维表面波包的Davey和Stewartson方程、海洋动力对流方程、水物理中的Boussinesq方程、不可压纳维-斯托克斯方程和经典的边界层方程。在线性偏微分方程中，我们着重于寻找所有的多项式解和解决初值问题。给出了非线性偏微分方程Lie对称性的直观推导。这些对称变换从相对简单的解生成具有更多参数的复杂解。它们也被用来简化我们寻找精确解的过程。在非线性偏微分方程的研究中，我们广泛地使用了活动标架、非对称条件、非线性项的稳定域、特殊函数和线性化。我们得到的精确解通常包含多个参数函数，而且大多数不是行波型的。
英文摘要：
Partial differential equations are fundamental tools in mathematics,sciences and engineering. This book is mainly an exposition of the various algebraic techniques of solving partial differential equations for exact solutions developed by the author in recent years, with emphasis on physical equations such as: the Calogero-Sutherland model of quantum many-body system in one-dimension, the Maxwell equations, the free Dirac equations, the generalized acoustic system, the Kortweg and de Vries (KdV) equation, the Kadomtsev and Petviashvili (KP) equation, the equation of transonic gas flows, the short-wave equation, the Khokhlov and Zabolotskaya equation in nonlinear acoustics, the equation of geopotential forecast, the nonlinear Schrodinger equation and coupled nonlinear Schrodinger equations in optics, the Davey and Stewartson equations of three-dimensional packets of surface waves, the equation of the dynamic convection in a sea, the Boussinesq equations in geophysics, the incompressible Navier-Stokes equations and the classical boundary layer equations. In linear partial differential equations, we focus on finding all the polynomial solutions and solving the initial-value problems. Intuitive derivations of Lie symmetry of nonlinear partial differential equations are given. These symmetry transformations generate sophisticated solutions with more parameters from relatively simple ones. They are also used to simplify our process of finding exact solutions. We have extensively used moving frames, asymmetric conditions, stable ranges of nonlinear terms, special functions and linearizations in our approaches to nonlinear partial differential equations. The exact solutions we obtained usually contain multiple parameter functions and most of them are not of traveling-wave type.
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