Abstract

All solutions to the Burgers, Hopf, Helmholtz, Klein–Gordon, sine-Gordon, Schrödinger, and Monge–Ampere equations having analytical complexity one (simple solutions) are described. It turns out that all simple solutions of the Burgers and Hopf equation are represented by elementary functions. An example of a family of solutions of complexity two to the Burgers equation is presented. Simple solutions to the Helmholtz (or Klein–Gordon) equation are expressed in terms of Bessel functions and elementary functions. For the Laplace and wave equations, an explicit description is given for the simple solutions that are expressed in terms of Jacobi elliptic functions. Open problems of the theory of analytic complexity (the analytical spectrum of an equation) are discussed.

中文翻译：

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关于一些数学物理方程的简单解

摘要

描述了具有解析复杂性1的Burgers，Hopf，Helmholtz，Klein-Gordon，sine-Gordon，Schrödinger和Monge-Ampere方程的所有解（简单解）。事实证明，Burgers和Hopf方程的所有简单解都由基本函数表示。给出了Burgers方程的一类复杂度为2的解的示例。亥姆霍兹（或Klein-Gordon）方程的简单解用贝塞尔函数和基本函数表示。对于拉普拉斯方程和波动方程，给出了用雅可比椭圆函数表示的简单解的明确描述。讨论了分析复杂性理论（方程的分析谱）的未解决问题。