双曲型偏微分方程组的条件对称与Riemann不变量
Conditional Symmetries and Riemann Invariants for Hyperbolic Systems of PDEs
摘要：
本文分析了秩k解的Riemann不变量，从两个概念之间的相互关系，即对称性约化方法和广义特征方法的一阶拟线性双曲方程组的偏微分方程在许多方面。提出了一种变形的条件对称方法获得这种类型的解决方案。构造了向量场的Lie模，向量场是由初始方程组和某些一阶微分约束定义的超定系统的对称。它表明，这个超定系统承认秩-k的解决方案表示的黎曼不变量。最后，详细讨论了该方法在（k+1）维流体动力学方程中的应用实例。获得了几个新的类孤立子解（其中包括扭结、凸块和多波解）。
英文摘要：
This paper contains an analysis of rank-k solutions in terms of Riemann invariants, obtained from interrelations between two concepts, that of the symmetry reduction method and of the generalized method of characteristics for first order quasilinear hyperbolic systems of PDEs in many dimensions. A variant of the conditional symmetry method for obtaining this type of solutions is proposed. A Lie module of vector fields, which are symmetries of an overdetermined system defined by the initial system of equations and certain first order differential constraints, is constructed. It is shown that this overdetermined system admits rank-k solutions expressible in terms of Riemann invariants. Finally, examples of applications of the proposed approach to the fluid dynamics equations in (k+1) dimensions are discussed in detail. Several new soliton-like solutions (among them kinks, bumps and multiple wave solutions) have been obtained.
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