关于微分方程保持守恒律的摄动
On perturbations retaining conservation laws of difftrential equations
摘要：
本文讨论了具有多个守恒律的方程的摄动问题。当在方程中加入一个小项时，它的守恒量通常以各自的速率衰减，这种现象称为选择性衰减。这些速率由使用守恒定律的生成函数和附加项的简单定律描述。然而，某些微扰可能保留特定的量，例如能量、动量和溶液的其他物理上重要的特征。我们介绍了一个程序，找到这样的扰动，并证明它的例子，包括KdV-Burgers方程和系统的磁动力学。揭示和讨论了此类扰动方程解的一些有趣性质。
英文摘要：
The paper deals with perturbations of the equation that have a number of conservation laws. When a small term is added to the equation its conserved quantities usually decay at individual rates, a phenomenon known as a selective decay. These rates are described by the simple law using the conservation laws' generating functions and the added term. Yet some perturbation may retain a specific quantity(s), such as energy, momentum and other physically important characteristics of solutions. We introduce a procedure for finding such perturbations and demonstrate it by examples including the KdV-Burgers equation and a system from magnetodynamics. Some interesting properties of solutions of such perturbed equations are revealed and discussed.
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