Conservation laws and normal forms of evolution equations
Roman O. Popovych, Artur Sergyeyev
TL;DR
This paper classifies local conservation laws for evolution equations, providing normal forms for those with low-order conservation laws and analyzing the structure of conservation laws for linear equations.
Contribution
It introduces normal forms for evolution equations with low-order conservation laws and characterizes conservation laws for linear equations, advancing understanding of their structure.
Findings
Normal forms for equations with one or two low-order conservation laws
All conservation laws for linear equations are at most quadratic in the dependent variable
Examples include Harry Dym, KdV-type, and Schwarzian KdV equations
Abstract
We study local conservation laws for evolution equations in two independent variables. In particular, we present normal forms for the equations admitting one or two low-order conservation laws. Examples include Harry Dym equation, Korteweg-de-Vries-type equations, and Schwarzian KdV equation. It is also shown that for linear evolution equations all their conservation laws are (modulo trivial conserved vectors) at most quadratic in the dependent variable and its derivatives.
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