Conservation laws and normal forms of evolution 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.

16 pages

References in corpus (7)

• Admissible Transformations and Normalized Classes of Nonlinear Schroedinger Equations
• Conservation Laws and Potential Symmetries of Linear Parabolic Equations
• Hierarchy of Conservation Laws of Diffusion--Convection Equations
• On recursion operators and nonlocal symmetries of evolution equations
• Local conservation laws of second-order evolution equations
• Potential Conservation Laws
• Conservation laws for multidimensional systems and related linear algebra problems

Cited by in corpus (13)

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• Well-posedness, travelling waves and geometrical aspects of generalizations of the Camassa-Holm equation
• Group classification of linear evolution equations
• Symmetries and conservation laws for the Karczewska--Rozmej--Rutkowski--Infeld equation
• Inverse problem on conservation laws
• A complete list of conservation laws for non-integrable compacton equations of $K(m,m)$ type
• Infinitely many nonlocal conservation laws for the $ABC$ equation with $A+B+C\ne 0$
• A family of wave equations with some remarkable properties
• Linearizability for third order evolution equations
• Generalized symmetries and conservation laws of (1+1)-dimensional Klein-Gordon equation
• Extended symmetry analysis of a "nonconservative Fokker-Plank equation"
• Generalized symmetries of remarkable (1+2)-dimensional Fokker-Planck equation
• Simplest potential conservation laws of linear evolution equations