Conservation Laws For Evolution Equations Research Articles

Inverse problem on conservation laws

The explicit formulation of the general inverse problem on conservation laws is presented for the first time. Within this problem, one aims to derive the general form of systems of differential equations that admit a prescribed set of conservation laws. The particular cases of the inverse problem on first integrals of ordinary differential equations and on conservation laws for evolution equations are studied. We also solve the inverse problem on conservation laws for differential equations admitting an infinite dimensional space of zeroth-order conservation-law characteristics. This particular case is further studied in the context of conservative first-order parameterization schemes for the two-dimensional incompressible Euler equations. We exhaustively classify conservative first-order parameterization schemes for the eddy-vorticity flux that lead to a class of closed, averaged Euler equations possessing conservation of generalized circulation, generalized momentum and energy.

Symmetries and conservation laws of evolution equations via multiplier and nonlocal conservation methods

Symmetries and conservation laws of evolution equations via multiplier and nonlocal conservation methods

Conservation laws and normal forms of evolution equations

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.

Conservation Laws of Evolution Equations: Generic Non-existence

Evolution-type partial differential equations in one space variable are formulated in terms of exterior differential systems. The space of conservation laws is discussed in this geometric context, and a familiar classical condition for conservation laws is derived. It is shown that the generic even-order evolution equation with one space variable possesses no conservation laws of order greater than the order of the equation.

On the derivation of fluxes for conservation laws in Hamiltonian systems

The use of Noether's theorem and Lie-group techniques provides a systematic method for investigating the conservation laws of evolution equations, that is, expressions of the form D,T + div F = 0. For Hamiltonian systems, the method gives only the density T, and the flux F must be computed indirectly using the evolution equations. Here, a direct procedure for calculating the flux is developed based on the density of the conserved quantity and on the associated symmetry generator. Simple expressions are given for specific classes of conservation laws. Particular attention is paid to Hamiltonian systems involving non-local operators.

Conservation laws for nonlinear evolution equations

A method to derive conservation laws for evolution equations that describe pseudospherical surfaces is introduced based on a geometrical property of these surfaces. A new third-order evolution equation is obtained as a first example for a nongeneric case in the classification given by Chern and Tenenblat [Stud. Appl. Math. 74, 1 (1986)].

On the correspondence between symmetries and conservation laws of evolution equations

There exists a well-defined correspondence between symmetries and conservation laws if an evolution equation admits a Hamiltonian formulation. We discuss whether the Hamiltonian structure is necessary for the correspondence.