Order Conservation Laws Research Articles

Evolution equations with high order conservation laws

An asymptotic algorithm presented in a previous paper is applied to investigate the possible structures of evolution equations ut=uM+K(u,...,uM−1), M=3,5, which could be compatible with the existence of a conserved density ρ0(u), depending only on u, and with the existence as well of conserved densities with arbitrarily high-order derivatives. For M=3 it is shown that the Calogero–Degasperis–Fokas equation is essentially the only nonpolynomial equation of that type. For the case ut=D[u4+Q], with Q(u,...,u3) a polynomial, we find a very narrow class of admissible structures for Q, typified by the few particular examples known up to date. Actually, there is in this case an essentially unique structure, modulo a Miura transformation.

Higher order conservation laws. II

Higher order conservation laws. II

Higher order conservation laws

Higher order conservation laws

Correlation theory of stationary electromagnetic fields. Part IV — Second order conservation laws

By imitating well-known techniques to obtain the energyequation in Maxwell’s theory, eight independent higher order conservation laws are found for the various second order correlation functions associated with stationary random electromagnetic fields. Two further conservation laws are deduced which involve the physically so significant energy coherence tensor and the flow coherence tensor.