Stochastic Conservation Laws?

We examine conservation laws, typically the conservation of linear momentum, in the light of a recent successful formulation of fermions as Kerr-Newman type Black Holes, which are created fluctuationally from a background Zero Point Field. We conclude that these conservation laws are to be taken in the spirit of thermodynamic laws.Comment: 5 pages, Te

Conservation laws for the classical Toda field theories

We have performed some explicit calculations of the conservation laws for classical (affine) Toda field theories, and some generalizations of these models. We show that there is a huge class of generalized models which have an infinite set of conservation laws, with their integrated charges being in involution. Amongst these models we find that only the $A_m$ and $A_m^{(1)}$ ($m\ge 2$) Toda field theories admit such conservation laws for spin-3. We report on our explicit calculations of spin-4 and spin-5 conservation laws in the (affine) Toda models. Our perhaps most interesting finding is that there exist conservation laws in the $A_m$ models ($m\ge4)$ which have a different origin than the exponents of the corresponding affine theory or the energy-momentum tensor of a conformal theory.Comment: 9 pages, Late

Potential Conservation Laws

We prove that potential conservation laws have characteristics depending only on local variables if and only if they are induced by local conservation laws. Therefore, characteristics of pure potential conservation laws have to essentially depend on potential variables. This statement provides a significant generalization of results of the recent paper by Bluman, Cheviakov and Ivanova [J. Math. Phys., 2006, V.47, 113505]. Moreover, we present extensions to gauged potential systems, Abelian and general coverings and general foliated systems of differential equations. An example illustrating possible applications of proved statements is considered. A special version of the Hadamard lemma for fiber bundles and the notions of weighted jet spaces are proposed as new tools for the investigation of potential conservation laws.Comment: 36 pages, extended versio

Conservation laws for self-adjoint first order evolution equations

In this work we consider the problem on group classification and conservation laws of the general first order evolution equations. We obtain the subclasses of these general equations which are quasi-self-adjoint and self-adjoint. By using the recent Ibragimov's Theorem on conservation laws, we establish the conservation laws of the equations admiting self-adjoint equations. We illustrate our results applying them to the inviscid Burgers' equation. In particular an infinite number of new symmetries of these equations are found and their corresponding conservation laws are established.Comment: This manuscript has been accepted for publication in Journal of Nonlinear Mathematical Physic

Integrability from an abelian subgroup of the diffeomorphism group

It has been known for some time that for a large class of non-linear field theories in Minkowski space with two-dimensional target space the complex eikonal equation defines integrable submodels with infinitely many conservation laws. These conservation laws are related to the area-preserving diffeomorphisms on target space. Here we demonstrate that for all these theories there exists, in fact, a weaker integrability condition which again defines submodels with infinitely many conservation laws. These conservation laws will be related to an abelian subgroup of the group of area-preserving diffeomorphisms. As this weaker integrability condition is much easier to fulfil, it should be useful in the study of those non-linear field theories.Comment: 13 pages, Latex fil

An Analysis of the invariance and conservation laws of some classes of nonlinear ostrovsky equations and related systems

A large class of partial differential equations in the modelling of ocean waves are due to Ostrovsky. We determine the invariance properties (through the Lie point symmetry generators) and construct classes of conservation laws for some of the models. In the latter case, the method involves finding the 'multipliers' associated with the conservation laws with a stronger emphasis on the 'higher-order' ones. The relationship between the symmetries and conservation laws is investigated by considering the invariance properties of the multipliers