We apply techniques of symmetry group analysis in solving two systems of conservation laws: a model of two strictly hyperbolic conservation laws and a zero pressure gas dynamics model, which both have no global solution, but whose solution consists of singular shock waves. We show that these shock waves are solutions in the sense of 1-strong association. Also, we compute all projectable symmetr...

1. Conservation laws For a system of partial differential equations, the existence of appropriate conservation laws is often a key ingredient in the investigation of its solutions and their properties. Conservation laws can be used in proving existence of solutions, decay and scattering properties, investigation of singularities, analysis of integrability properties of the system and so on. Rep...

We study the group-valued and semigroup-valued conservation laws in cellular automata (CA). We provide examples to distinguish between semigroup-valued, group-valued and real-valued conservation laws. We prove that, even in one-dimensional case, it is undecidable if a CA has any non-trivial conservation law of each type. For a fixed range, each CA has a most general (group-valued or semigroup-v...

Using the concept of variational tricomplex endowed with a presymplectic structure, we formulate the general notion of symmetry. We show that each generalized symmetry of a gauge system gives rise to a sequence of conservation laws that are represented by onshell closed forms of various degrees. This extends the usual Noether’s correspondence between global symmetries and conservation laws to t...

All possible linearly independent local conservation laws for n-dimensional diffusion–convection equations ut = (A(u))ii +(B i(u))i were constructed using the direct method and the composite variational principle. Application of the method of classification of conservation laws with respect to the group of point transformations [R.O. Popovych, N.M. Ivanova, J. Math. Phys., 2005, V.46, 043502 (m...

Equations on curved manifolds display interesting properties in a number of ways. In particular, the symmetries and, therefore, the conservation laws reduce depending on how curved the manifold is. Of particular interest are the wave and Gordon-type equations; we study the symmetry properties and conservation laws of these equations on the Milne and Bianchi type III metrics. Properties of reduc...

The paper compares computational aspects of four approaches to compute conservation laws of single differential equations (DEs) or systems of them, ODEs and PDEs. The only restriction, required by two of the four corresponding computer algebra programs, is that each DE has to be solvable for a leading derivative. Extra constraints for the conservation laws can be specified. Examples include new...