Construction of Conservation Laws Using Symmetries [chapter]

Nail H. Ibragimov
2014 Similarity and Symmetry Methods  
The concept of nonlinear self-adjointness of differential equations, introduced by the author in 2010, is discussed in detail. All linear equations and systems are nonlinearly self-adjoint. Moreover, the class of nonlinearly self-adjoint equations includes all nonlinear equations and systems having at least one local conservation law. It follows, in particular, that the integrable systems possessing infinite set of Lie-Bäcklund symmetries (higher-order tangent transformations) are nonlinearly
more » ... lf-adjoint. An explicit formula for conserved vectors associated with symmetries is provided for all nonlinearly self-adjoint differential equations and systems. The number of equations contained in the systems under consideration can be different from the number of dependent variables. A utilization of conservation laws for constructing exact solutions is discussed and illustrated by computing noninvariant solutions of the Chaplygin equations in gas dynamics.
doi:10.1007/978-3-319-08296-7_2 fatcat:2axnzj4drrfspo7ly5pfo6jr4y