Reduction of non-variational bi-Hamiltonian system of shallow-water waves propagation via symmetry approach

In this paper, non-variational bi-Hamiltonian system of shallow-water waves propagation is considered. Lie point generators are calculated and one dimensional optimal system of its subalgebras up to conjugacy classes are reported. Then similarity variables are computed by using these conjugacy classes which are further utilized for the reduction of considered system. Then, a transformation is used to convert the system from non-variational to variational system, thus standard Lagrangian is computed. Noether operators are calculated by using Noether approach and local conserved quantity is discussed for the new fourth order system of partial differential equations (PDEs). Further, inverse transformation is applied to get the corresponding local conserved quantity for the considered non-variational problem. Moreover, this local conservation law with the help of double reduction theorem is utilized to reduce the system.

Keywords:

Lagrangian, Noether operator Conserved quantities,

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