Invariant submodels and exact solutions of the generalization of the Leith model of the wave turbulence

Abstract

A generalization of Leith’s model of the phenomenological theory of the wave turbulence is investigated. With the methods of group analysis, the basic models possessing non-trivial symmetries are obtained. For each model, all the invariant submodels are found. For nonlinear differential equations describing these models, formulas for the production of new solutions containing arbitrary constants are obtained. By virtue of these formulas, each solution generates a family of the new solutions. In an explicit form, some invariant solutions (not connected by point transformations) describing invariant submodels are found. The physical meaning of these solutions is obtained. In particular, with the help of these solutions the turbulent processes for which there are “destructive waves” both with fixed wave numbers and with varying wave numbers are described. On the example of an invariant solution of rank 1, it was shown that the search of the invariant solutions of rank 1, which cannot be found explicitly, can be reduced to solve the integral equations. For this solution, turbulent processes are investigated for which at the initial instant of a time and for a fixed value of the wave number either the turbulence energy and rate of its change or the turbulence energy and its gradient are given. Under certain conditions, the existence and uniqueness of the solutions of the boundary value problems describing these processes are established.