### Chirkunov Y.A.

## Exact solutions of the generalization of Leith model of the wave turbulence

A generalization of the Leith’s model of the phenomenological theory of the wave turbulence are researched. With the methods of group analysis, the basic models possessing nontrivial 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 researched 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 that can not be found explicitly, can be reduced to solving of the integral equations. For this solution turbulent processes are researched 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. A generalization of the Leith’s model of the phenomenological theory of the wave turbulence are researched. With the methods of group analysis, the basic models possessing nontrivial 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 researched 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 that can not be found explicitly, can be reduced to solving of the integral equations. For this solution turbulent processes are researched 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. A generalization of the Leith’s model of the phenomenological theory of the wave turbulence are researched. With the methods of group analysis, the basic models possessing nontrivial 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 researched 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 that can not be found explicitly, can be reduced to solving of the integral equations. For this solution turbulent processes are researched 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. A generalization of the Leith’s model of the phenomenological theory of the wave turbulence are researched. With the methods of group analysis, the basic models possessing nontrivial 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 researched 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 that can not be found explicitly, can be reduced to solving of the integral equations. For this solution turbulent processes are researched 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. A generalization of the Leith’s model of the phenomenological theory of the wave turbulence are researched. With the methods of group analysis, the basic models possessing nontrivial 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 researched 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 that can not be found explicitly, can be reduced to solving of the integral equations. For this solution turbulent processes are researched 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. A generalization of the Leith’s model of the phenomenological theory of the wave turbulence are researched. With the methods of group analysis, the basic models possessing nontrivial 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 researched 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 that can not be found explicitly, can be reduced to solving of the integral equations. For this solution turbulent processes are researched 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. A generalization of the Leith’s model of the phenomenological theory of the wave turbulence are researched. With the methods of group analysis, the basic models possessing nontrivial 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 researched 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 that can not be found explicitly, can be reduced to solving of the integral equations. For this solution turbulent processes are researched 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. A generalization of the Leith’s model of the phenomenological theory of the wave turbulence are researched. With the methods of group analysis, the basic models possessing nontrivial 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 researched 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 that can not be found explicitly, can be reduced to solving of the integral equations. For this solution turbulent processes are researched 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. A generalization of the Leith’s model of the phenomenological theory of the wave turbulence are researched. With the methods of group analysis, the basic models possessing nontrivial 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 researched 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 that can not be found explicitly, can be reduced to solving of the integral equations. For this solution turbulent processes are researched 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. A generalization of the Leith’s model of the phenomenological theory of the wave turbulence are researched. With the methods of group analysis, the basic models possessing nontrivial 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 researched 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 that can not be found explicitly, can be reduced to solving of the integral equations. For this solution turbulent processes are researched 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. A generalization of the Leith’s model of the phenomenological theory of the wave turbulence are researched. With the methods of group analysis, the basic models possessing nontrivial 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 researched 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 that can not be found explicitly, can be reduced to solving of the integral equations. For this solution turbulent processes are researched 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. A generalization of the Leith’s model of the phenomenological theory of the wave turbulence are researched. With the methods of group analysis, the basic models possessing nontrivial 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 researched 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 that can not be found explicitly, can be reduced to solving of the integral equations. For this solution turbulent processes are researched 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. A generalization of the Leith’s model of the phenomenological theory of the wave turbulence are researched. With the methods of group analysis, the basic models possessing nontrivial 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 researched 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 that can not be found explicitly, can be reduced to solving of the integral equations. For this solution turbulent processes are researched 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. A generalization of the Leith’s model of the phenomenological theory of the wave turbulence are researched. With the methods of group analysis, the basic models possessing nontrivial 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 researched 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 that can not be found explicitly, can be reduced to solving of the integral equations. For this solution turbulent processes are researched 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.

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