Symmetry analysis of hydrodynamic-type systems

Opanasenko, Stanislav (2021) Symmetry analysis of hydrodynamic-type systems. Masters thesis, Memorial University of Newfoundland.

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Abstract

Using advantages of nonstandard computational techniques based on the light-cone variables, we explicitly find the algebra of generalized symmetries of the (1+1)-dimensional Klein{Gordon equation. This allows us to describe this algebra in terms of the universal enveloping algebra of the essential Lie invariance algebra of the Klein{Gordon equation. Then we single out variational symmetries of the corresponding Lagrangian and compute the space of local conservation laws of this equation, which turns out to be generated, up to the action of generalized symmetries, by a single first-order conservation law. We study the hydrodynamic-type system of differential equations modeling isothermal no-slip drift ux. Using the facts that the system is partially coupled and its essential subsystem reduces to the (1+1)-dimensional Klein{Gordon equation, we exhaustively describe generalized symmetries, cosymmetries and local conservation laws of this system. A generating set of local conservation laws under the action of generalized symmetries is proved to consist of two zeroth-order conservation laws. The subspace of translationinvariant conservation laws is singled out from the entire space of local conservation laws. The essential subsystem possesses three first-order hydrodynamic-type Hamiltonian operators, two of which are prolonged nonlocally to the entire system. The (1+2)-dimensional hydrodynamic-type system governing the shallow water model is studied from the symmetry-analysis point of view. Its complete point symmetry group is found with the help of the automorphism-based algebraic method. Lie reductions of both codimensions one and two are classified. We exhaustively describe the algebra of differential invariants of the point symmetry group of the system using the method of moving frames. We construct for the first time classes of differential equations with nontrivial generalized equivalence groups, i.e. whose equivalence-transformation components corresponding to independent and dependent variables locally depend on nonconstant arbitrary elements of the class. We rigourously construct extended generalized equivalence groups of several classes of differential equations as well. The new notion of effective generalized equivalence group is introduced.

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