Abstract

The Differential-Invariant Submodels

S.V.Khabirov (Differential Equations of Mechanics Laboratory, Institute of Mechanics URC RAS, Ufa, Russia)

e-mail: habirov@anrb.ru

The classification of submodels for the system of the differential equations is recommended with the help to Lie's group of the admitted transformations. The optimal system of the subalgebra of Lie's algebra corresponding Lie's group was constructed. The basis of the differential inveriants for a subalgebra and the operators of the invariant differantiation are calculated. The system of the differential equations is written as the algebraic menifold between some differential invariants, hence the independent differential invariants are selected from the basis. The differential-invariant submodel of the rank k is named k-dimensional submonifold of the algebraic menifold. The representation of the differential-invariant solution gives the k-dimensional surface in the space of the independent differential invariants. The number k is restricted by the number of the invariants depending only the independent variables and the numeber of the independent differential invariants. The compatible conditions for the representation of the differential-invariant solution and the algabraic menifold give the differential-invariant submodel. The submodel is reduced to the ordinary invariant submodel by the reduction theorem if the rank k was equal to the number of the invariants depending on the independent variables. The submodels containing the regular and irregular partial invariant submodels are obtained otherwise.

Section : 12