On nonlinear partial differential equations with an infinite-dimensional conditional symmetry

The invariance of nonlinear partial differential equations under a certain infinite-dimensional Lie algebra A_N(z) in N spatial dimensions is studied. The special case A_1(2) was introduced in J. Stat. Phys. {\bf 75}, 1023 (1994) and contains the Schr\"odinger Lie algebra sch_1 as a Lie subalgebra. It is shown that there is no second-order equation which is invariant under the massless realizations of A_N(z). However, a large class of strongly non-linear partial differential equations is found which are conditionally invariant with respect to the massless realization of A_N(z) such that the well-known Monge-Ampere equation is the required additional condition. New exact solutions are found for some representatives of this class.Comment: Latex2e, 14 pages, no figures; final for

Lie and conditional symmetries of a class of nonlinear (1+2)-dimensional boundary value problems

A new definition of conditional invariance for boundary value problems involving a wide range of boundary conditions (including initial value problems as a special case) is proposed. It is shown that other definitions worked out in order to find Lie symmetries of boundary value problems with standard boundary conditions, follow as particular cases from our definition. Simple examples of direct applicability to the nonlinear problems arising in applications are demonstrated. Moreover, the successful application of the definition for the Lie and conditional symmetry classification of a class of (1+2)-dimensional nonlinear boundary value problems governed by the nonlinear diffusion equation in a semi-infinite domain is realised. In particular, it is proved that there is a special exponent, $k=-2$, for the power diffusivity $u^k$ when the problem in question with non-vanishing flux on the boundary admits additional Lie symmetry operators compared to the case $k\not=-2$. In order to demonstrate the applicability of the symmetries derived, they are used for reducing the nonlinear problems with power diffusivity $u^k$ and a constant non-zero flux on the boundary (such problems are common in applications and describing a wide range of phenomena) to (1+1)-dimensional problems. The structure and properties of the problems obtained are briefly analysed. Finally, some results demonstrating how Lie invariance of the boundary value problem in question depends on geometry of the domain are presented.Comment: 25 pages; the main results were presented at the Conference Symmetry, Methods, Applications and Related Fields, Vancouver, Canada, May 13-16, 201