Conditional Lie Backlund Symmetries Research Articles

Invariant subspaces and conditional Lie-Bäcklund symmetries of inhomogeneous nonlinear diffusion equations

The inhomogeneous nonlinear diffusion equation is studied by invariant subspace and conditional Lie-Backlund symmetry methods. It is shown that the equations admit a class of invariant subspaces governed by the nonlinear ordinary differential equations, which is equivalent to a kind of higher-order conditional Lie-Backlund symmetries of the equations. As a consequence, a number of new solutions to the inhomogeneous nonlinear diffusion equations are constructed explicitly or reduced to solving finite-dimensional dynamical systems.

Solutions and symmetry reductions of the n-dimensional non-linear convection-diffusion equations

This paper discusses a wide class of n-dimensional non-linear convection―diffusion equations with source term. It is shown that the radially symmetric equations admit certain types of conditional Lie―Backlund symmetries. As a result, exact solutions and symmetry reductions to 2D dynamical systems of the resulting equations are obtained. Those solutions extend the known ones such as self-similar solutions and instantaneous source-type solutions of the porous medium equation with absorption term. The behaviour of extinction and blow-up to many of the solutions are described.

Conditional Lie-Backlund symmetry and reduction of evolution equations

We suggest a generalization of the notion of invariance of a given partial differential equation with respect to Lie-B\"acklund vector field. Such generalization proves to be effective and enables us to construct principally new Ans\"atze reducing evolution-type equations to several ordinary differential equations. In the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations $u_t=u_{xx}+F(u,u_x)$ with poor Lie symmetry and obtain their exact solutions. It is shown that these solutions can not be constructed by means of the symmetry reduction procedure.