Conditional Lie-Backlund symmetry and reduction of evolution equations

Abstract

We suggest a generalization of the notion of invariance of a given partial differential equation with respect to Lie-Bäcklund vector field. Such generalization proves to be effective and enables us to construct principally new Ansätze 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.then the LBVF (2) is equivalent to the usual Lie vector field and can be represented in the following equivalent form:

The Method of Linear Determining Equations to Evolution System and Application for Reaction-Diffusion System with Power Diffusivities

“…On the other hand, η 1 depends on u n , and η 2 depends on v n . Hence, (22) holds only for γ = 0, which yields nonlinear determining equation (16). In analogy with the discussion above, we can derive another nonlinear determining equation (17) if DC (3) is compatible with the system (4).…”

Nonclassical Symmetries for a Class of Reaction-Diffusion Equations: the Method of Heir-Equations

“…Fokas and Liu [23] and Zhdanov [58] independently introduced the method of generalised conditional symmetries, i.e., conditional Lie-Bäcklund symmetries. In [44] it was shown that the heirequations can retrieve all the conditional Lie-Bäcklund symmetries found by Zhdanov.…”

Finding symmetries by incorporating initial conditions as side conditions

“…Since 1969, there have been several generalisations of the classical Lie group method for symmetry reductions, such as the non-classical method (Bluman and Cole, 1969), and the method of generalised conditional symmetries (Fokas and Liu, 1994;Zhdanov, 1995). These generalisations were made in an effort to obtain a wider class of symmetries which will lead to new solutions not obtainable via the classical method (see e.g.…”