Conditional Lie–Bäcklund symmetries and differential constraints for inhomogeneous nonlinear diffusion equations due to linear determining equations

Abstract

The method of linear determining equations is developed to study conditional Lie-Bäcklund symmetries for evolution equations, which is more general than the classical determining equations for Lie's generators. As an application of this approach, the complete classification is presented for the inhomogeneous nonlinear diffusion equations which admit the second-order and third-order conditional Lie-Bäcklund symmetries. Several examples are given to illustrate the corresponding symmetry reductions due to the compatibility of the invariant conditions and the governing equations.

Conditional Lie-Bäcklund Symmetry Reductions and Exact Solutions of a Class of Reaction-Diffusion Equations

“…The conditional Lie-Bäcklund symmetry (CLBS) method introduced by Zhdanov [9] and Fokas and Liu [10,11] firstly has been proved to be very powerful to classify equations or specify the functions appeared in the equations and construct the corresponding group invariant solutions. Furthermore, authors have shown that CLBS is closely related to the invariant subspace; namely, exact solutions defined on invariant subspaces for equations or their variant forms can be obtained by using the CLBS method [12][13][14][15][16][17][18][19][20][21][22][23][24].…”

Solvable structures for evolution PDEs admitting differential constraints

“…One of the most useful methods for determining particular explicit solutions to a system of evolution PDEs of the form (2) is to reduce it to a system of ODEs. This can be done by enlarging the original system of PDEs appending compatible additional equations (called differential constraints or side conditions, see [14,15,17,20,21]). If we look at the system (2) as a submanifold E ⊂ J ∞ (M, R m ) the differential constraints method is equivalent to find a suitable finite-dimensional submanifold H of E. The particular form of equation (2) and the explicit expression of the generatorsD x ,D t of C Ē…”

“…An interesting generalization of this reduction method is provided by the differential constraints method, consisting in appending to (1) an overdetermined systems of PDEs of the form L = {L 1 (x, t, u, u σ ) = 0, ..., L k (x, t, u, u σ ) = 0} such that the system L admits a general finite dimensional solution and is compatible with (1). Many reduction methods, such as (conditional) Lie-Bäcklund and non classical symmetry reductions, direct method of Clarkson and Kruskal, Galaktionov's nonlinear separation method and others can be seen as particular instances of differential constraints method (see [7,9,12,14,15,17,18,20,23,26]).…”

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

“…Here, we will present the linear determining equations to identify DC (3) and CLBS (5) in the general form of second-order evolution system (4), which is exactly the extension of the results for the scalar evolution equation in [40][41][42].…”

“…It is clear that it is workable to find the DC with the general form (6) of evolution Equation (7) by solving linear determining equation (8) about η. The principal direction of the research on applying the method to second-order nonlinear diffusion equations [40][41][42] gains an appreciation of its usefulness. The two-component reaction-diffusion (RD) system with power law diffusivities u t = (u k u x ) x + P(u, v), v t = (v l v x ) x + Q(u, v)…”