Conditional Lie–Bäcklund Symmetries and Invariant Subspaces to Nonlinear Diffusion Equations with Convection and Source

Abstract

The conditional Lie-Bäcklund symmetry method is used to study the invariant subspace of the nonlinear diffusion equations with convection and source terms. We obtain a complete list of canonical forms for such equations which admit higher order conditional Lie-Bäcklund symmetries and multidimensional invariant subspaces. The functionally generalized separable solutions to the resulting equations are constructed due to the corresponding symmetry reductions. For most of the cases, they are reduced to solving finite-dimensional dynamical systems.

Conditional Lie–Bäcklund symmetry, second-order differential constraint and direct reduction of diffusion systems

“…It is important to present presumably the form of CLBS to study scalar diffusion equation. CLBSs related to separation of variables [50], sign-invariants [21] and invariant subspace [22,23] have been proved to be very effective to classify and seek for symmetry reductions of the considered equations. These form of CLBS can be extended to study diffusion systems and new results will be involved.…”

“…Indeed, since in principle, one can determine any solution by a suitably clever choice of CLBS unless one explicit know all possible solutions. An alternative tactic, which seems more practical, is to specify the CLBS by external considerations, for example, one might try CLBSs related to separation of variables [50], sign-invariants [21] and invariant subspaces [22,23] for scalar diffusion equations.…”

“…Zhdanov [13] considered CLBS of evolution equation for the general case. There are a lot of symmetry related methods such as the nonlinear separation method [14], the signinvariant method [15,16] and the invariant subspace method [17,18], which are all included within the framework of CLBS [19]- [23].…”

Second-Order Conditional Lie–Bäcklund Symmetries and Differential Constraints of Nonlinear Reaction–Diffusion Equations with Gradient-Dependent Diffusivity

“…is powerful to study classifications and reductions of second-order nonlinear diffusion equations [14][15][16]. In fact, the linear CLBS with the characteristic…”

Invariant Subspaces of the Two-Dimensional Nonlinear Evolution Equations

“…In [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], the extensions of the invariant subspace method and various applications to other nonlinear PDEs were also discussed. It is noticed that a large number of exact solutions, such as N-solitons of integrable equations, similarity solutions of nonlinear evolution equations and the generalized functional separable solutions to nonlinear PDEs, can be recovered by the invariant subspace methods [1,[21][22][23][24][25][26][27][28][29][30][31]. In the one-dimensional space case, the invariant subspace method can be implemented by the conditional Lie-Bäcklund symmetry introduced independently by Zhdanov [32] and Fokas-Liu [33].…”