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

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

“…Therefore, it is better to make oneself content with finding constraints in some classes, and these classes must be chosen using additional considerations. It is proved that DCs corresponding to the invariant surface conditions of the above CLBSs related to invariant subspace [14][15][16], sign-invariant [23,28] and separation of variables [24,25] are all very effective to study classifications and reductions of nonlinear diffusion equations.…”

“…The nonclassical symmetries and reductions of Arrhenius reaction-diffusion in n dimensions are considered in [41]. It has been known that the approaches of SI [42][43][44] and CLBS [14][15][16][17][18][19][20][21][22][23][24][25] have been successfully applied to obtain solutions and to explore the properties of Equation (3) with n = 1 or m = 1. Galaktionov [42][43][44] studied the existence of H-J SIs to the 1 + 1-dimensional nonlinear diffusion equations and the higher-dimensional case.…”

“…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].…”

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

“…It is known that the second-order nonlinear diffusion equations admit rich CLBSs [31,32,52,53]. The CLBS may provide symmetry interpretations for separation of variables [54], sign-invariants [55] and ISs [56]. More interestingly, the CLBS method can also be used to study symmetry reductions of initial-value problem of nonlinear evolution equations [57][58][59].…”