Invariant Subspaces of the Two-Dimensional Nonlinear Evolution Equations

Abstract

Abstract:In this paper, we develop the symmetry-related methods to study invariant subspaces of the two-dimensional nonlinear differential operators. The conditional Lie-Bäcklund symmetry and Lie point symmetry methods are used to construct invariant subspaces of two-dimensional differential operators. We first apply the multiple conditional Lie-Bäcklund symmetries to derive invariant subspaces of the two-dimensional operators. As an application, the invariant subspaces for a class of two-dimensional nonlinear quadratic operators are provided. Furthermore, the invariant subspace method in one-dimensional space combined with the Lie symmetry reduction method and the change of variables is used to obtain invariant subspaces of the two-dimensional nonlinear operators.

Invariant Subspaces of Short Pulse-Type Equations and Reductions

“…The invariant subspace method is powerful for studying nonlinear partial differential equations (PDEs). Various invariant subspaces to a number of nonlinear PDEs have been obtained (see [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], as well as the references therein). Accordingly, exact solutions stemming from this method play important roles in the study of their asymptotical behavior, blow up and geometric properties, etc.…”

Invariant subspace method to the initial and boundary value problem of the higher dimensional nonlinear time-fractional PDEs

“…In [49], Zhu and Qu have studied the exact solutions of the two-dimensional nonlinear evolution equations using the invariant subspace method. In 2021, Abdel Kader et al [44] have derived the exact solutions of the (2 + 1)-dimensional nonlinear time-fractional variable coefficient biological population model using the invariant subspace method along with the variable transformation.…”

“…The detailed study of the invariant subspace method was initially provided by Galaktionov and Svirshchevskii [41] for deriving the exact solutions of the integer-order nonlinear evolution PDEs, which is commonly known as the generalized separation of the variable method. Since then, this method has been further studied by Ma and many others [45][46][47][48][49][62][63][64] for scalar and coupled nonlinear PDEs. In recent days, Gazizov and Kasatkin [36], and others [29, 32-35, 37-40, 50] have extended this method for finding the exact solutions of fractional scalar and coupled nonlinear higher-dimensional PDEs.…”

Initial value problem for the two-dimensional time-fractional generalized convection-reaction-diffusion-wave equation: Invariant subspaces and exact solutions

“…Recently, the invariant subspace method was extended for scalar and coupled non-linear fractional PDEs [23][24][25][26][27][28][29][30][31][32][33][34][35]. Zhu and Qu [58] have extended this method to two-dimensional non-linear PDEs. However, to the best of our knowledge, no one has been extended the invariant subspace method to two-dimensional fractional non-linear PDEs.…”