Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms, II

Additional nonlocal symmetries of diffusion-convection equations and the Burgers equation are obtained. It is shown that these equations are connected via a generalized hodograph transformation and appropriate nonlocal symmetries arise from additional Lie symmetries of intermediate equations. Two entirely different techniques are used to search nonlocal symmetry of a given equation: the first is based on usage of the characteristic equations generated by additional operators, another technique assumes the reconstruction of a parametrical Lie group transformation from such operator. Some of them are based on the nonlocal transformations that contain new independent variable determined by an auxiliary differential equation and allow the interpretation as a nonlocal transformation with additional variables. The formulae derived for construction of exact solutions are used.

“…is a particular case of the more general result established in [26]. The results concerning linearization of Equation (11) were obtained by Rosen [27].…”

Section: Lie Symmetries Of Given and Intermediate Equationsmentioning

confidence: 70%

New Nonlocal Symmetries of Diffusion-Convection Equations and Their Connection with Generalized Hodograph Transformation

Tychynin

2015

“…For the interested reader, it could be worthwhile noticing that Equation (105) is a particular case of equation considered in [8,9]. Moreover, we can get the results obtained in [8,9] (see Table 1, Case 8 of both papers) by projection.…”

Section: Remarkmentioning

confidence: 96%

“…Moreover, we can get the results obtained in [8,9] (see Table 1, Case 8 of both papers) by projection. Indeed, the Lie symmetry generator (99) projected in the space t, x, u, v = − γ 3 γ 2 becomes the Lie symmetry T 1 of [8,9].…”

Section: Remarkmentioning

confidence: 99%

See 1 more Smart Citation

An Application of Equivalence Transformations to Reaction Diffusion Equations

Torrisi

Tracinà

2015

In this paper, we consider a quite general class of advection reaction diffusion systems. By using an equivalence generator, derived in a previous paper, the authors apply a projection theorem to determine some special forms of the constitutive functions that allow the extension by one of the two-dimensional principal Lie algebra. As an example, a special case is discussed at the end of the paper.

“…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].…”

Section: Introductionmentioning

confidence: 99%

Invariant Subspaces of the Two-Dimensional Nonlinear Evolution Equations

Zhu

2016

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.