Lie Symmetry Analysis of C 1 m , a , b Partial Differential Equations

In this article, we discussed the Lie symmetry analysis of C 1 m , a , b fractional and integer order differential equations. The symmetry algebra of both differential equations is obtained and utilized to find the similarity reductions, invariant solutions, and conservation laws. In both cases, the symmetry algebra is of low dimensions.

Conservation laws, classical symmetries and exact solutions of a (1 + 1)-dimensional fifth-order integrable equation

In this paper, we present a study of a fifth-order nonlinear partial differential equation, which was recently introduced in the literature. This equation can be used as a model for bidirectional water waves propagating in a shallow medium. Using elements of an optimal system of one-dimensional subalgebras, we perform similarity reductions culminating in analytic solutions. Rational, hyperbolic, power series and elliptic solutions are obtained. Furthermore, by using the multiple exponential function method we obtain one and two soliton solutions. Finally, local and low-order conserved quantities are derived by enlisting the multiplier approach.

Lie Symmetry Analysis, Self-Adjointness and Conservation Law for a Type of Nonlinear Equation

In the present paper, we mainly focus on the symmetry of the solutions of a given PDE via Lie group method. Meanwhile we transfer the given PDE to ODEs by making use of similarity reductions. Furthermore, it is shown that the given PDE is self-adjoining, and we also study the conservation law via multiplier approach.

Miura-Reciprocal Transformation and Symmetries for the Spectral Problems of KdV and mKdV

We present reciprocal transformations for the spectral problems of Korteveg de Vries (KdV) and modified Korteveg de Vries (mKdV) equations. The resulting equations, RKdV (reciprocal KdV) and RmKdV (reciprocal mKdV), are connected through a transformation that combines both Miura and reciprocal transformations. Lax pairs for RKdV and RmKdV are straightforwardly obtained by means of the aforementioned reciprocal transformations. We have also identified the classical Lie symmetries for the Lax pairs of RKdV and RmKdV. Non-trivial similarity reductions are computed and they yield non-autonomous ordinary differential equations (ODEs), whose Lax pairs are obtained as a consequence of the reductions.

Analytic study of solutions for a two-component Novikov system

Under investigation in this paper is a two-component Novikov system (also called Geng-Xue equation), which was proposed by Geng and Xue in 2009. Firstly, via the Lie symmetry method, infinitesimal generators, commutator table of Lie algebra and symmetry groups of the two-component Novikov system are presented. At the same time, some group invariant solutions are computed through similarity reductions. In particular, we construct peakon solution by applying the distribution theory. In addition, based on obtained group invariant solutions and symmetry transformations, we derive some new exact solutions, which include stationary solutions, smooth solutions, and a weak solution. The analytical properties to some of group invariant solutions and new exact solutions are discussed, such as decay, asymptotic behavior, and boundedness.

Similarities and exact solutions of transonic gas flow model

In this work, one of the important models in nonlinear wave theory and also in nonlinear acoustic, the Lin–Reissner–Tsien (LRT) equation is considered. For the homogeneous form of LRT equation, the exact solutions are obtained. For steady and non-steady state forms of the LRT equation with force terms, similarity reductions are obtained via the classical symmetry analysis method. Both of the considered problems are not seen in the literature. The results obtained in this paper are new solutions and believed to have a major role in the development of the model.

Classical Lie symmetries and similarity reductions of the (2 + 1)-dimensional dispersive long wave system

In the present work, classical Lie symmetry method is adopted to obtain the Lie symmetries and similarity reductions of the (2 + 1)-dimensional dispersive long wave equation. We also demonstrate that the symmetry algebra of this equation is an infinte dimensional, together with Kac–Moody–Virasoro type subalgebras.