A nonlocal Boussinesq equation: multiple-soliton solutions and symmetry analysis

A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method. To study various exact solutions of the nonlocal Boussinesq equation, it is converted into two local equations which contain the local Boussinesq equation. From the N-soliton solutions of the local Boussinesq equation, the N-soliton solutions of the nonlocal Boussinesq equation are obtained, among which the (N = 2, 3, 4)-soliton solutions are analyzed with graphs. Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from known solutions of the local Boussinesq equation. Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.

Electromagnetic control based on Lie symmetry transformation

An design method of electromagnetic metamaterial based on Lie symmetry of Maxwell's equation is proposed, which is applied to the modulation of electromagnetic wave / light. Firstly, the electromagnetic control model based on metamaterials is introduced, then according to the theory of Transformation Optics (TO), Lie symmetry analysis is applied to the coordinate transformation of material physical space, and the key core is the determining equations of Lie symmetry is derived. Secondly, the analytical forms of constitutive parameters (permittivity and permeability) of metamaterials are introduced, which can be used to design all kinds of electromagnetic metamaterials. Finally, the Lie symmetry method is applied to the control of electromagnetic beam width. The results show that the metamaterial based on Lie symmetry of Maxwell's equation have good field distribution, and it overcomes the single subjectivity of traditional coordinate transformation in optical transformation. The wave simulation by COMSOL Multiphysics software verify the correctness of Lie symmetry method.

Lie symmetry and μ-symmetry methods for nonlinear generalized Camassa–Holm equation

In this paper, a Lie symmetry method is used for the nonlinear generalized Camassa–Holm equation and as a result reduction of the order and computing the conservation laws are presented. Furthermore, μ-symmetry and μ-conservation laws of the generalized Camassa–Holm equation are obtained.

Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions

In this paper, we consider a member of an integrable family of generalized Camassa–Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new symmetries via the nonclassical method, which cannot be obtained by Lie symmetry method. We employ the multiplier method to construct conservation laws for this family of GCH equations. Using the conservation laws of the underlying equation, double reduction is also constructed. Finally, we investigate traveling waves of the GCH equations. We derive convergent series solutions both for the homoclinic and heteroclinic orbits of the traveling-wave equations, which correspond to pulse and front solutions of the original GCH equations, respectively.

New Exact Solutions of Date Jimbo Kashiwara Miwa Equation Using Lie Symmetry Groups

In this article, new exact solutions of 2 + 1 -dimensional Date Jimbo Kashiwara Miwa (DJKM) equation are constructed by applying the Lie symmetry method. By considering similarity variables obtained through Lie symmetry generators, considered 2 + 1 -dimensional DJKM equation is transformed into a linear partial differential equation with reduction of one independent variable. Afterwards by using Lie symmetry generators of this linear PDE, different invariant solutions involving exponential and logarithmic functions are explored which lead to the new exact solutions of the DJKM equation. Graphical representations of the obtained solutions are also presented to show the significance of the current work.

Invariance Analysis, Exact Solution and Conservation Laws of (2 + 1) Dim Fractional Kadomtsev-Petviashvili (KP) System

In this work, a Lie group reduction for a (2 + 1) dimensional fractional Kadomtsev-Petviashvili (KP) system is determined by using the Lie symmetry method with Riemann Liouville derivative. After reducing the system into a two-dimensional nonlinear fractional partial differential system (NLFPDEs), the power series (PS) method is applied to obtain the exact solution. Further the obtained power series solution is analyzed for convergence. Then, using the new conservation theorem with a generalized Noether’s operator, the conservation laws of the KP system are obtained.

Lie Symmetry Analysis, Exact Solutions, and Conservation Laws for the Generalized Time-Fractional KdV-Like Equation

In this paper, the problem of constructing the Lie point symmetries group of the nonlinear partial differential equation appeared in mathematical physics known as the generalized KdV-Like equation is discussed. By using the Lie symmetry method for the generalized KdV-Like equation, the point symmetry operators are constructed and are used to reduce the equation to another fractional ordinary differential equation based on Erdélyi-Kober differential operator. The symmetries of this equation are also used to construct the conservation Laws by applying the new conservation theorem introduced by Ibragimov. Furthermore, another type of solutions is given by means of power series method and the convergence of the solutions is provided; also, some graphics of solutions are plotted in 3D.

Lie symmetries of Generalized Equal Width wave equations

<abstract><p>Lie symmetry analysis of differential equations proves to be a powerful tool to solve or atleast to reduce the order and non-linearity of the equation. The present article focuses on the solution of Generalized Equal Width wave (GEW) equation using Lie group theory. Over the years, different solution methods have been tried for GEW but Lie symmetry analysis has not been done yet. At first, we obtain the infinitesimal generators, commutation table and adjoint table of Generalized Equal Width wave (GEW) equation. After this, we find the one dimensional optimal system. Then we reduce GEW equation into non-linear ordinary differential equation (ODE) by using the Lie symmetry method. This transformed equation can take us to the solution of GEW equation by different methods. After this, we get the travelling wave solution of GEW equation by using the Sine-cosine method. We also give graphs of some solutions of this equation.</p></abstract>

Construction of invariant solutions and conservation laws to the $(2+1)$-dimensional integrable coupling of the KdV equation

Under investigation in this paper is the $(2+1)$ ( 2 + 1 ) -dimensional integrable coupling of the KdV equation which has applications in wave propagation on the surface of shallow water. Firstly, based on the Lie symmetry method, infinitesimal generators and an optimal system of the obtained symmetries are presented. At the same time, new analytical exact solutions are computed through the tanh method. In addition, based on Ibragimov’s approach, conservation laws are established. In the end, the objective figures of the solutions of the coupling of the KdV equation are performed.