Group properties and conservation laws for nonlocal shallow water wave equation

Symmetry groups, symmetry reductions, optimal system, conservation laws and invariant solutions of the shallow water wave equation with nonlocal term are studied. First, Lie symmetries based on the invariance criterion for nonlocal equations and the solution approach for nonlocal determining equations are found and then the reduced equations and optimal system are obtained. Finally, new conservation laws are generated and some similarity solutions for symmetry reduction forms are discussed.     

Group-Invariant Solutions and Conservation Laws of One-Dimensional Nonlinear Wave Equation

Based on classical Lie symmetry method, the one-dimensional nonlinear wave equation is investigated. By using four-dimensional subalgebras of the equation, the invariant groups and commutator table are constructed. Furthermore, optimal system of the equation is obtained, and the exact solutions can be gained by solving reduced equations. Finally, a complete derivation of the conservation law is given by using conservation multipliers.     

Conservation laws and symmetries of semilinear radial wave equations

Classifications of symmetries and conservation laws are presented for a variety of physically and analytically interesting wave equations with power nonlinearities in n spatial dimensions: a radial hyperbolic equation, a radial Schrödinger equation and its derivative variant, and two proposed radial generalizations of modified Korteweg-de Vries equations, as well as Hamiltonian variants. The mains results classify all admitted local point symmetries and all admitted local conserved densities depending on up to first order spatial derivatives, including any that exist only for special powers or dimensions. All such cases for which these wave equations admit, in particular, dilational energies or conformal energies and inversion symmetries are determined. In addition, potential systems arising from the classified conservation laws are used to determine nonlocal symmetries and nonlocal conserved quantities admitted by these equations. As illustrative applications, a discussion is given of energy norms, conserved Hs norms, critical powers for blow-up solutions, and one-dimensional optimal symmetry groups for invariant solutions.     

Group-invariant solutions, non-group-invariant solutions and conservation laws of Qiao equation

This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators.     

Motion of curves and solutions of two multi-component mKdV equations

Two classes of multi-component mKdV equations have been shown to be integrable. One class called the multi-component geometric mKdV equation is exactly the system for curvatures of curves when the motion of the curves is governed by the mKdV flow. In this paper, exact solutions including solitary wave solutions of the two- and three-component mKdV equations are obtained, the symmetry reductions of the two-component geometric mKdV equation to ODE systems corresponding to it’s Lie point symmetry groups are also given. Curves and their behavior corresponding to solitary wave solutions of the two-component geometric mKdV equation are presented.     

Conservation Laws For the (1+2)-Dimensional Wave Equation in Biological Environment

The derivation of conservation laws for the wave equation on sphere, cone and flat space is considered. The partial Noether approach is applied for wave equation on curved surfaces in terms of the coefficients of the first fundamental form (FFF) and the partial Noether operator's determining equations are derived. These determining equations are then used to construct the partial Noether operators and conserved vectors for the wave equation on different surfaces. The conserved vectors for the wave equation on the sphere, cone and flat space are simplified using the Lie point symmetry generators of the equation and conserved vectors with the help of the symmetry conservation laws relation.     

Lie symmetries and conservation laws of the Hirota–Ramani equation

In this paper, Lie symmetry method is performed for the Hirota–Ramani (H–R) equation. We will find the symmetry group and optimal systems of Lie subalgebras. Furthermore, preliminary classification of its group invariant solutions, symmetry reduction and nonclassical symmetries are investigated. Finally conservation laws of the H–R equation are presented.     

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations is performed. First, complete infinite-order approximate symmetry classification of the equation is obtained by means of the method originated by Fushchich and Shtelen. An optimal system of one-dimensional subalgebras is derived and used to construct general formulas of approximate symmetry reductions and similarity solutions. Second, we study approximate homotopy symmetry of the equation and construct connections between the two symmetry methods for the first-order and higher-order cases, respectively. The series solutions derived by the two methods are compared.     

热方程的非古典势对称群与不变解

主要研究了热方程与波方程的非古典势对称群生成元及相应的群不变解．研究表明对于守恒形式的偏微分方程,可通过其伴随系统求得的非古典势对称群生成元来构造其显式解．这些显式解不能由方程本身的Lie对称群生成元或Lie-B?cklund对称群生成元构造得到．     

Combined sinh-cosh-Gordon equation: Symmetry reductions,exact solutions and conservation laws

Abstract

In this paper we study the combined sinh-cosh-Gordon equation, which arises in mathematical physics and has a wide range of scientific applications that range from chemical reactions to water surface gravity waves. We employ Lie symmetry analysis along with the simplest equation method to obtain exact solutions based on the optimal systems of one-dimensional subalgebras for the combined sinh-cosh-Gordon equation. Furthermore, conservation laws for the combined sinh-cosh-Gordon equation are derived by employing two different methods; the direct method and new conservation theorem.     

Local conservation laws,symmetries, and exact solutions for a Kudryashov‐Sinelshchikov equation

In this paper, we consider a Kudryashov‐Sinelshchikov equation that describes pressure waves in a mixture of a liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer between liquid and gas bubbles. We show that this equation is rich in conservation laws. These conservation laws have been found by using the direct method of the multipliers. We apply the Lie group method to derive the symmetries of this equation. Then, by using the optimal system of 1‐dimensional subalgebras we reduce the equation to ordinary differential equations. Finally, some exact wave solutions are obtained by applying the simplest equation method.     

The Lie Theory of Two-Variable Hypergeometric Functions

We develop a Lie-algebraic method that associates with each of the 34 distinct second-order hypergeometric functions in two variables a canonical system of partial differential equations. The special functions arise by partial separation of variables in these simple systems. Some consequences are a demonstration that all such functions appear as solutions of the 4-variable wave equation and a classification of the possible imbeddings. In each case the functions are characterized by first- and second-order operators in the enveloping algebra of the conformal symmetry algebra for the wave equation. In some cases the 3-variable wave and heat equations and the 2-variable Helmholtz equation also arise. This intimate relationship between Horn functions and some fundamental equations of mathematical physics shows that these functions are more interesting than was previously recognized and permits use of the powerful tools of Lie theory and separation of variables to obtain properties of the functions.     

Approximate Noether-type symmetries and conservation laws via partial Lagrangians for PDEs with a small parameter

We show how one can construct approximate conservation laws of approximate Euler-type equations via approximate Noether-type symmetry operators associated with partial Lagrangians. The ideas of the procedure for a system of unperturbed partial differential equations are extended to a system of perturbed or approximate partial differential equations. These approximate Noether-type symmetry operators do not form a Lie algebra in general. The theory is applied to the perturbed linear and nonlinear (1+1) wave equations and the Maxwellian tails equation. We have also obtained new approximate conservation laws for these equations.     

Travelling Wave Solutions and Conservation Laws of the (2+1)-dimensional Broer-Kaup-Kupershmidt Equatio

The travelling wave solutions and conservation laws of the (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK) equation are considered in this paper. Under the travelling wave frame, the BKK equation is transformed to a system of ordinary differential equations (ODEs) with two dependent variables. Therefore, it happens that one dependent variable $u$ can be decoupled into a second order ODE that corresponds to a Hamiltonian planar dynamical system involving three arbitrary constants. By using the bifurcation analysis, we obtain the bounded travelling wave solutions $u$, which include the kink, anti-kink and periodic wave solutions. Finally, the conservation laws of the BBK equation are derived by employing the multiplier approach.     

Lie symmetries and conservation laws of the Fokker-Planck equation with power diffusion

We concentrate on Lie symmetries and conservation laws of the Fokker-Planck equation with power diffusion describing the growth of cell populations. First, we perform a complete symmetry classification of the equation, and then we find some interesting similarity solutions by means of the symmetries and the variable coefficient heat equation. Local dynamical behaviors are analyzed via the solutions for the growing cell populations. Second, we show that the conservation law multipliers of the equation take the form Λ=Λ(t,x,u), which satisfy a linear partial differential equation, and then give the general formula of conservation laws. Finally, symmetry properties of the conservation law are investigated and used to construct conservation laws of the reduced equations.     

Complete Group Classification and Exact Solutions to the Generalized Short Pulse Equation

In this paper, the complete group classification is performed on the generalized short pulse equation, which includes a lot of important nonlinear wave equations as its special cases. In the sense of geometric symmetry, all of the vector fields of the equation are obtained in terms of the arbitrary functions. Then, the symmetry reductions and exact solutions to the equations are investigated. Especially, we develop the analytic power series method for constructing the exact power series solutions to the short pulse types of equations.     

Second-order approximate symmetry classification and optimal system of a class of perturbed nonlinear wave equations

A complete second-order approximate symmetry classification of a class of perturbed nonlinear wave equations with a arbitrary function is performed by means of the method originated from Fushchich and Shtelen. An optimal system of one-dimensional subalgebras is given. Based on the Lie symmetry reduction method, large classes of approximate reductions and invariant solutions of the equations are constructed.     

Symmetry group analysis and similarity solutions of variant nonlinear long-wave equations

Symmetry group properties and similarity solutions of the variant nonlinear long-wave equations in the form of system of nonlinear partial differential equations are analyzed. Lie symmetry group analysis of the variant nonlinear long-wave equations presents that the system has only two-parameter point symmetry group that corresponds to only traveling wave solutions. The symmetry groups yield the general reduced similarity form of the system, which is in the system of nonlinear ordinary differential equations. By using the improved tanh method the similarity solutions are obtained from the reduced system of equations. In addition, some graphical representations of the solitary and periodic solutions are presented.     

Conservation Laws and Potential Symmetries of Linear Parabolic Equations

We carry out an extensive investigation of conservation laws and potential symmetries for the class of linear (1+1)-dimensional second-order parabolic equations. The group classification of this class is revised by employing admissible transformations, the notion of normalized classes of differential equations and the adjoint variational principle. All possible potential conservation laws are described completely. They are in fact exhausted by local conservation laws. For any equation from the above class the characteristic space of local conservation laws is isomorphic to the solution set of the adjoint equation. Effective criteria for the existence of potential symmetries are proposed. Their proofs involve a rather intricate interplay between different representations of potential systems, the notion of a potential equation associated with a tuple of characteristics, prolongation of the equivalence group to the whole potential frame and application of multiple dual Darboux transformations. Based on the tools developed, a preliminary analysis of generalized potential symmetries is carried out and then applied to substantiate our construction of potential systems. The simplest potential symmetries of the linear heat equation, which are associated with single conservation laws, are classified with respect to its point symmetry group. Equations possessing infinite series of potential symmetry algebras are studied in detail.     

The Theory of Linear G-Difference Equations

We introduce the notion of difference equations defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants of the action of the symmetry group. Linear equations are modules over the skew group algebra, solutions are morphisms relating a given equation to other equations, symmetries of an equation are module endomorphisms, and conserved structures are invariants in the tensor algebra of the given equation.We show that the equations and their solutions can be described through representations of the isotropy group of the symmetry group of the underlying set. We relate our notion of difference equation and solutions to systems of classical difference equations and their solutions and show that out notions include these as a special case.