Classical and nonclassical symmetries for the Krichever-Novikov equation

We study the Krichever-Novikov equation from the standpoint of the theory of symmetry reductions in partial differential equations. We obtain a Lie group classification. Moreover, we obtain some exact solutions, and we apply the nonclassical method.     

On double reductions from symmetries and conservation laws

We present the theory of double reductions of PDEs with two independent variables that admit a Lie point symmetry and a conserved vector invariant under the symmetry. The theory is applied to a third order nonlinear partial differential equation which describes the filtration of a visco-elastic liquid with relaxation through a porous medium.     

Symmetries and solutions for a Fisher equation with a proliferation term involving tumor development

We consider a generalized Fisher equation involving tumor development from the point of view of the theory of symmetry reductions in partial differential equations. The study of this equation is relevant as it includes generalizations within the proliferation rate, being interpreted in terms of the total mass of the tumor. Classical Lie point symmetries admitted by the equation are determined. Finally, we obtain some biologically meaningful solutions in terms of a hyperbolic tangent function, which describes the tumor dynamics.     

A Lie symmetry analysis and explicit solutions of the two‐dimensional ∞‐Polylaplacian

In this work, we consider the Lie point symmetry analysis of a strongly nonlinear partial differential equation of third order, the ∞‐Polylaplacian, in two spatial dimensions. This equation is a higher order generalization of the ∞‐Laplacian, also known as Aronsson's equation, and arises as the analog of the Euler–Lagrange equations of a second‐order variational principle in L^∞. We obtain its full symmetry group, one‐dimensional Lie subalgebras and the corresponding symmetry reductions to ordinary differential equations. Finally, we use the Lie symmetries to construct new invariant ∞‐Polyharmonic functions.     

Lie symmetries and travelling wave solutions of the nonlinear waves in the inhomogeneous Fisher-Kolmogorov equation

In this work, we consider a Fisher-Kolmogorov equation depending on two exponential functions of the spatial variables. We study this equation from the point of view of symmetry reductions in partial differential equations. Through two-dimensional abelian subalgebras, the equation is reduced to ordinary differential equations. New solutions have been derived and interpreted.     

Exceptional symmetry reductions of Burgers' equation in two and three spatial dimensions

Lie point symmetry analysis of the general class of nonlinear diffusion-convection equations in two and three dimensions has shown that only for Burgers' equation (that isD(u)=const,K(u)=quadratic) is a full symmetry reduction to an ordinary differential equation possible. The optimal system of symmetry operators is determined to ensure that a minimal complete set of reductions is obtained. For each reduced partial differential equation, classical Lie group analysis has been performed and further reductions obtained. In this manner, all possible reductions to an ordinary differential equation are found, leading to exact solutions to both the two and three dimensional Burgers' equation.     

Nonclassical symmetry reductions of the Boussinesq equation

In this paper we discuss symmetry reductions and exact solutions of the Boussinesq equation using the classical Lie method of infinitesimals, the direct method due to Clarkson and Kruskal and the nonclassical method due to Bluman and Cole. In particular, we compare and contrast the application of these three methods. We discuss the use of symbolic manipulation programs in the implementation of these methods and differential Gröbner bases as a technique for solving the overdetermined systems of equations that arise. The relationship between the direct and nonclassical methods and other ansatz-based methods for deriving exact solutions of partial differential equations are also mentioned. To conclude we describe some of the important open problems in the field of symmetry analysis of differential equations.     

Conservation laws and exact solutions of the Whitham-type equations

In this paper, we study conservation laws for some partial differential equations. It is shown that interesting conserved quantities arise from multipliers by using homotopy operator that is a powerful algorithmic tool. Furthermore, the invariance properties of the conserved flows with respect to the Lie point symmetry generators are investigated via the symmetry action on the multipliers. Furthermore, the similarity reductions and some exact solutions are provided.     

mKdV方程的对称与群不变解

主要考虑mKdV方程的一些简单对称及其构成的李代数,并利用对称约化的方法将mKdV方程化为常微分方程,从而得到该方程的群不变解,这是对该方程群不变解的进一步扩展.     

Group analysis of the modified generalized Vakhnenko equation

In this paper the Lie symmetry group, the corresponding symmetry reductions and invariant solutions of the modified generalized Vakhnenko equation are determined. Moreover a numerical algorithm that is based on a Lie symmetry group preserving scheme is applied to the ordinary differential equations obtained by symmetry reduction.     

Symmetry analysis and exact explicit solutions for Kadomtsev-Petviashvili-Burgers equation

We apply the group theory to Kadomtsev-Petviashvili-Burgers (KPBII) equation which is a natural model for the propagation of the two-dimensional damped waves. In correspondence with the generators of the symmetry group allowed by the equation, new types of symmetry reductions are performed. Some new exact solutions are obtained, which can be in the form of solitary waves and periodic waves. Specially, our solutions indicate that the equation may have time-dependent nonlinear shears. Such exact explicit solutions and symmetry reductions are important in both applications and the theory of nonlinear science.     

直接法的推广与广义Burgers方程的条件对称约化

A generalization of the direct method of Clarkson and Kruskal for finding similarity reductions of partial differential equations with arbitrary functions is found and discussed for the generalized Burgers equation. The corresponding reductions and the exact solutions due to the methods of the ordinary differential equations are then given by the methods. The results given here answer partially an open problem proposed by Clarkson, that is how to develop the direct method to seek symmetry reductions of nonlinear PDEs with arbitrary functions.     

Bianchi Permutability for the Anti‐Self‐Dual Yang‐Mills Equations

The anti‐self‐dual Yang‐Mills equations are known to have reductions to many integrable differential equations. A general Bäcklund transformation (BT) for the anti‐self‐dual Yang‐Mills (ASDYM) equations generated by a Darboux matrix with an affine dependence on the spectral parameter is obtained, together with its Bianchi permutability equation. We give examples in which we obtain BTs of symmetry reductions of the ASDYM equations by reducing this ASDYM BT. Some discrete integrable systems are obtained directly from reductions of the ASDYM Bianchi system.     

Comparison of approximate symmetry and approximate homotopy symmetry to the Cahn–Hilliard equation

Complete infinite order approximate symmetry and approximate homotopy symmetry classifications of the Cahn–Hilliard equation are performed and the reductions are constructed by an optimal system of one-dimensional subalgebras. Zero order similarity reduced equations are nonlinear ordinary differential equations while higher order similarity solutions can be obtained by solving linear variable coefficient ordinary differential equations. The relationship between two methods for different order are studied and the results show that the approximate homotopy symmetry method is more effective to control the convergence of series solutions than the approximate symmetry one.     

Lie group analysis of two-dimensional variable-coefficient Burgers equation

The modern group analysis of differential equations is used to study a class of two-dimensional variable coefficient Burgers equations. The group classification of this class is performed. Equivalence transformations are also found that allow us to simplify the results of classification and to construct the basis of differential invariants and operators of invariant differentiation. Using equivalence transformations, reductions with respect to Lie symmetry operators and certain non-Lie ansätze, we construct exact analytical solutions for specific forms of the arbitrary elements. Finally, we classify the local conservation laws.     

Lie group analysis of two-dimensional variable-coefficient Burgers equation

The modern group analysis of differential equations is used to study a class of two-dimensional variable coefficient Burgers equations. The group classification of this class is performed. Equivalence transformations are also found that allow us to simplify the results of classification and to construct the basis of differential invariants and operators of invariant differentiation. Using equivalence transformations, reductions with respect to Lie symmetry operators and certain non-Lie ans?tze, we construct exact analytical solutions for specific forms of the arbitrary elements. Finally, we classify the local conservation laws.     

Classical Lie point symmetry analysis of nonlinear diffusion equations describing thermal energy storage

In this paper, we employed the linear transformation group approach to time dependent nonlinear diffusion equations describing thermal energy storage problem. Symmetry analysis of the governing equation resulted in admitted large Lie symmetry algebras for some special cases of the arbitrary constants and the source term. Some transformations that lead to equations with fewer arbitrary parameters are applied and classical Lie point symmetry methods are employed to analyze the transformed equations. Some symmetry reductions are performed and wherever possible the reduced ordinary differential equations are completely solved subject to realistic boundary conditions.     

Symmetry reductions and traveling wave solutions for the Krichever–Novikov equation

In this paper, we study the Krichever–Novikov equation from the point of view of the theory of symmetry reductions in PDEs. By using this theory, we find that for the Krichever–Novikov equation some similarity solutions are solutions with physical interest: solitons, kinks, antikinks, and compactons. Copyright © 2012 John Wiley & Sons, Ltd.     

The four‐dimensional Martínez Alonso–Shabat equation: nonlinear self‐adjointness and conservation laws

We show that the four‐dimensional Martínez Alonso–Shabat equation is nonlinearly self‐adjoint with differential substitution and the required differential substitution is just the admitted adjoint symmetry and vice versa. By means of computer algebra system, we obtain a number of local and nonlocal symmetries admitted by the equations under study. Then such symmetries are used to construct conservation laws of the equation under study and its reductions. Copyright © 2016 John Wiley & Sons, Ltd.