Continuous and discrete transformations of a one-dimensional porous medium equation

We consider the one-dimensional porous medium equation $u_t=\left (u^nu_x \right )_x+\frac{\mu}{x}u^nu_x$. We derive point transformations of a general class that map this equation into itself or into equations of a similar class. In some cases this porous medium equation is connected with well known equations. With the introduction of a new dependent variable this partial differential equation can be equivalently written as a system of two equations. Point transformations are also sought for this auxiliary system. It turns out that in addition to the continuous point transformations that may be derived by Lie's method, a number of discrete transformations are obtained. In some cases the point transformations which are presented here for the single equation and for the auxiliary system form cyclic groups of finite order

Conservation laws for nonlinear telegraph equations

AbstractA complete conservation law classification is given for nonlinear telegraph (NLT) systems with respect to multipliers that are functions of independent and dependent variables. It turns out that a very large class of NLT systems admits four nontrivial local conservation laws. The results of this work are summarized in tables which display all multipliers, fluxes and densities for the corresponding conservation laws. A physical example is considered for possible applications

Classical and nonclassical symmetries of a generalized Boussinesq equation

We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions $f(u)$ for which this equation admit either the classical or the nonclassical method. The reductions obtained are derived. Some new exact solutions can be derived

Nonlinear Dirac and diffusion equations in 1 + 1 dimensions from stochastic considerations

We generalize the method of obtaining the fundamental linear partial differential equations such as the diffusion and Schrodinger equation, Dirac and telegrapher's equation from a simple stochastic consideration to arrive at certain nonlinear form of these equations. The group classification through one parameter group of transformation for two of these equations is also carried out.Comment: 18 pages, Latex file, some equations corrected and group analysis in one more case adde

Lie group classifications and exact solutions for time-fractional Burgers equation

Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests a fractional Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained.Comment: 9 pp, accepte

Contact symmetry of time-dependent Schr\"odinger equation for a two-particle system: symmetry classification of two-body central potentials

Symmetry classification of two-body central potentials in a two-particle Schr\"{o}dinger equation in terms of contact transformations of the equation has been investigated. Explicit calculation has shown that they are of the same four different classes as for the point transformations. Thus in this problem contact transformations are not essentially different from point transformations. We have also obtained the detailed algebraic structures of the corresponding Lie algebras and the functional bases of invariants for the transformation groups in all the four classes

Equivalence of conservation laws and equivalence of potential systems

We study conservation laws and potential symmetries of (systems of) differential equations applying equivalence relations generated by point transformations between the equations. A Fokker-Planck equation and the Burgers equation are considered as examples. Using reducibility of them to the one-dimensional linear heat equation, we construct complete hierarchies of local and potential conservation laws for them and describe, in some sense, all their potential symmetries. Known results on the subject are interpreted in the proposed framework. This paper is an extended comment on the paper of J.-q. Mei and H.-q. Zhang [Internat. J. Theoret. Phys., 2006, in press].Comment: 10 page

Differential constraints compatible with linearized equations

Differential constraints compatible with the linearized equations of partial differential equations are examined. Recursion operators are obtained by integrating the differential constraints

On asymptotic nonlocal symmetry of nonlinear Schr\"odinger equations

A concept of asymptotic symmetry is introduced which is based on a definition of symmetry as a reducibility property relative to a corresponding invariant ansatz. It is shown that the nonlocal Lorentz invariance of the free-particle Schr\"odinger equation, discovered by Fushchych and Segeda in 1977, can be extended to Galilei-invariant equations for free particles with arbitrary spin and, with our definition of asymptotic symmetry, to many nonlinear Schr\"odinger equations. An important class of solutions of the free Schr\"odinger equation with improved smoothing properties is obtained

Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction-diffusion equations of the general form $f(x)u_t=(g(x)u_x)_x+h(x)u^m$ ($m\ne0,1$) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with $m=2$ is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case $m\ne2$. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).Comment: 40 pages, this is version published in Acta Applicanda Mathematica