Roman Cherniha

Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms

New results concerning Lie symmetries of nonlinear reaction-diffusion-convection equations, which supplement in a natural way the results published in the European Journal of Applied Mathematics ( 9 (1998) 527–542) are presented.

New non-Lie ansätze and exact solutions of nonlinear reaction-diffusion-convection equations

A constructive method for obtaining new exact solutions of nonlinear evolution equations is further developed. The method is based on the consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high-order ordinary differential equation. Using this method new non-Lie ansatze and exact solutions are obtained for two classes of diffusion equations with power and exponential nonlinearities, which describe real processes in physics, chemistry, and biology. The analysis of the found solutions and the relation of the proposed method to some approaches, which have been suggested in several recently published papers, are presented.

The Galilean relativistic principle and nonlinear partial differential equations

The second order partial differential equations invariant under transformations of Galilei, rotation, scale and projection are described.

The exotic conformal Galilei algebra and nonlinear partial differential equations

Abstract The conformal Galilei algebra ( cga ) and the exotic conformal Galilei algebra ( ecga ) are applied to construct partial differential equations (PDEs) and systems of PDEs, which admit these algebras. We show that there are no single second-order PDEs invariant under the cga but systems of PDEs can admit this algebra. Moreover, a wide class of nonlinear PDEs exists, which are conditionally invariant under cga . It is further shown that there are systems of nonlinear PDEs admitting ecga with the realisation obtained very recently in [D. Martelli, Y. Tachikawa, Comments on Galilei conformal field theories and their geometric realisation, preprint, arXiv:0903.5184v2 [hep-th] , 2009]. Moreover, wide classes of nonlinear systems, invariant under two different 10-dimensional subalgebras of ecga are explicitly constructed and an example with possible physical interpretation is presented.

On non-linear partial differential equations with an infinite-dimensional conditional symmetry

Abstract The invariance of non-linear partial differential equations under a certain infinite-dimensional Lie algebra A N ( z ) in N spatial dimensions is studied. The special case A 1 ( 2 ) was introduced in [J. Stat. Phys. 75 (1994) 1023] and contains the Schrodinger Lie algebra sch 1 as a Lie subalgebra. It is shown that there is no second-order equation which is invariant under the massless realizations of A N ( z ) . However, a large class of strongly non-linear partial differential equations is found which are conditionally invariant with respect to the massless realization of A N ( z ) such that the well-known Monge–Ampere equation is the required additional condition. New exact solutions are found for some representatives of this class.

Nonlinear systems of the Burgers-type equations: Lie and Q-conditional symmetries, Ansätze and solutions

Abstract A class of nonlinear diffusion–convection systems containing two Burgers-type equations is considered. A complete description of Lie symmetries is obtained for these systems. New results of finding Q-conditional symmetries are also presented. Moreover, a variety of Lie and non-Lie Ansatze and exact solutions of a diffusion–convection system invariant under the generalized Galilei algebra are constructed.

Exact and numerical solutions of the generalized fisher equation

Abstract Multiparameter families of exact solutions to the generalized Fisher equation (which is a simplification of the known coupled reaction-diffusion system describing spatial segregation of interacting species) have been found using the classical Lie approach and the method of additional generating conditions. The exact solutions are applied next to solving a nonlinear boundary-value problem with the zero Neumann conditions. Analytical results are compared with numerical calculations by using the finite elements method. It is concluded that the obtained exact solutions play an important role for the generalized Fisher equation.

A constructive method for obtaining new exact solutions of nonlinear evolution equations

Abstract New constructive method is suggested for obtaining exact solutions of nonlinear equations. The method is based on the consideration of a fixed nonlinear PDE (system of PDEs) together with additional condition in the form of a linear ODE (system of ODEs). With the help of this method new exact solutions are obtained for a nonlinear generalizations of the Burgers equations and some nonlinear evolution systems, which describe real processes in physics and chemistry.

New conditional symmetries and exact solutions of reaction-diffusion systems with power diffusivities

A wide range of new Q-conditional symmetries for reaction–diffusion systems with power diffusivities are constructed. The relevant non-Lie ansatze to reduce the reaction–diffusion systems to ODE systems and examples of exact solutions are obtained. The relation of the solutions obtained with the development of spatially inhomogeneous structures is discussed.

Lie symmetries of nonlinear boundary value problems

Abstract Nonlinear boundary value problems (BVPs) by means of the classical Lie symmetry method are studied. A new definition of Lie invariance for BVPs is proposed by the generalization of existing those on much wider class of BVPs. A class of two-dimensional nonlinear boundary value problems, modeling the process of melting and evaporation of metals, is studied in details. Using the definition proposed, all possible Lie symmetries and the relevant reductions (with physical meaning) to BVPs for ordinary differential equations are constructed. An example how to construct exact solution of the problem with correctly-specified coefficients is presented and compared with the results of numerical simulations published earlier.