Christodoulos Sophocleous

On form-preserving point transformations of partial differential equations

New identities are presented relating arbitrary order partial derivatives of and for the general point transformation , . These identities are used to study the nature of those point transformations which preserve the general form of a wide class of 1 + 1 partial differential equations. These results facilitate the search for point symmetries, both discrete and continuous (Lie), and assist the search for point transformations which reduce equations to canonical, but similar, form. A simple test for the existence of hodograph-type transformations between equations of similar form is given.

Enhanced group analysis and conservation laws of variable coefficient reaction–diffusion equations with power nonlinearities

A class of variable coefficient (1+1)-dimensional nonlinear reaction–diffusion equations of the general form f(x)ut=(g(x)unux)x+h(x)um is investigated. Different kinds of equivalence groups are constructed including ones with transformations which are nonlocal with respect to arbitrary elements. For the class under consideration the complete group classification is performed with respect to convenient equivalence groups (generalized extended and conditional ones) and with respect to the set of all local transformations. Usage of different equivalences and coefficient gauges plays the major role for simple and clear formulation of the final results. The corresponding set of admissible transformations is described exhaustively. Then, using the most direct method, we classify local conservation laws. Some exact solutions are constructed by the classical Lie method.

Group analysis of variable coefficient diffusion-convection equations. I. Enhanced group classification

We discuss the classical statement of group classification problem and some its extensions in the general case. After that, we carry out the complete extended group classification for a class of (1 + 1)-dimensional nonlinear diffusion-convection equations with coefficients depending on the space variable. At first, we construct the usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements. The extended equivalence group has interesting structure since it contains a non-trivial subgroup of non-local gauge equivalence transformations. The complete group classification of the class under consideration is carried out with respect to the extended equivalence group and with respect to the set of all point transformations. Usage of extended equivalence and correct choice of gauges of arbitrary elements play the major role for simple and clear formulation of the final results. The set of admissible transformations of this class is preliminary investigated.

Potential symmetries of nonlinear diffusion-convection equations

In this paper potential symmetries are sought for the nonlinear diffusion - convection equations . The functional forms of f(u) and k(u) that admit such symmetries are completely classified.

Extended group analysis of variable coefficient reaction–diffusion equations with exponential nonlinearities

Abstract The group classification of a class of variable coefficient reaction–diffusion equations with exponential nonlinearities is carried out up to both the equivalence generated by the corresponding generalized equivalence group and the general point equivalence. The set of admissible transformations of this class is exhaustively described via finding the complete family of maximal normalized subclasses and associated conditional equivalence groups. Limit processes between variable coefficient reaction–diffusion equations with power nonlinearities and those with exponential nonlinearities are simultaneously studied with limit processes between objects related to these equations (including Lie symmetries, exact solutions and conservation laws).

Equivalence transformations in the study of integrability

We discuss how point transformations can be used for the study of integrability, in particular, for deriving classes of integrable variable-coefficient differential equations. The procedure of finding the equivalence groupoid of a class of differential equations is described and then specified for the case of evolution equations. A class of fifth-order variable-coefficient Korteweg–de Vries-like equations is studied within the framework suggested.

Classification of potential symmetries of generalised inhomogeneous nonlinear diffusion equations

We consider the class of generalised nonlinear diffusion equations f(x)ut=[g(x)unux]x which are of considerable interest in mathematical physics. We classify the nonlocal symmetries, which are known as potential symmetries, for these equations. It turns out that potential symmetries exist only if the parameter n takes the values −2 or −23. Also certain relations must be satisfied by the functions f(x) and g(x). For the cases where we obtain infinite-parameter potential symmetries, linearising mappings are constructed. Furthermore we employ the potential symmetries to derive similarity solutions.

Symmetries of Hamiltonian systems with two degrees of freedom

We classify the Lie point symmetry groups for an autonomous Hamiltonian system with two degrees of freedom. With the exception of the harmonic oscillator or a free particle where the dimension is 15, we obtain all dimensions between 1 and 7. For each system in the classification we examine integrability.

Group analysis of nonlinear fin equations

Group classification of a class of nonlinear fin equations is carried out exhaustively. Additional equivalence transformations and conditional equivalence groups are also found. These enable us to simplify the classification results and their further applications. The derived Lie symmetries are used to construct exact solutions of truly nonlinear equations for the class under consideration. Nonclassical symmetries of the fin equations are discussed. Adduced results complete and essentially generalize recent works on the subject [M. Pakdemirli and A.Z. Sahin, Similarity analysis of a nonlinear fin equation, Appl. Math. Lett. 19 (2006) 378–384; A.H. Bokhari, A.H. Kara and F.D. Zaman, A note on a symmetry analysis and exact solutions of a nonlinear fin equation, Appl. Math. Lett. 19 (2006) 1356–1360].

Symmetries and form-preserving transformations of generalised inhomogeneous nonlinear diffusion equations

We consider the variable coefficient inhomogeneous nonlinear diffusion equations of the form f(x)ut=[g(x)unux]x. We present a complete classification of Lie symmetries and form-preserving point transformations in the case where f(x)=1 which is equivalent to the original equation. We also introduce certain nonlocal transformations. When f(x)=xp and g(x)=xq we have the most known form of this class of equations. If certain conditions are satisfied, then this latter equation can be transformed into a constant coefficient equation. It is also proved that the only equations from this class of partial differential equations that admit Lie–Backlund symmetries is the well-known nonlinear equation ut=[u−2ux]x and an equivalent equation. Finally, two examples of new exact solutions are given.