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New $Q$-conditional symmetries and exact solutions of some reaction-diffusion-convection equations arising in mathematical biology. (English) Zbl 1160.35031

Nonlinear reaction-diffusion-convection equations arise, with various constraints, in mathematical biology, chemical dynamics, physiology and fluid dynamics (to mention but a few applications). This class of equations is not amenable to solution by means of the inverse scattering method. Consequently the use of the Lie theory of extended groups is the natural mode of attack. Unfortunately a number of the important equations of this class, such as the Fisher and Fitzhugh-Nagumo equations, are poorly endowed with Lie point symmetries. To overcome this deficiency a number of stratagems has been proposed to increase the number of symmetries available. Thus we have nonclassical symmetries, conditional symmetries and generalised conditional symmetries together with various methods such as that of heir-equations. In this paper the author confines his attention to a particular class of the general reaction-diffusion-convection equation with the form ${u}_{t}={u}_{xx}+\lambda u{u}_{x}+C\left(u\right)$, where $\lambda$ is a real parameter. This equation admits more than two Lie point symmetries for nontrivial $\lambda$ for only three forms of the function $C\left(u\right)$ up to equivalence classes. The author uses the method of $Q$-conditional symmetries introduced by W. I. Fushchich, W. M. Shtelen and N. I. Serov [Symmetry analysis and exact solutions of equations of nonlinear mathematical physics, Dordrecht: Kluwer Academic Publishers (1993; Zbl 0838.58043)]. The advantage of this method is that there is a constructive algorithm to determine the symmetries. This is not to claim that the process of their determination is easy, but at least one knows where one fails. The author shows the cases for which the equation above admits $Q$-conditional symmetries.

The second half of the paper is concerned with the determination of closed-form solutions for some reaction-diffusion-convection equations arising in applications which belong to the class of equations admitting $Q$-conditional symmetries. The discussion of the methods of solution of the resulting ordinary differential equations is quite interesting and could doubtless lead to further investigations by interested parties.

This paper once again raises the question of what lies behind the ability to present a closed-form solution of a differential equation. The genius of Lie’s approach was to unify the different known methods of solution of differential equations under the guiding-light of symmetry. Lie commenced from a geometric viewpoint and it was natural for the infinitesimal transformations at the basis of his theory of symmetry to have a specific geometric interpretation. The demands for methods of solution of equations which did not fit into this geometrical mould led to the abandonment of geometry with the introduction of generalised and nonlocal symmetries. Particularly in the case of partial differential equations these generalisations appear not to have been sufficient for the needs of practitioners and several extensions of Lie’s method have been introduced. Even so we see that there are famous equations, such as the Painlevé Six, which remain impervious to the symmetric approach. If one believes that symmetry, in an obviously so-far vaguely defined manifestation, is at the basis of integrability, it would seem that a closer and deeper examination of the basis of symmetry is required. When Lie began his work, there were many disparate methods for the resolution of differential equations. Lie united them. In our present epoch there are many types of symmetry. Who will unite them?

##### MSC:
 35K57 Reaction-diffusion equations 58J70 Invariance and symmetry properties