NEW EXACT SOLUTIONS OF SPATIALLY AND TEMPORALLY VARYING REACTION-DIFFUSION EQUATIONS

Abstract. This paper considers reaction diffusion equations from a new point of view, by including spatiotemporal dependence in the source terms. We show for the first time that solutions are given in terms of the classical Painleve ́ tran-scendents. We consider reaction diffusion equations with cubic and quadratic source terms. A new feature of our analysis is that the coefficient functions are also solutions of differential equations, including the Painleve ́ equations. Special cases arise with elliptic functions as solutions. Additional solutions given in terms of equations that do not have the Painleve ́ property are also considered. Solutions are constructed using a Lie symmetry approach. 1