Introduction

Abstract

Nonlinear partial differential equations (PDEs) do not, in general, admit exact solutions; these solutions are even more rare when initial/boundary conditions are imposed. There are exceptional circumstances when the PDEs enjoy certain symmetries: they are invariant to a class of finite or infinitesimal transformations (Sachdev (2000). When this is the case, the PDEs are exactly reducible to ordinary differential equations (ODEs) if they are functions of two independent variables; the ODEs may occasionally be integrated in a closed form. Alternatively, one may study their qualitative properties and obtain the actual solutions numerically with reference to appropriate initial/boundary conditions. These solutions are called and belong to one of the two classes, first kind and second kind (Zel’dovich 1956, Zel’dovich and Raizer 1967, Barenblatt and Zel’dovich 1972, Sachdev 2000), and solve some degenerate problems for which ‘all, or at least some, of the constant parameters in the initial and boundary conditions of the problem, having the dimensionality of independent variables, tend to zero or infinity.’ These solutions describe those properties of the phenomena that do not depend on the details of the initial and boundary conditions; they do involve some nondimensional parameters which, in some integral sense, represent the memory of initial/boundary conditions. Exceptionally, there may not be any nondimensional parameter of the problem in the asymptotic solution (Barenblatt and Zel’dovich 1972). These special solutions do not describe equilibrium states; they describe intermediate stages when the process of evolution of the solution is continuing and yet the details of initial/boundary conditions have already disappeared. These solutions satisfy some singular, delta functions like, initial conditions.

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