A nonlinear second order field equation – similarity solutions and relation to a Bellmann-type equation - Applications to Maxwellian Molecules

In this paper Lie’s formalism is applied to deduce classes of solutions of a nonlinear partial differential equation (nPDE) of second order with quadratic nonlinearity. The equation has the meaning of a field equation appearing in the formulation of kinetic models. Similarity solutions and transformations are given in a most general form derived to the first time in terms of reciprocal Jacobian elliptic functions. By using a special transformation the first derivative of the equation can be transformed off leading to a further nPDE. The latter equation is also studied as well as algebraic properties and group invariant solutions could be derived. This new classes of solutions obtained are closely related to solutions of the kinetic model and so far, expressions for a generating function considering normalized moments are also deduced. Finally, the connection to Painlevé’s first equation is shown whereby these classes of solutions are solutions due to the invariant properties too. For practical use in numerical calculations some series representations are given explicitly. In view of the point of novelty it is further shown how to derive a Bellman-type equation to the first time and asymptotic classes of solutions result by appropriate transformations. The importance of the present paper is the relation to the Boltzmann Equation which describes the one particle distribution function in a gas of particles interacting only through binary collisions. Since transformations remain an equation invariant, solutions of the new transformed equation also generates solutions of physical relevance. Normalized moments are discussed finally.