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 partialdifferential equation (npde) of second order with quadratic nonlinearity. the equation has themeaning of a field equation appearing in the formulation of kinetic models. similaritysolutions and transformations are given in a most general form derived to the first time interms of reciprocal jacobian elliptic functions.by using a special transformation the first derivative of the equation can be transformed offleading to a further npde. the latter equation is also studied as well as algebraic propertiesand group invariant solutions could be derived. this new classes of solutions obtained areclosely related to solutions of the kinetic model and so far, expressions for a generatingfunction considering normalized moments are also deduced.finally, the connection to painlevé’s first equation is shown whereby these classes ofsolutions are solutions due to the invariant properties too. for practical use in numericalcalculations some series representations are given explicitly. in view of the point of novelty itis further shown how to derive a bellman-type equation to the first time and asymptoticclasses of solutions result by appropriate transformations. the importance of the present paperis the relation to the boltzmann equation which describes the one particle distribution functionin a gas of particles interacting only through binary collisions. since transformations remainan equation invariant, solutions of the new transformed equation also generates solutions ofphysical relevance. normalized moments are discussed finally.