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We present results of a group analysis of the multidimensional Boltzmann and Vlasov equations. For the Boltzmann equation, we obtain the equivalence group and classifying relations for the symmetry group and study these relations in the case where external forces are absent. We discover a scale paradox: we show that for any collision integral, an equation that is invariant with respect to the shift group does not admit uniform dilations, because the left- and right-hand sides of the equation scale differently. In particular, this holds for the classical Boltzmann equation. For the Vlasov equation, we also obtain the equivalence group and classifying relations for the symmetry group and classify the interparticle interactions for which the Vlasov equation admits groups containing the Galilean group in the case where external forces are absent. [ABSTRACT FROM AUTHOR]

We consider the one-dimensional Boltzmann equation f t + c f x + (F f) c = 0 with a function F depending on (t,x,c,f) and obtain the complete group classification of such equations in the class of point changes of whole set of variables (t,x,c,f). for this, we impose additional conditions on the transformations for the invariance of (a) the relations dx = c dt and d c = F d t , (b) the lines dt = dx = 0, and (c) the form f dx dc, which fix the physical meaning of the used variables and the relations between them. [ABSTRACT FROM AUTHOR]

We study the two-dimensional eikonal equation ψ²x + ψ²y = 1/v²(x, y). We carry out the group analysis of the equation, establish a connection between the group properties and geometric characteristics of the Riemannian space with the metric ds² = [dx² + dy²]/v² (x, y). We select the most important classes of equations and derive some conditions for reducibility of a given equation to an equation of one of those classes. We find a condition for two equations to be equivalent (the theorem of seven invariants). For the equations corresponding to Riemannian spaces of constant curvature, we obtain explicit formulas for the solutions describing the wave front for a point source and also the ray equations. [ABSTRACT FROM AUTHOR]

Studies the relaxation oscillations for a single van der Pol equations and for a pair of van der Pol equations. Focuses on the use of constructive nonstandard analysis in investigating relaxation oscillations; Computation of slow and rapid motions in relaxation oscillations; Description of rapid and slow trajectories in a very complicated problem; Derivation of conditions providing the motions on the stable part of slow manifold.