Conditional Symmetries and Riemann Invariants for Hyperbolic Systems of PDEs

This paper contains an analysis of rank-k solutions in terms of Riemann invariants, obtained from interrelations between two concepts, that of the symmetry reduction method and of the generalized method of characteristics for first order quasilinear hyperbolic systems of PDEs in many dimensions. A variant of the conditional symmetry method for obtaining this type of solutions is proposed. A Lie module of vector fields, which are symmetries of an overdetermined system defined by the initial system of equations and certain first order differential constraints, is constructed. It is shown that this overdetermined system admits rank-k solutions expressible in terms of Riemann invariants. Finally, examples of applications of the proposed approach to the fluid dynamics equations in (k+1) dimensions are discussed in detail. Several new soliton-like solutions (among them kinks, bumps and multiple wave solutions) have been obtained

Elliptic solutions of isentropic ideal compressible fluid flow in (3 + 1) dimensions

A modified version of the conditional symmetry method, together with the classical method, is used to obtain new classes of elliptic solutions of the isentropic ideal compressible fluid flow in (3+1) dimensions. We focus on those types of solutions which are expressed in terms of the Weierstrass P-functions of Riemann invariants. These solutions are of special interest since we show that they remain bounded even when these invariants admit the gradient catastrophe. We describe in detail a procedure for constructing such classes of solutions. Finally, we present several examples of an application of our approach which includes bumps, kinks and multi-wave solutions

R\'eductions d'un syst\`eme bidimensionnel de sine-Gordon \`a la sixi\`eme \'equation de Painlev\'e

We derive all the reductions of the system of two coupled sine-Gordon equations introduced by Konopelchenko and Rogers to ordinary differential equations. All these reductions are degeneracies of a master reduction to an equation found by Chazy "curious for its elegance", an algebraic transform of the most general sixth equation of Painlev\'e. -- -- Nous \'etablissons toutes les r\'eductions du syst\`eme de deux \'equations coupl\'ees de sine-Gordon introduit par Konopelchenko et Rogers \`a des \'equations diff\'erentielles ordinaires. Ces r\'eductions sont toutes des d\'eg\'en\'erescences d'une r\'eduction ma{\^\i}tresse \`a une \'equation jug\'ee par Chazy "curieuse en raison de [son] \'el\'egance", transform\'ee alg\'ebrique de la sixi\`eme \'equation de Painlev\'e la plus g\'en\'erale.Comment: 17 pages, no figure, in French. To appear, Bulletin des sciences math\'ematique

A reduction of the resonant three-wave interaction to the generic sixth Painleve' equation

Among the reductions of the resonant three-wave interaction system to six-dimensional differential systems, one of them has been specifically mentioned as being linked to the generic sixth Painleve' equation P6. We derive this link explicitly, and we establish the connection to a three-degree of freedom Hamiltonian previously considered for P6.Comment: 13 pages, 0 figure, J. Phys. A Special issue "One hundred years of Painleve' VI