Solutions de rang k et invariants de Riemann pour les systèmes de type hydrodynamique multidimensionnels
In this work, the conditional symmetry method is adapted in order to construct solutions expressed in terms of Riemann invariants.  Nonelliptic quasilinear homogeneous systems of multidimensional partial differential equations of hydrodynamic type are considered.  A detailed description of the necessary and sufficient conditions for the local existence of these types of solutions is given.  The relationship between the structure of integral elements and the possibility of constructing certain classes of rank-k solutions is discussed.   These classes of solutions include nonlinear superpositions of Riemann waves and multisolitonic solutions.  This approach is generalized to first-order inhomogeneous hyperbolic quasilinear systems.  These methods are applied to the equations describing an isentropic fluid flow in (3+1) dimensions.  Several new classes of rank-2 and rank-3 solutions are obtained which contain double and triple solitonic solutions.  New nonlinear and linear superpositions of Riemann waves are described.  Finally, certain aspects of the construction of rank-2 solutions through an application to the dispersionless Kadomtsev-Petviashvili equation are discussed.
Riemann invariants Hydrodynamic type systems Symmetry methods Conditional symmetries Rank-k solutions