Two new approaches to solving first-order quasilinear elliptic systems ofPDEs in many dimensions are proposed. The first method is based on an analysisof multimode solutions expressible in terms of Riemann invariants, based onlinks between two techniques, that of the symmetry reduction method and of thegeneralized method of characteristics. A variant of the conditional symmetrymethod for constructing this type of solution is proposed. A specific featureof that approach is an algebraic-geometric point of view, which allows theintroduction of specific first-order side conditions consistent with theoriginal system of PDEs, leading to a generalization of the Riemann invariantmethod for solving elliptic homogeneous systems of PDEs. A furthergeneralization of the Riemann invariants method to the case of inhomogeneoussystems, based on the introduction of specific rotation matrices, enables us toweaken the integrability condition. It allows us to establish a connectionbetween the structure of the set of integral elements and the possibility ofconstructing specific classes of simple mode solutions. These theoreticalconsiderations are illustrated by the examples of an ideal plastic flow in itselliptic region and a system describing a nonlinear interaction of waves andparticles. Several new classes of solutions are obtained in explicit form,including the general integral for the latter system of equations.
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