Multimode Solutions of First-Order Elliptic Quasilinear Systems

Abstract

Two new approaches to solving first-order quasilinear elliptic systems of PDEs in many dimensions are proposed. The first method is based on an analysis of multimode solutions expressible in terms of Riemann invariants, based on links between two techniques, that of the symmetry reduction method and of the generalized method of characteristics. A variant of the conditional symmetry method for constructing this type of solution is proposed. A specific feature of that approach is an algebraicgeometric point of view, which allows the introduction of specific firstorder side conditions consistent with the original system of PDEs, leading to a generalization of the Riemann invariant method for solving elliptic homogeneous systems of PDEs. A further generalization of the Riemann invariants method to the case of inhomogeneous systems, based on the introduction of specific rotation matrices, enables us to weaken the integrability condition. It allows us to establish a connection between the structure of the set of integral elements and the possibility of constructing specific classes of simple mode solutions. These theoretical considerations are illustrated by the examples of an ideal plastic flow in its elliptic region and a system describing a nonlinear interaction of waves and particles. Several new classes of solutions are obtained in explicit form, including the general integral for the latter system of equations.

Solutions of First-Order Quasilinear Systems Expressed in Riemann Invariants

“…This approach is applied to hydrodynamic-type equations in their elliptic and hyperbolic regions and has been studied through both the GMC and the CSM. This constitutes the subject of this work, which is a follow up of the paper [11].…”

“…The algebraization that has been performed in [11] for the elliptic system (1) allows us to construct more general classes of solutions, namely the k-multimode solutions (a superposition of k simple mode solutions). We look for real solutions of the form…”

“…The CSM is not limited to hyperbolic systems, but can also be applied to elliptic systems. The links between the two methods is an interesting problem which was addressed in [10,11]. The adaptation of the GMC in the context of the analysis of the symmetry group of first-order quasilinear elliptic systems requires the introduction of complex integral elements instead of simple real integral elements (which are used in the construction of solutions of hyperbolic systems [3,15,24,26]).…”

Nonlinear wave interaction problems in the three-dimensional case

“…Therefore it is straightforward to ascertain that multiplying by dR k dτ the corresponding equation of the set (8), by taking the sum over k = 1, ...N, a further integration allows us to express A N (τ ) in terms of A 1 (τ ) , ....A N −1 (τ ) as follows…”

Application of conservation laws to Dirichlet problem for elliptic quasilinear systems

“…Let us mention the recent article [5], where the so-called symmetry reduction method (using CL) is applied to solve firstorder quasilinear elliptic systems.…”