Simple waves in quasilinear hyperbolic systems. II. Riemann invariants for the problem of simple wave interactions

Abstract

In this paper a generalization of the Riemann invariant method to the case of a nonhomogeneous system of equations has been formulated. We have discussed in detail the necessary and sufficient conditions for the existence of Riemann invariants. We perform the analysis using the apparatus of differential forms and Cartan theory of systems in involution. The problem has been reduced to examining Pfaff forms. We have considered the connections between the structure of the set of integral elements and the possibility of a construction of special classes of solutions depending on k arbitrary functions of one variable. These solutions can be interpreted physically as the interactions between k simple waves on a simple state. We have proven that, in the case of interaction of many simple waves described by Riemann invariants, a conservation law for the type and quantity of waves holds. It has been also shown that such a solution, resulting from the interaction of many simple waves propagating on the simple state, decay for a large time in an exact way into simple waves (of the same kind as those entering the interaction) on the state. The Cauchy problem for the nonlinear superposition of k-sample waves has been formulated. A couple of theorems useful for this problem have been given in the Sec. III. The functorial properties of the system of equations determining Riemann invariants have been described. The last part of the work contains an analysis of some examples of the solutions of nonhomogeneous magnetohydrodynamic equations from the point of view of the method described above.

The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type

“…Solutions of this type, known as nonlinear interactions of n planar simple waves, were discussed in a series of publications [2, 3,26,15]. Later, they were investigated by Gibbons and Tsarev in the context of the dispersionless KP hierarchy [10,11,12,13], see also [22], and the theory of Egorov's integrable hydrodynamic chains [24,25].…”

On the Integrability of (2+1)-Dimensional Quasilinear Systems

“…Thus, the original 2+1-dimensional system (1) is decoupled into a pair of diagonal 1+1-dimensional systems. Solutions of this type, known as nonlinear interactions of n planar simple waves, were extensively investigated in gas dynamics and magnetohydrodynamics in a series of publications [5,6,7,36,37,9,20]. Later, they appeared in the context of the dispersionless KP hierarchy [15,16,17,18,21,31,29,30] and the theory of integrable hydrodynamictype chains [34,35].…”

The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice

“…At last, equations (5.13) and (5.15) imply the following relation between the circularity property and its dual: 25) which implies that also equation (5.16) is satisfied. The proof of: iii) ⇒ i) is similar and is left to the reader.…”

“…We finally remark that the equations characterizing the above lattices are potentially relevant also in physics, being integrable discretizations of equations arising in hydrodynamics [19,25,22,39] and in quantum field theory [20,18,9]. …”