Equations of hydrodynamic type from the point of view of the theory of webs (Q1181610)

Every solution of a system \(u_ x^ i + v^ i_ j(u)u_ t^ j = 0, i, j = 1, \ldots, n\), of hydrodynamic type carries an \(n\)-web of characteristics \(dt - \lambda^ i dx = 0\) where \(\lambda^ i(u)\) are eigenvalues of the matrix \((v^ i_ j)\). The author considers only systems that can be reduced to the diagonal form \(R_ x^ i + \lambda^ i(R) R_ i^ t = 0\), where the variables \(R^ i\) are the Riemann invariants, and he assumes that all eigenvalues \(\lambda^ i(u)\) are real and distinct. For such diagonalizable systems the characteristics are level curves of the Riemann invariants. The author earlier described the systems of three equations whose three-web of characteristics is hexagonal [see the author, Funkt. Anal. Appl. 23, No. 2, 151-153 (1989), hausl. from Funkts. Anal. Prilozh. 23, No. 2, 79-80 (1989; Zbl 0714.35070)] and proved that all such systems are weakly nonlinear and by means of an appropriate change of variables can be reduced to constant eigenvalues. This result was generalized for \(n > 3\) in [the author, Mat. Sb. 181, No. 9, 1220-1235 (1990)] where it was proved that on any solution of a weakly nonlinear semi-Hamiltonian system its \(n\)-web of characteristics has nontrivial rank which is equal to \(n - 2\). If \(n = 3\), three-webs of rank one are hexagonal three-webs. However, most of the known systems of hydrodynamic type are not weakly nonlinear. In the paper under review the author describes all solutions with a hexagonal 3-web of characteristics for a system of hydrodynamic type arising in gas chromatography. For such a system, he proves that all solutions with a hexagonal three-web of characteristics form a finite- dimensional manifold and can be parametrized by algebraic cubic curves. This result is generalized for \(n >3\): all solutions with an \(n\)-web of characteristics of maximal rank are described.