Integration of weekly nonlinear semi-hamiltonian systems of hydrodynamic type by methods of the theory of~webs

E. V. Ferapontov

Sbornik. Mathematics, Tome 71 (1992) no. 1, pp. 65-79

Résumé

Weakly nonlinear semi-Hamiltonian systems of $n$ differential equations of hydrodynamic type in Riemann invariants are considered, and the geometry of the $(n+2)$-web formed by the characteristics and the level lines of the independent variables are studied. It is shown that the rank of this web on the general solution of the system is equal to $n$. This result is used to obtain formulas for the general integral of the systems under consideration, with the necessary arbitrariness in $n$ functions of a single argument. Separate consideration is given to the cases $n=3$ and $n=4$, for which it is possible not only to integrate the corresponding systems, but also to give a complete classification of them to within so-called transformations via a solution (reciprocal transformations). It turns out that for $n=3$ they can all be linearized (and are thus equivalent), while for $n=4$ there exist exactly five mutually nonequivalent systems, and any other system can be reduced to one of them by a transformation via a solution. There is a discussion of the connection between weakly nonlinear semi-Hamiltonian systems and Dupin cyclides-hypersurfaces of Euclidean space whose principal curvatures are constant along the corresponding principal directions. Some unsolved problems are formulated at the end of the paper.