Quasiperiodic solutions of the negative-order Korteweg–de Vries hierarchy

Jinbing Chen

Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 3, pp. 372-398

Résumé

We develop a complete algorithm for deriving quasiperiodic solutions of the negative-order KdV (nKdV) hierarchy using the backward Neumann systems. Starting with the nonlinearization of a Lax pair, the nKdV hierarchy reduces to a family of backward Neumann systems via separating temporal and spatial variables. We show that the backward Neumann systems are integrable in the Liouville sense and their involutive solutions yield finite-parameter solutions of the nKdV hierarchy. We present the negative-order Novikov equation, which specifies a finite-dimensional invariant subspace of nKdV flows. Using the Abel–Jacobi variable, we integrate the nKdV flows with Abel–Jacobi solutions on the Jacobian variety of a Riemann surface. Finally, we study the Riemann–Jacobi inversion of the Abel–Jacobi solutions, whence we obtain some quasiperiodic solutions of the nKdV hierarchy.

Détail

Keywords: nKdV hierarchy, backward Neumann system
Mots-clés : quasiperiodic solution.

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