Geodesic
Quasiperiodic solutions of the~negative-order Korteweg--de~Vries hierarchy
Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 3, pp. 372-398

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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.
Keywords: nKdV hierarchy, backward Neumann system
Mots-clés : quasiperiodic solution. 
@article{TMF_2019_199_3_a2,
     author = {Jinbing Chen},
     title = {Quasiperiodic solutions of the~negative-order {Korteweg--de~Vries} hierarchy},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {372--398},
     publisher = {mathdoc},
     volume = {199},
     number = {3},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2019_199_3_a2/}
}
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Jinbing Chen. Quasiperiodic solutions of the~negative-order Korteweg--de~Vries hierarchy. Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 3, pp. 372-398. http://geodesic.mathdoc.fr/item/TMF_2019_199_3_a2/