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 Zap. Nauchn. Sem. POMI, 2018, Volume 473, Pages 205–227 (Mi znsl6663)

Two-phase periodic solutions to the AKNS hierarchy equations

V. B. Matveevabc, A. O. Smirnovcba

a St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia
b Institut de Mathématiques de Bourgogne (IMB), Université de Bourgogne - Franche Comté, BP 47870, 21078, Dijon, France
c Saint-Petersburg State University of Aerospace Instrumentation, Saint-Petersburg, Russia

Abstract: In this paper we investigate the genus $2$ algebro-geometric solutions of the AKNS hierarchy equations, strictly periodic with respect to the space variable $x$. In general position these solutions, expressed by means of two-dimensional Riemann theta functions are not strictly periodic in $x$. We show that $x$ periodic solutions can be obtained by appropriate choice of the hyperelliptic spectral curves, having a structure of covering over elliptic curve. For odd number members of AKNS hierarchy these solutions might be made periodic also with respect to the corresponding time variables of the AKNS hierarchy, imposing further restrictions on the structure of the spectral curve pointed out in the paper. The related solutions are especially interesting from the point of view of potential applications to study the signals propagation in nonlinear optical fibers.

Key words and phrases: AKNS hierarchy, Nonlinear Schrödinger equation, Hirota equation, theta-functions, spectral curves.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00518_à

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English version:
Journal of Mathematical Sciences (New York), 2019, 242:5, 722–741

UDC: 517.9

Citation: V. B. Matveev, A. O. Smirnov, “Two-phase periodic solutions to the AKNS hierarchy equations”, Questions of quantum field theory and statistical physics. Part 25, Zap. Nauchn. Sem. POMI, 473, POMI, St. Petersburg, 2018, 205–227; J. Math. Sci. (N. Y.), 242:5 (2019), 722–741

Citation in format AMSBIB
\Bibitem{MatSmi18} \by V.~B.~Matveev, A.~O.~Smirnov \paper Two-phase periodic solutions to the AKNS hierarchy equations \inbook Questions of quantum field theory and statistical physics. Part~25 \serial Zap. Nauchn. Sem. POMI \yr 2018 \vol 473 \pages 205--227 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6663} \transl \jour J. Math. Sci. (N. Y.) \yr 2019 \vol 242 \issue 5 \pages 722--741 \crossref{https://doi.org/10.1007/s10958-019-04510-8} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85074207698}