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Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Finite gap conditions and small dispersion asymptotics for the classical periodic Benjamin–Ono equation

Author: Alexander Moll
Journal: Quart. Appl. Math. 78 (2020), 671-702
MSC (2010): Primary 37K10, 37K40, 47B35
Published electronically: March 4, 2020
MathSciNet review: 4148823
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Abstract: In this paper we characterize the Nazarov–Sklyanin hierarchy for the classical periodic Benjamin–Ono equation in two complementary degenerations: for the multiphase initial data (the periodic multisolitons) at fixed dispersion and for bounded initial data in the limit of small dispersion. First, we express this hierarchy in terms of a piecewise-linear function of an auxiliary real variable which we call a dispersive action profile and whose regions of slope $\pm 1$ we call gaps and bands, respectively. Our expression uses Kerov’s theory of profiles and Kreĭn’s spectral shift functions. Next, for multiphase initial data, we identify Baker–Akhiezer functions in Dobrokhotov–Krichever and Nazarov–Sklyanin and prove that multiphase dispersive action profiles have finitely many gaps determined by the singularities of their Dobrokhotov–Krichever spectral curves. Finally, for bounded initial data independent of the coefficient of dispersion, we show that in the small dispersion limit, the dispersive action profile concentrates weakly on a convex profile which encodes the conserved quantities of the dispersionless equation. To establish the weak limit, we reformulate Szegő’s first theorem for Toeplitz operators using spectral shift functions. To illustrate our results, we identify the dispersive action profile of sinusoidal initial data with a profile found by Nekrasov–Pestun–Shatashvili and its small dispersion limit with the convex profile found by Vershik–Kerov and Logan–Shepp.

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Additional Information

Alexander Moll
Affiliation: Department of Mathematics, Northeastern University, 550 Nightingale, Boston, Massachussetts 02115

Received by editor(s): November 15, 2019
Received by editor(s) in revised form: December 3, 2019
Published electronically: March 4, 2020
Additional Notes: This work was supported by the Andrei Zelevinsky Research Instructorship at Northeastern University and by the National Science Foundation RTG in Algebraic Geometry and Representation Theory under grant DMS-1645877.
Article copyright: © Copyright 2020 Brown University