Finite Gap Conditions and Small Dispersion Asymptotics for the Classical Periodic BenjaminOno Equation
Abstract
In this paper we characterize the NazarovSklyanin hierarchy for the classical periodic BenjaminOno 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 piecewiselinear 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 BakerAkhiezer functions in DobrokhotovKrichever and NazarovSklyanin and prove that multiphase dispersive action profiles have finitelymany gaps determined by the singularities of their DobrokhotovKrichever 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 NekrasovPestunShatashvili and its small dispersion limit with the convex profile found by VershikKerov and LoganShepp.
 Publication:

arXiv eprints
 Pub Date:
 January 2019
 arXiv:
 arXiv:1901.04089
 Bibcode:
 2019arXiv190104089M
 Keywords:

 Mathematical Physics;
 Mathematics  Analysis of PDEs;
 Mathematics  Spectral Theory;
 Nonlinear Sciences  Pattern Formation and Solitons;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 This is a revised version of the paper submitted on 13 January 2019 to appear in the Quarterly of Applied Mathematics