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On Peller's characterization of trace class Hankel operators and smoothness of KdV solutions

Author: Alexei Rybkin
Journal: Proc. Amer. Math. Soc. 146 (2018), 1627-1637
MSC (2010): Primary 34B20, 37K15, 47B35
Published electronically: November 7, 2017
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Abstract: In the context of the Cauchy problem for the Korteweg-de Vries equation we put forward a new effective method to link smoothness of the solution with the rate of decay of the initial data. Our approach is based on the Peller characterization of trace class Hankel operators.

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  • [1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 (1993), no. 3, 209-262. MR 1215780,
  • [2] Amy Cohen and Thomas Kappeler, Solutions to the Korteweg-de Vries equation with initial profile in $ L^1_1({\bf R})\cap L^1_N({\bf R}^+)$, SIAM J. Math. Anal. 18 (1987), no. 4, 991-1025. MR 892486,
  • [3] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Sharp global well-posedness for KdV and modified KdV on $ \mathbb{R}$ and $ \mathbb{T}$, J. Amer. Math. Soc. 16 (2003), no. 3, 705-749. MR 1969209,
  • [4] P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), no. 2, 121-251. MR 512420,
  • [5] E. M. Dynkin, Pseudoanalytic continuation of smooth functions. Uniform scale, Mathematical programming and related questions (Proc. Seventh Winter School, Drogobych, 1974) Central Èkonom.-Mat. Inst. Akad. Nauk SSSR, Moscow, 1976, pp. 40-73 (Russian). MR 0587795
  • [6] M. S. P. Eastham, Semi-bounded second-order differential operators, Proc. Roy. Soc. Edinburgh Sect. A 72 (1974), no. 1, 9-16 (1974). MR 0417485
  • [7] Zihua Guo, Global well-posedness of Korteweg-de Vries equation in $ H^{-3/4}(\mathbb{R})$, J. Math. Pures Appl. (9) 91 (2009), no. 6, 583-597 (English, with English and French summaries). MR 2531556,
  • [8] John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. MR 2261424
  • [9] J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation, SIAM J. Math. Anal. 20 (1989), no. 6, 1388-1425. MR 1019307,
  • [10] Sergei Grudsky, Christian Remling, and Alexei Rybkin, The inverse scattering transform for the KdV equation with step-like singular Miura initial profiles, J. Math. Phys. 56 (2015), no. 9, 091505, 14. MR 3395045,
  • [11] Sergei Grudsky and Alexei Rybkin, Soliton theory and Hankel operators, SIAM J. Math. Anal. 47 (2015), no. 3, 2283-2323. MR 3356985,
  • [12] Thomas Kappeler, Peter Perry, Mikhail Shubin, and Peter Topalov, The Miura map on the line, Int. Math. Res. Not. 50 (2005), 3091-3133. MR 2189502,
  • [13] Carlos E. Kenig, Gustavo Ponce, and Luis Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), no. 2, 573-603. MR 1329387,
  • [14] Carlos E. Kenig, Gustavo Ponce, and Luis Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001), no. 3, 617-633. MR 1813239,
  • [15] Nobu Kishimoto, Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $ \overline u{}^2$, Commun. Pure Appl. Anal. 7 (2008), no. 5, 1123-1143. MR 2410871,
  • [16] S. N. Kruzhkov and A. V. Faminskiĭ, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, Mat. Sb. (N.S.) 120(162) (1983), no. 3, 396-425 (Russian). MR 691986
  • [17] Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz; Translated from the French by Andreas Hartmann. MR 1864396
  • [18] Vladimir V. Peller, Hankel operators and their applications, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. MR 1949210
  • [19] Alexei Rybkin, Spatial analyticity of solutions to integrable systems. I. The KdV case, Comm. Partial Differential Equations 38 (2013), no. 5, 802-822. MR 3046294,
  • [20] Alexei Rybkin, The Hirota $ \tau$-function and well-posedness of the KdV equation with an arbitrary step-like initial profile decaying on the right half line, Nonlinearity 24 (2011), no. 10, 2953-2990. MR 2842104,
  • [21] Alexei Rybkin, Meromorphic solutions to the KdV equation with non-decaying initial data supported on a left half line, Nonlinearity 23 (2010), no. 5, 1143-1167. MR 2630095,
  • [22] Alexei Rybkin, Some new and old asymptotic representations of the Jost solution and the Weyl $ m$-function for Schrödinger operators on the line, Bull. London Math. Soc. 34 (2002), no. 1, 61-72. MR 1866429,

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Additional Information

Alexei Rybkin
Affiliation: Department of Mathematics and Statistics, University of Alaska Fairbanks, P.O. Box 756660, Fairbanks, Alaska 99775

Keywords: KdV equation, Hankel operators.
Received by editor(s): April 21, 2017
Received by editor(s) in revised form: May 21, 2017
Published electronically: November 7, 2017
Additional Notes: The author was supported in part by the NSF grant DMS-1411560
Dedicated: This paper is dedicated to the memory of Ludwig Faddeev, one of the founders of soliton theory
Communicated by: Joachim Krieger
Article copyright: © Copyright 2017 American Mathematical Society

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