Estimates for solutions of KDV on the phase space of periodic distributions in terms of action variables

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April 2011, 30(1): 219-225. doi: 10.3934/dcds.2011.30.219

Estimates for solutions of KDV on the phase space of periodic distributions in terms of action variables

1.	St. Petersburg State Univ., Russian Federation

Received December 2009 Revised June 2010 Published February 2011

Abstract
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We consider the KdV equation on the Sobolev space of periodic distributions. We obtain estimates of the solution of the KdV in terms of action variables.

Keywords: Periodic KDV, action variables, estimates..

Mathematics Subject Classification: Primary: 37K05; Secondary: 35Q53, 37K1.

Citation: Evgeny L. Korotyaev. Estimates for solutions of KDV on the phase space of periodic distributions in terms of action variables. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 219-225. doi: 10.3934/dcds.2011.30.219

References:

[1]	J. Bourgain, Periodic Korteweg - de Vries equation with measures as initial data,, Selecta Math., 3 (1997), 115. doi: 10.1007/s000290050008.
[2]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on R and T,, J. Amer. Math. Soc., 16 (2003), 705. doi: 10.1090/S0894-0347-03-00421-1.
[3]	H. Flaschka and D. McLaughlin, Canonically conjugate variables for the Korteveg- de Vries equation and the Toda lattice with periodic boundary conditions,, Prog. of Theor. Phys., 55 (1976), 438. doi: 10.1143/PTP.55.438.
[4]	J. Garnett and E. Trubowitz, Gaps and bands of one dimensional periodic Schrödinger operators,, Comment. Math. Helv., 59 (1984), 258. doi: 10.1007/BF02566350.
[5]	A. Jenkins, "Univalent Functions and Conformal Mapping,", Berlin, (1958).
[6]	T. Kappeler and P. Topalov, Global wellposedness of KdV in $H^-1(\T,\R)$,, Duke Math. J., 135 (2006), 327. doi: 10.1215/S0012-7094-06-13524-X.
[7]	T.Kappeler and J. Pöschel, "Kdv & Kam,", Springer, (2003).
[8]	T. Kappeler, C. Möhr and P. Topalov, Birkhoff coordinates for KdV on phase spaces of distributions,, Selecta Math. (N.S.), 11 (2005), 37.
[9]	P. Kargaev and E. Korotyaev, The inverse problem for the Hill operator, a direct method,, Invent. Math., 129 (1997), 567. doi: 10.1007/s002220050173.
[10]	E. Korotyaev, Estimates for the Hill operator. I,, Journal Diff. Eq., 162 (2000), 1. doi: 10.1006/jdeq.1999.3684.
[11]	E. Korotyaev, Estimate for the Hill operator. II,, Journal Diff. Eq., 223 (2006), 229. doi: 10.1016/j.jde.2005.04.017.
[12]	E. Korotyaev, Characterization of the spectrum of Schrödinger operators with periodic distributions,, Int. Math. Res. Not., 2003 (2003), 2019. doi: 10.1155/S1073792803209107.
[13]	E. Korotyaev, The estimates of periodic potentials in terms of effective masses,, Commun. Math. Phys., 183 (1997), 383. doi: 10.1007/BF02506412.
[14]	E. Korotyaev, Periodic "weighted" operators,, J. Differential Equations, 189 (2003), 461. doi: 10.1016/S0022-0396(02)00154-7.
[15]	E. Korotyaev, Estimates of KdV Hamiltomian in terms of actions,, preprint, (2009).
[16]	S. Kuksin, "Analysis of Hamiltonian PDEs,", Oxford Lecture Series in Mathematics and its Applications, 19 (2000).
[17]	H. McKean and E. Trubowitz, Hill's surfaces and their theta functions,, Bull. Am. Math. Soc., 84 (1978), 1042. doi: 10.1090/S0002-9904-1978-14542-X.
[18]	V. Marchenko and I. Ostrovski, A characterization of the spectrum of the Hill operator,, Math. USSR Sb., 26 (1975), 493. doi: 10.1070/SM1975v026n04ABEH002493.
[19]	V. Marchenko and I. Ostrovski, Approximation of periodic by finite-zone potentials,, Selecta Math. Sovietica., 6 (1987), 101.
[20]	A. Veselov and S. Novikov, Poisson brackets and complex tori,, Proc. Steklov Inst. Math., 165 (1985), 53.

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References:

[1]	J. Bourgain, Periodic Korteweg - de Vries equation with measures as initial data,, Selecta Math., 3 (1997), 115. doi: 10.1007/s000290050008.
[2]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on R and T,, J. Amer. Math. Soc., 16 (2003), 705. doi: 10.1090/S0894-0347-03-00421-1.
[3]	H. Flaschka and D. McLaughlin, Canonically conjugate variables for the Korteveg- de Vries equation and the Toda lattice with periodic boundary conditions,, Prog. of Theor. Phys., 55 (1976), 438. doi: 10.1143/PTP.55.438.
[4]	J. Garnett and E. Trubowitz, Gaps and bands of one dimensional periodic Schrödinger operators,, Comment. Math. Helv., 59 (1984), 258. doi: 10.1007/BF02566350.
[5]	A. Jenkins, "Univalent Functions and Conformal Mapping,", Berlin, (1958).
[6]	T. Kappeler and P. Topalov, Global wellposedness of KdV in $H^-1(\T,\R)$,, Duke Math. J., 135 (2006), 327. doi: 10.1215/S0012-7094-06-13524-X.
[7]	T.Kappeler and J. Pöschel, "Kdv & Kam,", Springer, (2003).
[8]	T. Kappeler, C. Möhr and P. Topalov, Birkhoff coordinates for KdV on phase spaces of distributions,, Selecta Math. (N.S.), 11 (2005), 37.
[9]	P. Kargaev and E. Korotyaev, The inverse problem for the Hill operator, a direct method,, Invent. Math., 129 (1997), 567. doi: 10.1007/s002220050173.
[10]	E. Korotyaev, Estimates for the Hill operator. I,, Journal Diff. Eq., 162 (2000), 1. doi: 10.1006/jdeq.1999.3684.
[11]	E. Korotyaev, Estimate for the Hill operator. II,, Journal Diff. Eq., 223 (2006), 229. doi: 10.1016/j.jde.2005.04.017.
[12]	E. Korotyaev, Characterization of the spectrum of Schrödinger operators with periodic distributions,, Int. Math. Res. Not., 2003 (2003), 2019. doi: 10.1155/S1073792803209107.
[13]	E. Korotyaev, The estimates of periodic potentials in terms of effective masses,, Commun. Math. Phys., 183 (1997), 383. doi: 10.1007/BF02506412.
[14]	E. Korotyaev, Periodic "weighted" operators,, J. Differential Equations, 189 (2003), 461. doi: 10.1016/S0022-0396(02)00154-7.
[15]	E. Korotyaev, Estimates of KdV Hamiltomian in terms of actions,, preprint, (2009).
[16]	S. Kuksin, "Analysis of Hamiltonian PDEs,", Oxford Lecture Series in Mathematics and its Applications, 19 (2000).
[17]	H. McKean and E. Trubowitz, Hill's surfaces and their theta functions,, Bull. Am. Math. Soc., 84 (1978), 1042. doi: 10.1090/S0002-9904-1978-14542-X.
[18]	V. Marchenko and I. Ostrovski, A characterization of the spectrum of the Hill operator,, Math. USSR Sb., 26 (1975), 493. doi: 10.1070/SM1975v026n04ABEH002493.
[19]	V. Marchenko and I. Ostrovski, Approximation of periodic by finite-zone potentials,, Selecta Math. Sovietica., 6 (1987), 101.
[20]	A. Veselov and S. Novikov, Poisson brackets and complex tori,, Proc. Steklov Inst. Math., 165 (1985), 53.

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