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

Evgeny L. Korotyaev ^1,

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 and Continuous Dynamical Systems, 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), 115-159. 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), 705-749. 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), 438-456. 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), 258-312 doi: 10.1007/BF02566350.
[5]	A. Jenkins, "Univalent Functions and Conformal Mapping," Berlin, Göttingen, Heidelberg: Springer, 1958.
[6]	T. Kappeler and P. Topalov, Global wellposedness of KdV in $H^-1(\T,\R)$, Duke Math. J., 135 (2006), 327-360. 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), 37-98.
[9]	P. Kargaev and E. Korotyaev, The inverse problem for the Hill operator, a direct method, Invent. Math., 129 (1997), 567-593. doi: 10.1007/s002220050173.
[10]	E. Korotyaev, Estimates for the Hill operator. I, Journal Diff. Eq., 162 (2000), 1-26. doi: 10.1006/jdeq.1999.3684.
[11]	E. Korotyaev, Estimate for the Hill operator. II, Journal Diff. Eq., 223 (2006), 229-260. 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), 2019-2031. doi: 10.1155/S1073792803209107.
[13]	E. Korotyaev, The estimates of periodic potentials in terms of effective masses, Commun. Math. Phys., 183 (1997), 383-400. doi: 10.1007/BF02506412.
[14]	E. Korotyaev, Periodic "weighted" operators, J. Differential Equations, 189 (2003), 461-486. 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, Oxford University Press, Oxford, 2000.
[17]	H. McKean and E. Trubowitz, Hill's surfaces and their theta functions, Bull. Am. Math. Soc., 84 (1978), 1042-1085. 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), 493-554. doi: 10.1070/SM1975v026n04ABEH002493.
[19]	V. Marchenko and I. Ostrovski, Approximation of periodic by finite-zone potentials, Selecta Math. Sovietica., 6 (1987), 101-136.
[20]	A. Veselov and S. Novikov, Poisson brackets and complex tori, Proc. Steklov Inst. Math., 165 (1985), 53-65.

show all references

References:

[1]	J. Bourgain, Periodic Korteweg - de Vries equation with measures as initial data, Selecta Math., 3 (1997), 115-159. 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), 705-749. 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), 438-456. 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), 258-312 doi: 10.1007/BF02566350.
[5]	A. Jenkins, "Univalent Functions and Conformal Mapping," Berlin, Göttingen, Heidelberg: Springer, 1958.
[6]	T. Kappeler and P. Topalov, Global wellposedness of KdV in $H^-1(\T,\R)$, Duke Math. J., 135 (2006), 327-360. 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), 37-98.
[9]	P. Kargaev and E. Korotyaev, The inverse problem for the Hill operator, a direct method, Invent. Math., 129 (1997), 567-593. doi: 10.1007/s002220050173.
[10]	E. Korotyaev, Estimates for the Hill operator. I, Journal Diff. Eq., 162 (2000), 1-26. doi: 10.1006/jdeq.1999.3684.
[11]	E. Korotyaev, Estimate for the Hill operator. II, Journal Diff. Eq., 223 (2006), 229-260. 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), 2019-2031. doi: 10.1155/S1073792803209107.
[13]	E. Korotyaev, The estimates of periodic potentials in terms of effective masses, Commun. Math. Phys., 183 (1997), 383-400. doi: 10.1007/BF02506412.
[14]	E. Korotyaev, Periodic "weighted" operators, J. Differential Equations, 189 (2003), 461-486. 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, Oxford University Press, Oxford, 2000.
[17]	H. McKean and E. Trubowitz, Hill's surfaces and their theta functions, Bull. Am. Math. Soc., 84 (1978), 1042-1085. 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), 493-554. doi: 10.1070/SM1975v026n04ABEH002493.
[19]	V. Marchenko and I. Ostrovski, Approximation of periodic by finite-zone potentials, Selecta Math. Sovietica., 6 (1987), 101-136.
[20]	A. Veselov and S. Novikov, Poisson brackets and complex tori, Proc. Steklov Inst. Math., 165 (1985), 53-65.

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