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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

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, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Jenkins, "Univalent Functions and Conformal Mapping," Berlin, Göttingen, Heidelberg: Springer, 1958. Google Scholar

[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.  Google Scholar

[7]

T.Kappeler and J. Pöschel, "Kdv & Kam," Springer, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. Korotyaev, Estimates for the Hill operator. I, Journal Diff. Eq., 162 (2000), 1-26. doi: 10.1006/jdeq.1999.3684.  Google Scholar

[11]

E. Korotyaev, Estimate for the Hill operator. II, Journal Diff. Eq., 223 (2006), 229-260. doi: 10.1016/j.jde.2005.04.017.  Google Scholar

[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.  Google Scholar

[13]

E. Korotyaev, The estimates of periodic potentials in terms of effective masses, Commun. Math. Phys., 183 (1997), 383-400. doi: 10.1007/BF02506412.  Google Scholar

[14]

E. Korotyaev, Periodic "weighted" operators, J. Differential Equations, 189 (2003), 461-486. doi: 10.1016/S0022-0396(02)00154-7.  Google Scholar

[15]

E. Korotyaev, Estimates of KdV Hamiltomian in terms of actions, preprint, 2009. Google Scholar

[16]

S. Kuksin, "Analysis of Hamiltonian PDEs," Oxford Lecture Series in Mathematics and its Applications, 19, Oxford University Press, Oxford, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

V. Marchenko and I. Ostrovski, Approximation of periodic by finite-zone potentials, Selecta Math. Sovietica., 6 (1987), 101-136.  Google Scholar

[20]

A. Veselov and S. Novikov, Poisson brackets and complex tori, Proc. Steklov Inst. Math., 165 (1985), 53-65.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Jenkins, "Univalent Functions and Conformal Mapping," Berlin, Göttingen, Heidelberg: Springer, 1958. Google Scholar

[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.  Google Scholar

[7]

T.Kappeler and J. Pöschel, "Kdv & Kam," Springer, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. Korotyaev, Estimates for the Hill operator. I, Journal Diff. Eq., 162 (2000), 1-26. doi: 10.1006/jdeq.1999.3684.  Google Scholar

[11]

E. Korotyaev, Estimate for the Hill operator. II, Journal Diff. Eq., 223 (2006), 229-260. doi: 10.1016/j.jde.2005.04.017.  Google Scholar

[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.  Google Scholar

[13]

E. Korotyaev, The estimates of periodic potentials in terms of effective masses, Commun. Math. Phys., 183 (1997), 383-400. doi: 10.1007/BF02506412.  Google Scholar

[14]

E. Korotyaev, Periodic "weighted" operators, J. Differential Equations, 189 (2003), 461-486. doi: 10.1016/S0022-0396(02)00154-7.  Google Scholar

[15]

E. Korotyaev, Estimates of KdV Hamiltomian in terms of actions, preprint, 2009. Google Scholar

[16]

S. Kuksin, "Analysis of Hamiltonian PDEs," Oxford Lecture Series in Mathematics and its Applications, 19, Oxford University Press, Oxford, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

V. Marchenko and I. Ostrovski, Approximation of periodic by finite-zone potentials, Selecta Math. Sovietica., 6 (1987), 101-136.  Google Scholar

[20]

A. Veselov and S. Novikov, Poisson brackets and complex tori, Proc. Steklov Inst. Math., 165 (1985), 53-65.  Google Scholar

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