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Dynamics of the $p$-adic shift and applications
Estimates for solutions of KDV on the phase space of periodic distributions in terms of action variables
1. | St. Petersburg State Univ., Russian Federation |
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] | |
[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] | |
[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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