American Institute of Mathematical Sciences

Advanced
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 - 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.  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.  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.  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.  doi: 10.1007/BF02566350.  Google Scholar

[5]

A. Jenkins, "Univalent Functions and Conformal Mapping,", Berlin, (1958).   Google Scholar

[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.  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.   Google Scholar

[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.  Google Scholar

[10]

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

[11]

E. Korotyaev, Estimate for the Hill operator. II,, Journal Diff. Eq., 223 (2006), 229.  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.  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.  doi: 10.1007/BF02506412.  Google Scholar

[14]

E. Korotyaev, Periodic "weighted" operators,, J. Differential Equations, 189 (2003), 461.  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 (2000).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

show all references

References:
[1]

J. Bourgain, Periodic Korteweg - de Vries equation with measures as initial data,, Selecta Math., 3 (1997), 115.  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.  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.  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.  doi: 10.1007/BF02566350.  Google Scholar

[5]

A. Jenkins, "Univalent Functions and Conformal Mapping,", Berlin, (1958).   Google Scholar

[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.  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.   Google Scholar

[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.  Google Scholar

[10]

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

[11]

E. Korotyaev, Estimate for the Hill operator. II,, Journal Diff. Eq., 223 (2006), 229.  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.  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.  doi: 10.1007/BF02506412.  Google Scholar

[14]

E. Korotyaev, Periodic "weighted" operators,, J. Differential Equations, 189 (2003), 461.  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 (2000).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[1]

Paolo Perfetti. Fixed point theorems in the Arnol'd model about instability of the action-variables in phase-space. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 379-391. doi: 10.3934/dcds.1998.4.379
[2]

Alessandro Fonda, Antonio J. Ureña. Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 169-192. doi: 10.3934/dcds.2011.29.169
[3]

Jan-Cornelius Molnar. On two-sided estimates for the nonlinear Fourier transform of KdV. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3339-3356. doi: 10.3934/dcds.2016.36.3339
[4]

Yingte Sun, Xiaoping Yuan. Quasi-periodic solution of quasi-linear fifth-order KdV equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6241-6285. doi: 10.3934/dcds.2018268
[5]

Brandon Seward. Every action of a nonamenable group is the factor of a small action. Journal of Modern Dynamics, 2014, 8 (2) : 251-270. doi: 10.3934/jmd.2014.8.251
[6]

Michael Hutchings. Mean action and the Calabi invariant. Journal of Modern Dynamics, 2016, 10: 511-539. doi: 10.3934/jmd.2016.10.511
[7]

Valentin Ovsienko, MichaeL Shapiro. Cluster algebras with Grassmann variables. Electronic Research Announcements, 2019, 26: 1-15. doi: 10.3934/era.2019.26.001
[8]

Helmut Kröger. From quantum action to quantum chaos. Conference Publications, 2003, 2003 (Special) : 492-500. doi: 10.3934/proc.2003.2003.492
[9]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75
[10]

S. A. Krat. On pairs of metrics invariant under a cocompact action of a group. Electronic Research Announcements, 2001, 7: 79-86.

[11]

Alexandre Rocha, Mário Jorge Dias Carneiro. A dynamical condition for differentiability of Mather's average action. Journal of Geometric Mechanics, 2014, 6 (4) : 549-566. doi: 10.3934/jgm.2014.6.549
[12]

Wei Xu, Liying Yu, Gui-Hua Lin, Zhi Guo Feng. Optimal switching signal design with a cost on switching action. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019068
[13]

Raphael Stuhlmeier. KdV theory and the Chilean tsunami of 1960. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 623-632. doi: 10.3934/dcdsb.2009.12.623
[14]

Juan-Ming Yuan, Jiahong Wu. The complex KdV equation with or without dissipation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 489-512. doi: 10.3934/dcdsb.2005.5.489
[15]

Jean-Paul Chehab, Georges Sadaka. On damping rates of dissipative KdV equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1487-1506. doi: 10.3934/dcdss.2013.6.1487
[16]

Nate Bottman, Bernard Deconinck. KdV cnoidal waves are spectrally stable. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1163-1180. doi: 10.3934/dcds.2009.25.1163
[17]

Philippe G. LeFloch, Seiji Ukai. A symmetrization of the relativistic Euler equations with several spatial variables. Kinetic & Related Models, 2009, 2 (2) : 275-292. doi: 10.3934/krm.2009.2.275
[18]

Tatiana Odzijewicz. Generalized fractional isoperimetric problem of several variables. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2617-2629. doi: 10.3934/dcdsb.2014.19.2617
[19]

Sebastián Ferrer, Francisco J. Molero. Andoyer's variables and phases in the free rigid body. Journal of Geometric Mechanics, 2014, 6 (1) : 25-37. doi: 10.3934/jgm.2014.6.25
[20]

Robert I McLachlan, Christian Offen, Benjamin K Tapley. Symplectic integration of PDEs using Clebsch variables. Journal of Computational Dynamics, 2019, 6 (1) : 111-130. doi: 10.3934/jcd.2019005

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences