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Symplectic nonsqueezing of the korteweg-de vries flow

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References
28
[2]
—, Approximation of solution of the cubic nonlinear Schrödinger equations by finite-dimensional equations and nonsqueezing properties.Int. Math. Res. Not., 1994:2 (1994), 79–88. 10.1155/s1073792894000103
[3]
—, Aspects of long time behaviour of solutions of nonlinear Hamiltonian evolution equations.Geom. Funct. Anal., 5 (1995), 105–140. 10.1007/bf01895664
[4]
—, Periodic Korteweg-de Vries equation with measures as initial data.Selecta Math. 3 (1997), 115–159. 10.1007/s000290050008
[5]
—,Global Solutions of Nonlinear Schrödinger Equations. Amer. Math. Soc. Colloq. Publ., 46. Amer. Math. Soc., Providence, RI, 1999. 10.1090/coll/046
[6]
Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations

Michael Christ, James Colliander, Terrence Tao

American Journal of Mathematics 2003 10.1353/ajm.2003.0040
[7]
Colliander, J., Keel, M., Staffilani, G. Takaoka, H., &Tao, T., Global well-posedness result for KdV in Sobolev spaces of negative index.Electron. J. Differential Equations, 2001:26 (2001), 1–7.
[8]
—, A rifined global well-posedness result for Schrödinger equations with derivative.Siam J. Math. Anal., 34 (2002), 64–86. 10.1137/s0036141001394541
[9]
Sharp global well-posedness for KdV and modified KdV on ℝ and 𝕋

J. Colliander, M. Keel, G. Staffilani et al.

Journal of the American Mathematical Society 2003 10.1090/s0894-0347-03-00421-1
[10]
—, Multilinear estimates for periodic KdV equations, and applications.J. Funct. Anal., 211 (2004), 173–218. 10.1016/s0022-1236(03)00218-0
[11]
Dickey, L. A.,Soliton Equations and Hamiltonian Systems. Adv. Ser. Math. Phys., 12. World Sci. Publishing, River Edge, NJ, 1991. 10.1142/1109
[12]
Gardner, C. S., Korteweg-de Vries equation and generalizations, IV.J. Math. Phys., 12 (1971), 1548–1551. 10.1063/1.1665772
[13]
Gromov, M., Pseudo-holomorphic curves in symplectic manifolds.Invent. Math., 82 (1985), 307–347. 10.1007/bf01388806
[14]
Hofer, H., &Zehnder, E., A new capacity for symplectic manifolds, inAnalysis, et cetera, pp. 405–427. Academic Press Boston, MA, 1990. 10.1016/b978-0-12-574249-8.50023-7
[15]
—,Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser, Basel, 1994. 10.1007/978-3-0348-8540-9
[16]
Kappeler, T. & Topalov, P., Global well-posedness of KdV inH −1 (T,R). Preprint 2003. 10.1007/978-3-662-08054-2_1
[17]
— Global fold structure of the Miura map onL 2(T).Int. Math. Res. Not., 2004:39 (2004), 2039–2068. 10.1155/s1073792804133205
[18]
The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices

Carlos E. Kenig, Gustavo Ponce, Luis Vega

Duke Mathematical Journal 1993 10.1215/s0012-7094-93-07101-3
[19]
A bilinear estimate with applications to the KdV equation

Carlos Kenig, Gustavo Ponce, Luis Vega

Journal of the American Mathematical Society 1996 10.1090/s0894-0347-96-00200-7
[20]
—, On the ill-posedness of some canonical dispersive, equations.Duke Math. J., 106 (2001), 617–633. 10.1215/s0012-7094-01-10638-8
[21]
Kuksin, S., Infinite-dimensional symplectic capacities and a squeezing theorem for Hamiltonian PDEs.Comm. Math. Phys., 167 (1995), 531–552. 10.1007/bf02101534
[22]
—,Analysis of Hamiltonian PDEs. Oxford Lecture Ser. Math. Appl., 19. Oxford Univ. Press, Oxford, 2000. 10.1093/oso/9780198503958.001.0001
[23]
Magri, F. A simple model of the integrable Hamiltonian equation.J. Math. Phys., 19 (1978), 1156–1162. 10.1063/1.523777
[24]
Miura, R. M., Korteweg-de Vries equation and generalizations, I. A remarkable explicit nonlinar transformation.J. Math. Phys., 9 (1968), 1202–1204. 10.1063/1.1664700
[25]
Olver, P. J.,Applications of Lie Groups to Differential Equations, 2nd edition. Graduate Texts in Math., 107. Springer, New York, 1993. 10.1007/978-1-4612-4350-2
[26]
Sjöberg, A., On the Korteweg-de Vries equation: existence and uniqueness.J. Math. Anal. Appl., 29 (1970), 569–579. 10.1016/0022-247x(70)90068-5
[27]
Takaoka, H., &Tsutsumi, Y., Well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition.Int. Math. Res. Not., 2004:56 (2004), 3009–3040. 10.1155/s1073792804140555
[28]
Zakharov, V. E., &Faddeev, L. D., The Korteweg-de Vries equation is a completely integrable Hamiltonian system.Funktsional. Anal. i. Prilozhen., 5:4 (1971), 18–27.
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Details
Published
Jan 01, 2005
Vol/Issue
195(2)
Pages
197-252
Cite This Article
James Colliander, Gigliola Staffilani, Markus Keel, et al. (2005). Symplectic nonsqueezing of the korteweg-de vries flow. Acta Mathematica, 195(2), 197-252. https://doi.org/10.1007/bf02588080
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