Notes on symplectic non-squeezing of the KdV flow

J. Colliander[1]; M. Keel[2]; G. Staffilani[3]; H. Takaoka[4]; T. Tao[5]

  • [1] University of Toronto
  • [2] University of Minnesota
  • [3] M.I.T.
  • [4] Kobe University and the University of Chicago
  • [5] University of California, Los Angeles

Journées Équations aux dérivées partielles (2005)

  • page 1-15
  • ISSN: 0752-0360

Abstract

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We prove two finite dimensional approximation results and a symplectic non-squeezing property for the Korteweg-de Vries (KdV) flow on the circle 𝕋 . The nonsqueezing result relies on the aforementioned approximations and the finite-dimensional nonsqueezing theorem of Gromov [14]. Unlike the work of Kuksin [22] which initiated the investigation of non-squeezing results for infinite dimensional Hamiltonian systems, the nonsqueezing argument here does not construct a capacity directly. In this way our results are similar to those obtained for the NLS flow by Bourgain [3]. A major difficulty here though is the lack of any sort of smoothing estimate which would allow us to easily approximate the infinite dimensional KdV flow by a finite-dimensional Hamiltonian flow. To resolve this problem we invert the Miura transform and work on the level of the modified KdV (mKdV) equation, for which smoothing estimates can be established.

How to cite

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Colliander, J., et al. "Notes on symplectic non-squeezing of the KdV flow." Journées Équations aux dérivées partielles (2005): 1-15. <http://eudml.org/doc/10605>.

@article{Colliander2005,
abstract = {We prove two finite dimensional approximation results and a symplectic non-squeezing property for the Korteweg-de Vries (KdV) flow on the circle $\mathbb\{T\}$. The nonsqueezing result relies on the aforementioned approximations and the finite-dimensional nonsqueezing theorem of Gromov [14]. Unlike the work of Kuksin [22] which initiated the investigation of non-squeezing results for infinite dimensional Hamiltonian systems, the nonsqueezing argument here does not construct a capacity directly. In this way our results are similar to those obtained for the NLS flow by Bourgain [3]. A major difficulty here though is the lack of any sort of smoothing estimate which would allow us to easily approximate the infinite dimensional KdV flow by a finite-dimensional Hamiltonian flow. To resolve this problem we invert the Miura transform and work on the level of the modified KdV (mKdV) equation, for which smoothing estimates can be established.},
affiliation = {University of Toronto; University of Minnesota; M.I.T.; Kobe University and the University of Chicago; University of California, Los Angeles},
author = {Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.},
journal = {Journées Équations aux dérivées partielles},
language = {eng},
month = {6},
pages = {1-15},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Notes on symplectic non-squeezing of the KdV flow},
url = {http://eudml.org/doc/10605},
year = {2005},
}

TY - JOUR
AU - Colliander, J.
AU - Keel, M.
AU - Staffilani, G.
AU - Takaoka, H.
AU - Tao, T.
TI - Notes on symplectic non-squeezing of the KdV flow
JO - Journées Équations aux dérivées partielles
DA - 2005/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 15
AB - We prove two finite dimensional approximation results and a symplectic non-squeezing property for the Korteweg-de Vries (KdV) flow on the circle $\mathbb{T}$. The nonsqueezing result relies on the aforementioned approximations and the finite-dimensional nonsqueezing theorem of Gromov [14]. Unlike the work of Kuksin [22] which initiated the investigation of non-squeezing results for infinite dimensional Hamiltonian systems, the nonsqueezing argument here does not construct a capacity directly. In this way our results are similar to those obtained for the NLS flow by Bourgain [3]. A major difficulty here though is the lack of any sort of smoothing estimate which would allow us to easily approximate the infinite dimensional KdV flow by a finite-dimensional Hamiltonian flow. To resolve this problem we invert the Miura transform and work on the level of the modified KdV (mKdV) equation, for which smoothing estimates can be established.
LA - eng
UR - http://eudml.org/doc/10605
ER -

References

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  1. J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part II, Geometric and Funct. Anal. 3 (1993), 209-262. Zbl0787.35098MR1215780
  2. J. Bourgain, Aspects of longtime behaviour of solutions of nonlinear Hamiltonian evolution equations, GAFA 5 (1995), 105–140. Zbl0879.35024MR1334864
  3. J. Bourgain, Approximation of solutions of the cubic nonlinear Schrödinger equations by finite-dimensional equations and nonsqueezing properties, Int. Math. Res. Notices, 1994, no. 2, (1994), 79–90. Zbl0818.35112MR1264931
  4. J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Selecta Math. (N.S.) 3 (1997), 115-159. Zbl0891.35138MR1466164
  5. J. Bourgain, Global solutions of nonlinear Schrödinger equations, AMS Publications, 1999. Zbl0933.35178MR1691575
  6. M. Christ, J. Colliander, T. Tao, Illposedness for canonical defocussing equations below the endpoint regularity, to appear. 
  7. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness result for KdV in Sobolev spaces of negative index, Elec. J. Diff. Eq. 2001 (2001) No 26, 1–7. Zbl0967.35119MR1824796
  8. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Sharp global well-posedness for periodic and non-periodic KdV and mKdV on ℝ and 𝕋 , J. Amer. Math. Soc. 16 (2003), 705–749. Zbl1025.35025MR1969209
  9. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Multilinear estimates for periodic KdV equations, and applications, Journ. Funct. Analy. 211 (2004), no. 1, 173–218. Zbl1062.35109MR2054622
  10. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, , preprint (2004). 
  11. J. Colliander, G. Staffilani, H. Takaoka, Global well-posedness of the KdV equation below L 2 , Math Res. Letters 6 (1999), 755-778. Zbl0959.35144MR1739230
  12. L. Dickey, Soliton equations and Hamiltonian systems, World Scientific, 1991. Zbl1140.35012MR1147643
  13. C.S. Gardner, Korteweg-de Vries equation and generalizations IV, J. Math. Phys. 12 (1971), no. 8, 1548–1551. Zbl0283.35021MR286402
  14. M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. math., 82 (1985), 307–347. Zbl0592.53025MR1554036
  15. H. Hofer, E. Zehnder, A new capacity for symplectic manifolds., in Analysis et cetera, Academic Press (1990), 405–428. Edited by P. Rabinowitz and E. Zehnder. Zbl0702.58021MR1039354
  16. H. Hofer, E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Verlag, 1994. Zbl0805.58003MR1306732
  17. T. Kappeler, P. Topalov, Global well-posedness of KdV in H - 1 ( 𝕋 , ℝ ) , preprint 2003. 
  18. T. Kappeler, P. Topalov, Global fold structure of the Miura map on L 2 , IMRN 2004:39 (2004), 2039–2068. Zbl1076.35111MR2062735
  19. C. Kenig, G. Ponce, L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), 573–603. Zbl0848.35114MR1329387
  20. C. Kenig, G. Ponce, L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001), no 3, 617–633.. Zbl1034.35145MR1813239
  21. C. Kenig, G. Ponce, L.Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1–21. Zbl0787.35090MR1230283
  22. S. Kuksin, Infinite-dimensional symplectic capacities and a squeezing theorem for Hamiltonian PDE’s, CMP 167 (1995), 521–552. Zbl0827.35121MR1316759
  23. S. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and Its Applications, 19, Oxford Univ. Press, 2000. Zbl0960.35001MR1857574
  24. F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), no. 5, 1156–1162. Zbl0383.35065MR488516
  25. R. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Mathematical. Phys. 9 (1968), 1202–1204. Zbl0283.35018MR252825
  26. P. Olver, Applications of Lie groups to differential equations, Springer, 1997. Zbl0588.22001MR1240056
  27. A. Sjöberg, On the Korteweg-de Vries equation: existence and uniqueness, J. Math. Anal. Appl. 29 (1970), 569–579. Zbl0179.43101MR410135
  28. H. Takaoka, Y. Tsutsumi, Well-posedness of the Cauchy Problem for the modified KdV equation with periodic boundary condition, Internat. Math. Res. Notices 2004, no. 56, 3009–3040. Zbl1154.35442MR2097834
  29. T. Tao, Multilinear weighted convolution of L 2 functions, and applications to non-linear dispersive equations, Amer. J. Math. 123 (2001), 839–908. Zbl0998.42005MR1854113
  30. T. Tao, Global regularity of wave maps I. Small critical Sobolev norm in high dimension, IMRN 7 (2001), 299–328. Zbl0983.35080MR1820329
  31. T. Tao, Global regularity of wave maps II. Small energy in two dimensions, Comm. Math. Phys. 224 (2001), 443–544. Zbl1020.35046MR1869874
  32. V.E. Zakharov, L.D. Faddeev, The Korteweg-de Vries equation is a completely integrable Hamiltonian System, Funkz. Anal. Priloz. 5 (1971), no. 4, 18–27. Zbl0257.35074MR303132

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