Global dynamics beyond the ground state energy for nonlinear dispersive equations

Kenji Nakanishi[1]

  • [1] Department of Mathematics Kyoto University Kyoto 606-8502 Japan

Journées Équations aux dérivées partielles (2012)

  • page 1-6
  • ISSN: 0752-0360

Abstract

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This is a brief introduction to the joint work with Wilhelm Schlag and Joachim Krieger on the global dynamics for nonlinear dispersive equations with unstable ground states. We prove that the center-stable and the center-unstable manifolds of the ground state solitons separate the energy space by the dynamical property into the scattering and the blow-up regions, respectively in positive time and in negative time. The transverse intersection of the two manifolds yields nine sets of global dynamics, which include stable transition from blow-up to scattering and vice versa.

How to cite

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Nakanishi, Kenji. "Global dynamics beyond the ground state energy for nonlinear dispersive equations." Journées Équations aux dérivées partielles (2012): 1-6. <http://eudml.org/doc/275654>.

@article{Nakanishi2012,
abstract = {This is a brief introduction to the joint work with Wilhelm Schlag and Joachim Krieger on the global dynamics for nonlinear dispersive equations with unstable ground states. We prove that the center-stable and the center-unstable manifolds of the ground state solitons separate the energy space by the dynamical property into the scattering and the blow-up regions, respectively in positive time and in negative time. The transverse intersection of the two manifolds yields nine sets of global dynamics, which include stable transition from blow-up to scattering and vice versa.},
affiliation = {Department of Mathematics Kyoto University Kyoto 606-8502 Japan},
author = {Nakanishi, Kenji},
journal = {Journées Équations aux dérivées partielles},
keywords = {nonlinear wave equations; nonlinear Schrödinger equation; nonlinear Klein-Gorond equation; solitons; scattering theory; blow-up; invariant manifolds},
language = {eng},
pages = {1-6},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Global dynamics beyond the ground state energy for nonlinear dispersive equations},
url = {http://eudml.org/doc/275654},
year = {2012},
}

TY - JOUR
AU - Nakanishi, Kenji
TI - Global dynamics beyond the ground state energy for nonlinear dispersive equations
JO - Journées Équations aux dérivées partielles
PY - 2012
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 6
AB - This is a brief introduction to the joint work with Wilhelm Schlag and Joachim Krieger on the global dynamics for nonlinear dispersive equations with unstable ground states. We prove that the center-stable and the center-unstable manifolds of the ground state solitons separate the energy space by the dynamical property into the scattering and the blow-up regions, respectively in positive time and in negative time. The transverse intersection of the two manifolds yields nine sets of global dynamics, which include stable transition from blow-up to scattering and vice versa.
LA - eng
KW - nonlinear wave equations; nonlinear Schrödinger equation; nonlinear Klein-Gorond equation; solitons; scattering theory; blow-up; invariant manifolds
UR - http://eudml.org/doc/275654
ER -

References

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  1. K. Nakanishi J. Krieger, W. Schlag, Global dynamics above the ground state energy for the one-dimensional NLKG equation, to appear in Math. Z. Zbl1263.35002MR2968226
  2. K. Nakanishi J. Krieger, W. Schlag, Global dynamics away from the ground state for the energy-critical nonlinear wave equation, to appear in Amer. J. Math. Zbl1307.35170MR3086065
  3. K. Nakanishi, W. Schlag, Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, J. Differential Equations 250 (2011), 2299-2333 Zbl1213.35307MR2756065
  4. K. Nakanishi, W. Schlag, Invariant manifolds and dispersive Hamiltonian evolution equations, (2011), European Mathematical Society, Zürich Zbl1235.37002MR2847755
  5. K. Nakanishi, W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. Partial Differential Equations 44 (2012), 1-45 Zbl1237.35148MR2898769
  6. K. Nakanishi, W. Schlag, Global dynamics above the ground state for the nonlinear Klein-Gordon equation without a radial assumption, Arch. Ration. Mech. Anal. 203 (2012), 809-851 Zbl1256.35138MR2928134
  7. K. Nakanishi, W. Schlag, Invariant manifolds around soliton manifolds for the nonlinear Klein-Gordon equation, SIAM J. Math. Anal. 44 (2012), 1175-1210 Zbl1261.35037MR2914265

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