# 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

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topNakanishi, 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

top- 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
- 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
- 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
- K. Nakanishi, W. Schlag, Invariant manifolds and dispersive Hamiltonian evolution equations, (2011), European Mathematical Society, Zürich Zbl1235.37002MR2847755
- 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
- 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
- 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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