Abstract
We consider a ground state (soliton) of a Hamiltonian PDE. We prove that if the soliton is orbitally stable, then it is also asymptotically stable. The main assumptions are transversal nondegeneracy of the manifold of the ground states, linear dispersion (in the form of Strichartz estimates) and nonlinear Fermi Golden Rule. We allow the linearization of the equation at the soliton to have an arbitrary number of eigenvalues. The theory is tailor made for the application to the translational invariant NLS in space dimension 3. The proof is based on the extension of some tools of the theory of Hamiltonian systems (reduction theory, Darboux theorem, normal form) to the case of systems invariant under a symmetry group with unbounded generators.
Similar content being viewed by others
References
Bambusi D., Cuccagna S.: On dispersion of small energy solutions of the nonlinear Klein Gordon equation with a potential. Amer. J. Math. 133(5), 1421–1468 (2011)
Beceanu M.: New estimates for a time-dependent Schrödinger equation. Duke Math. J. 159(3), 417–477 (2011)
Buslaev V.S., Perelman G.S.: Scattering for the nonlinear Schrödinger equation: states that are close to a soliton. Alge. i Anal. 4(6), 63–102 (1992)
Cuccagna S., Mizumachi T.: On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations. Commun. Math. Phys. 284(1), 51–77 (2008)
Cuccagna S.: Stabilization of solutions to nonlinear Schrödinger equations. Commun. Pure Appl. Math. 54(9), 1110–1145 (2001)
Cuccagna S.: The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states. Commun. Math. Phys. 305(2), 279–331 (2011)
Cuccagna, S.: On asymptotic stability of moving ground states of the nonlinear Schrödinger equation. Preprint: http://arxiv.org/abs/1107.4954v4 [math.Ap], 2012
Fröhlich J., Gustafson S., Jonsson B.L.G., Sigal I.M.: Solitary wave dynamics in an external potential. Commun. Math. Phys. 250(3), 613–642 (2004)
Gustafson, S., Nakanishi, K., Tsai, T.-P.: Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves. Int. Math. Res. Not. no. 66, 3559–3584 (2004)
Gang Z., Sigal I.M.: Relaxation of solitons in nonlinear Schrödinger equations with potential. Adv. Math. 216(2), 443–490 (2007)
Gang Z., Weinstein M.I.: Dynamics of nonlinear Schrödinger/Gross-Pitaevskii equations: mass transfer in systems with solitons and degenerate neutral modes. Anal. PDE 1(3), 267–322 (2008)
Perelman G.S.: Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations. Comm. Part. Diff. Eqs. 29(7-8), 1051–1095 (2004)
Perelman, G.S.: Asymptotic stability in H 1 of NLS. One soliton case. Personal communication, 2011
Schmid, R.: Infinite-dimensional Hamiltonian systems, Monographs and Textbooks in Physical Science. Lecture Notes, Vol. 3, Naples: Bibliopolis, 1987
Sigal I.M.: Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions. Commun. Math. Phys. 153(2), 297–320 (1993)
Soffer A., Weinstein M.I.: Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136(1), 9–74 (1999)
Yajima K.: The W k,p continuity of wave operators for Schrödinger operators. J. Math. Soc. Japan 47, 551–581 (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Gallavotti
Rights and permissions
About this article
Cite this article
Bambusi, D. Asymptotic Stability of Ground States in Some Hamiltonian PDEs with Symmetry. Commun. Math. Phys. 320, 499–542 (2013). https://doi.org/10.1007/s00220-013-1684-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-013-1684-3