## On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations.(English)Zbl 1155.35092

Summary: We consider nonlinear Schrödinger equations
$iu_t +\Delta u +\beta (|u|^2)u=0\quad \text{for }(t,x)\in \mathbb{R}\times \mathbb{R}^d,$
where $$d\geq 3$$ and $$\beta$$ is smooth. We prove that symmetric finite energy solutions close to orbitally stable ground states converge to a sum of a ground state and a dispersive wave as $$t\to\infty$$, assuming the so-called Fermi Golden Rule (FGR) hypothesis. We improve the “sign condition” required in a recent paper by Gang Zhou and I. M. Sigal.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35B35 Stability in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

### Keywords:

asymptotic stability; Strichartz estimate
Full Text:

### References:

 [1] Arnold, V.I.: Geometrical methods in the theory of ordinary differential equations. Grundlehren der Mathematischen Wissenschaften 250, New York: Springer-Verlag, 1983 · Zbl 0507.34003 [2] Buslaev V.S., Perelman G.S.: Scattering for the nonlinear Schrödinger equation: states close to a soliton. St. Petersburg Math. J. 4, 1111–1142 (1993) [3] Buslaev, V.S., Perelman, G.S.: On the stability of solitary waves for nonlinear Schrödinger equations. In: Nonlinear evolution equations, N.N. Uraltseva, ed. Transl. Ser. 2, 164, Providence, RI: Amer. Math. Soc., 1995, pp 75–98 · Zbl 0841.35108 [4] Buslaev V.S., Sulem C.: On the asymptotic stability of solitary waves of Nonlinear Schrödinger equations. Ann. Inst. H. Poincaré. An. Nonlin. 20, 419–475 (2003) · Zbl 1028.35139 [5] Cazenave, T.: Semilinear Schrodinger equations. Courant Lecture Notes in Mathematics 10, New York University, Courant Institute of Mathematical Sciences, Providence, RI: Amer. Math. Soc., 2003 · Zbl 1055.35003 [6] Cazenave T., Lions P.L.: Orbital stability of standing waves for nonlinear Schrödinger equations. Commun. Math. Phys. 85, 549–561 (1982) · Zbl 0513.35007 [7] Cuccagna, S.: Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure App. Math. 54, 1110–1145 (2001); Comm. Pure Appl. Math. 58, 147 (2005) · Zbl 1031.35129 [8] Cuccagna S.: On asymptotic stability of ground states of NLS. Rev. Math. Phys. 15, 877–903 (2003) · Zbl 1084.35089 [9] Cuccagna, S.: Dispersion for Schrödinger equation with periodic potential in 1D. To appear J. Diff. Eq. · Zbl 1163.35032 [10] Cuccagna, S.: On instability of excited states of the nonlinear Schrödinger equation. http://arxiv.org/abs/0801.4237v2[math.AP] , 2008 · Zbl 1163.35032 [11] Cuccagna S., Pelinovsky D.: Bifurcations from the endpoints of the essential spectrum in the linearized nonlinear Schrodinger problem. J. Math. Phys. 46, 053520 (2005) · Zbl 1110.35082 [12] Cuccagna S., Pelinovsky D., Vougalter V.: Spectra of positive and negative energies in the linearization of the NLS problem. Comm. Pure Appl. Math. 58, 1–29 (2005) · Zbl 1064.35181 [13] Cuccagna, S., Tarulli, M.: On asymptotic stability in energy space of ground states of NLS in 2D. http://arxiv.org/abs/0801.1277v1[math.AP] , 2008 · Zbl 1171.35470 [14] Dancer E.N.: A note on asymptotic uniqueness for some nonlinearities which change sign. Bull. Austral. Math. Soc. 61, 305–312 (2000) · Zbl 0945.35031 [15] Fibich G., Wang X.P.: Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities. Physica D 175, 96–108 (2003) · Zbl 1098.74614 [16] Grillakis M., Shatah J., Strauss W.: Stability of solitary waves in the presence of symmetries, I. J. Funct. An. 74, 160–197 (1987) · Zbl 0656.35122 [17] Grillakis M., Shatah J., Strauss W.: Stability of solitary waves in the presence of symmetries, II. Jour. Funct. An. 94, 308–348 (1990) · Zbl 0711.58013 [18] 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. Notices 66, 3559–3584 (2004) · Zbl 1072.35167 [19] Kabeya Y., Tanaka K.: Uniqueness of positive radial solutions of semilinear elliptic equations in R N and Sere’s non-degeneracy condition. Comm. Partial Differ. Eqs. 24, 563–598 (1999) · Zbl 0930.35064 [20] Keel M., Tao T.: Endpoint Strichartz estimates. Amer. J. Math. 120, 955–980 (1998) · Zbl 0922.35028 [21] Kwong M.K.: Uniqueness of positive solutions of {$$\Delta$$}u u + u p = 0 in $${\mathbb{R}^n}$$ . Arch. Rat. Mech. Anal. 105, 243–266 (1989) · Zbl 0676.35032 [22] McLeod K.: Uniqueness of positive radial solutions of {$$\Delta$$}u + f(u) = 0 in $${\mathbb{R}^n}$$ , II. Trans. Amer. Math. Soc. 339, 495–505 (1993) · Zbl 0804.35034 [23] Mizumachi, T.: Asymptotic stability of small solitons to 1D NLS with potential. http://arxiv.org/abs/math.AP/0605031 , 2006, to appear in J. Math. Kyoto Univ · Zbl 1091.35093 [24] Mizumachi, T.: Asymptotic stability of small solitons for 2D Nonlinear Schrödinger equations with potential. http://arxiv.org/abs/math.AP/0609323 , 2006 · Zbl 1146.35085 [25] Pillet C.A., Wayne C.E.: Invariant manifolds for a class of dispersive, Hamiltonian partial differential equations. J. Diff. Eq. 141, 310–326 (1997) · Zbl 0890.35016 [26] Perelman G.S.: Asymptotic stability of solitons for nonlinear Schrödinger equations. Comm. in PDE 29, 1051–1095 (2004) · Zbl 1067.35113 [27] Rodnianski, I., Schlag, W., Soffer, A.: Asymptotic stability of N-soliton states of NLS. http://arxiv.org/abs/math.AP/0309114 , 2003 [28] Shatah J., Strauss W.: Instability of nonlinear bound states. Commun. Math. Phys. 100, 173–190 (1985) · Zbl 0603.35007 [29] Sigal I.M.: Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasi- periodic solutions. Commun. Math. Phys. 153, 297–320 (1993) · Zbl 0780.35106 [30] Stuart D.M.A.: Modulation approach to stability for non topological solitons in semilinear wave equations. J. Math. Pures Appl. 80, 51–83 (2001) · Zbl 1158.35389 [31] Soffer A., Weinstein M.: Multichannel nonlinear scattering II. The case of anisotropic potentials and data. J. Diff. Eq. 98, 376–390 (1992) · Zbl 0795.35073 [32] Soffer A., Weinstein M.: Selection of the ground state for nonlinear Schrödinger equations. Rev. Math. Phys. 16, 977–1071 (2004) · Zbl 1111.81313 [33] Soffer A., Weinstein M.: Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136, 9–74 (1999) · Zbl 0910.35107 [34] Tsai T.P.: Asymptotic dynamics of nonlinear Schrödinger equations with many bound states. J. Diff. Eq. 192, 225–282 (2003) · Zbl 1038.35128 [35] Tsai T.P., Yau H.T.: Asymptotic dynamics of nonlinear Schrödinger equations: resonance dominated and radiation dominated solutions. Comm. Pure Appl. Math. 55, 153–216 (2002) · Zbl 1031.35137 [36] Tsai T.P., Yau H.T.: Relaxation of excited states in nonlinear Schrödinger equations. Int. Math. Res. Not. 31, 1629–1673 (2002) · Zbl 1011.35120 [37] Tsai T.P., Yau H.T.: Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data. Adv. Theor. Math. Phys. 6, 107–139 (2002) · Zbl 1033.81034 [38] Weder R.: Center manifold for nonintegrable nonlinear Schrödinger equations on the line. Commun. Math. Phys. 170, 343–356 (2000) · Zbl 1003.37045 [39] Weinstein M.: Lyapunov stability of ground states of nonlinear dispersive equations. Comm. Pure Appl. Math. 39, 51–68 (1986) · Zbl 0594.35005 [40] Weinstein M.: Modulation stability of ground states of nonlinear Schrödinger equations. Siam J. Math. Anal. 16, 472–491 (1985) · Zbl 0583.35028 [41] Wei J., Winter M.: On a cubic-quintic Ginzburg-Landau equation with global coupling. Proc. Amer. Math. Soc. 133, 1787–1796 (2005) · Zbl 1058.35035 [42] Yajima K.: The W k,p -continuity of wave operators for Schrödinger operators. J. Math. Soc. Japan 47, 551–581 (1995) · Zbl 0837.35039 [43] Yajima K.: The W k,p -continuity of wave operators for Schrödinger operators III. J. Math. Sci. Univ. Tokyo 2, 311–346 (1995) · Zbl 0841.47009 [44] Zhou, G.: Perturbation Expansion and N th Order Fermi Golden Rule of the Nonlinear Schrödinger Equations. http://arxiv.org/abs/math.AP/0610381 , 2006 [45] Zhou, G., Sigal, I.M.: Relaxation of Solitons in Nonlinear Schrödinger Equations with Potential. http://arxiv.org/abs/math-ph/0603060 , 2006 · Zbl 1126.35065
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.