## An invariant set in energy space for supercritical NLS in 1D.(English)Zbl 1160.35534

Summary: We consider radial solutions of a mass supercritical monic NLS and we prove the existence of a set, which looks like a hypersurface, in the space of finite energy functions, invariant for the flow and formed by solutions which converge to ground states.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations)

### Keywords:

ground state; supercritical NLS
Full Text:

### References:

 [1] Berestycki, H.; Cazenave, T., Instabilité des états stationnaires des LES équations de Schrödinger et de Klein Gordon non-linéaires, C. R. math. acad. sci. Paris, 293, 489-492, (1981) · Zbl 0492.35010 [2] Beceanu, M., A centre-stable manifold for the focussing cubic NLS in $$R^{1 + 3}$$, Comm. math. phys., 280, 145-205, (2008) · Zbl 1148.35082 [3] Buslaev, V.S.; Perelman, G.S., On the stability of solitary waves for nonlinear Schrödinger equations, (), 75-98 · Zbl 0841.35108 [4] Cote, R., Construction of solutions to the L2-critical KdV equation with a given asymptotic behaviour, Duke math. J., 138, 487-532, (2007) · Zbl 1130.35112 [5] Cuccagna, S., On asymptotic stability in energy space of ground states of NLS in 1D, J. differential equations, 245, 653-691, (2008) · Zbl 1185.35251 [6] Cuccagna, S., On instability of excited states of the nonlinear Schrödinger equation · Zbl 1161.35500 [7] Cuccagna, S., A revision of “on asymptotic stability in energy space of ground states of NLS in 1D” · Zbl 1185.35251 [8] Cuccagna, S., Stabilization of solutions to nonlinear Schrödinger equations, Comm. pure appl. math., 54, 1110-1145, (2001), Errata: 58 (2005) 147 · Zbl 1031.35129 [9] Cuccagna, S., Erratum: stabilization of solutions to nonlinear Schrödinger equations, Comm. pure appl. math., 58, 147, (2005) [10] 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 [11] S. Cuccagna, N. Visciglia, On asymptotic stability of ground states of NLS with a finite bands periodic potential in 1D, http://arxiv.org/abs/0809.4775 · Zbl 1298.35191 [12] Kato, T., Wave operators and similarity for some non-selfadjoint operators, Math. ann., 162, 258-269, (1966) · Zbl 0139.31203 [13] Krieger, J.; Schlag, W., Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension, J. amer. math. soc., 19, 815-920, (2006) · Zbl 1281.35077 [14] Martel, I., Asymptotic N-soliton-like solutions of subcritical and critical generalized KdV equations, Amer. J. math., 127, 1103-1140, (2005) · Zbl 1090.35158 [15] Mizumachi, T., Asymptotic stability of small solitons to 1D NLS with potential [16] Schlag, W., Stable manifolds for an orbitally unstable NLS, (2004) [17] Tsai, T.P.; Yau, H.T., Stable directions for excited states of nonlinear Schrödinger equations, Comm. partial differential equations, 27, 2363-2402, (2002) · Zbl 1021.35113 [18] Yajima, K., The $$W^{k, p}$$ continuity of wave operators for Schrödinger operators, J. math. soc. Japan, 47, 551-581, (1995) · Zbl 0837.35039 [19] Yajima, K., The $$W^{k, p}$$ continuity of wave operators for Schrödinger operators III, even dimensional case $$m \geqslant 4$$, J. math. sci. univ. Tokyo, 2, 311-346, (1995) · Zbl 0841.47009 [20] Weinstein, M.I., Modulation stability of ground states of nonlinear Schrödinger equations, SIAM J. math. anal., 16, 472-491, (1985) · Zbl 0583.35028
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