## Asymptotic stability of solitons to 1D nonlinear Schrödinger equations in subcritical case.(English)Zbl 1465.35360

In this paper, the author studies the asymptotic stability of solitary waves to 1D nonlinear Schrödinger equations(NLS) $$i\partial_tu+\partial_x^2 u=F(|u|^2)u$$ in the subcritical case with symmetry and spectrum assumptions.
The asymptotic stability of solitons for (NLS) started from V. S. Buslaev and G. S. Perel’man [St. Petersbg. Math. J. 4, No. 6, 63–102 (1992; Zbl 0853.35112); translation from Algebra Anal. 4, No. 6, 63–102 (1992)] where the one dimension (NLS) case was considered and further refined in [V. S. Buslaev and G. S. Perel’man Transl., Ser. 2, Am. Math. Soc. 164, 75–98 (1995; Zbl 0841.35108)] and [V. S. Buslaev and C. Sulem, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20, No. 3, 419–475 (2003; Zbl 1028.35139)]. Their work was also extended to high dimensions and multi-solitons by many authors.
The aim of this paper is to prove asymptotic stability of solitons for (NLS) by using the vector field method to obtain more decay of the solution to the linearized equation with respect to time. Meanwhile, the author applied the polynomial growth of the high Sobolev norms of solutions to 1D Schrödinger equations obtained by G. Staffilani [Duke Math. J. 86, No. 1, 109–142 (1997; Zbl 0874.35114)] to control the high moments of the solutions emerging from the vector fields method. The vector field method appeared originally in [S. Klainerman, Commun. Pure Appl. Math. 33, 43–101 (1980; Zbl 0405.35056)] where the small data global well-posedness of quasilinear wave equations was solved.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35C08 Soliton solutions 35B40 Asymptotic behavior of solutions to PDEs 35B35 Stability in context of PDEs
Full Text:

### References:

 [1] 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 [2] Beceanu, M., New estimates for a time-dependent Schrödinger equation, Duke Math J, 159, 417-477 (2011) · Zbl 1229.35224 [3] Beceanu, M., A critical center-stable manifold for Schrödinger’s equation in three dimensions, Comm Pure Appl Math, 65, 431-507 (2012) · Zbl 1234.35240 [4] Berestycki, H.; Lions, P. L., Nonlinear scalar field equations. I. Existence of a ground state, Arch Ration Mech Anal, 62, 313-345 (1983) · Zbl 0533.35029 [5] Bourgain J. On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE. Int Math Res Not IMRN, 1996: 277-304 · Zbl 0934.35166 [6] Bourgain, J., Global Solutions of Nonlinear Schrödinger Equations, Amer Math Soc Colloq Publ, Vol 46 (1999), Providence: Amer Math Soc, Providence · Zbl 0933.35178 [7] Buslaev, V.; Perelman, G. S., Scattering for the nonlinear Schrödinger equations: states close to a soliton, St Petersburgh Math J, 4, 6, 1111-1142 (1993) [8] Buslaev, V.; Perelman, G. S., On the stability of solitary waves for nonlinear Schrödinger equation, Amer Math Soc Transl Ser 2, 2, 164, 75-99 (1995) [9] Buslaev, V.; Sulem, C., On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann Inst H Poincaré Anal Non Lunéaire, 20, 3, 419-475 (2003) · Zbl 1028.35139 [10] Cazenave, T., Semilinear Schrödinger Equations, Courant Lect Notes Math, Vol 10 (2003), Providence: Amer Math Soc, Providence · Zbl 1055.35003 [11] Costin, O.; Huang, M.; Schlag, W., On the spectral properties of L_± in three dimensions, Nonlinearity, 25, 125-164 (2012) · Zbl 1232.35106 [12] Cuccagna, S., Stabilization of solutions to nonlinear Schroödinger equations, Comm Pure Appl Math, 4, 9, 1110-1145 (2001) · Zbl 1031.35129 [13] Cuccagna, S., An invariant set in energy space for supercritical NLS in 1D, J Math Anal Appl, 352, 634-644 (2009) · Zbl 1160.35534 [14] Cuccagna, S., The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states, Comm Math Phys, 305, 2, 279-320 (2011) · Zbl 1222.35183 [15] Cuccagna, S.; Georgiev, V.; Visciglia, N., Decay and scattering of small solutions of pure power NLS in ℝ with p > 3 and with a potential, Comm Pure Appl Math, 6, 957-980 (2013) [16] Cuccagna, S.; Maeda, M., On small energy stabilization in the NLS with a trapping potential, Anal PDE, 8, 6, 1289-1349 (2015) · Zbl 1326.35335 [17] Cuccagna, S.; Mizumachi, T., On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations, Comm Math Phys, 284, 51-77 (2008) · Zbl 1155.35092 [18] Grillakis, M.; Shatah, J.; Strauss, W., Stability theory of solitary waves in the presence of symmetry I, J Funct Anal, 74, 1, 160-197 (1987) · Zbl 0656.35122 [19] Grillakis, M.; Shatah, J.; Strauss, W., Stability of solitary waves in presence of symmetry II, J Funct Anal, 94, 2, 308-384 (1990) · Zbl 0711.58013 [20] 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 IMRN, 66, 3559-3584 (2004) · Zbl 1072.35167 [21] Kenig, C. E.; Ponce, G.; Vega, L., Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm Pure Appl Math, 46, 527-620 (1993) · Zbl 0808.35128 [22] Kirr, E.; Mizrak, O., Asymptotic stability of ground states in 3D nonlinear Schrödinger equation including subcritical cases, J Funct Anal, 257, 3691-3747 (2009) · Zbl 1187.35238 [23] Kirr, E.; Zarnescu, A., On the asymptotic stability of bound states in 2D cubic Schrödinger equation, Comm Math Phys, 272, 443-468 (2007) · Zbl 1194.35416 [24] Klainerman, S., Global existence for nonlinear wave equations, Comm Pure Appl Math, 33, 1, 43-101 (1980) · Zbl 0405.35056 [25] Klainerman, S., Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm Pure Appl Math, 38, 3, 321-332 (1985) · Zbl 0635.35059 [26] 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 [27] Krieger, J.; Schlag, W., Non-generic blow-up solutions for the critical focusing NLS in 1-D, J Eur Math Soc (JEMS), 11, 1, 1-125 (2009) · Zbl 1163.35035 [28] Maeda, M., Stability of bound states of Hamiltonian PDEs in the degenerate cases, J Funct Anal, 263, 2, 511-528 (2012) · Zbl 1244.35008 [29] Martel, Y.; Merle, F.; Tsai, T. P., Stability in H^1 of the sum of K solitary waves for some nonlinear Schrödinger equations, Duke Math J, 133, 3, 405-466 (2006) · Zbl 1099.35134 [30] McKean, H. P.; Shatah, J., The nonlinear Schrödinger equation and the nonlinear heat equation reduction to linear form, Comm Pure Appl Math, 44, 8-9, 1067-1080 (1991) · Zbl 0773.35075 [31] Mizumachi, T., Asymptotic stability of small solitary waves to 1D nonlinear Schrödinger equations with potential, Kyoto J Math, 48, 471-497 (2008) · Zbl 1175.35138 [32] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Berlin: Springer-Verlag, Berlin · Zbl 0516.47023 [33] Perelman, G. S., Asymptotic stability of solitons for nonlinear Schrödinger equations, Comm Partial Differential Equations, 29, 1051-1095 (2004) · Zbl 1067.35113 [34] Pusateri F, Soffer A. Bilinear estimates in the presence of a large potential and a critical NLS in 3d. arXiv: 2003.00312 [35] Rodnianski I, Schlag W, Soffer A. Asymptotic stability of N-soliton states of NLS. arXiv: 0309114 · Zbl 1130.81053 [36] Schlag, W., Stable manifolds for an orbitally unstable nonlinear Schrödinger equation, Ann of Math, 169, 139-227 (2009) · Zbl 1180.35490 [37] Sigal, I. M., Nonlinear wave and Schrödinger equations, I. Instability of periodic and quasi-periodic solutions, Comm Math Phys, 153, 297-320 (1993) · Zbl 0780.35106 [38] Soffer, A.; Weinstein, M. I., Multichannel nonlinear scattering theory for nonintegrable equations, Comm Math Phys, 342, 312-327 (1989) · Zbl 0712.35074 [39] Soffer, A.; Weinstein, M. I., Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent Math, 136, 1, 9-74 (1999) · Zbl 0910.35107 [40] Soffer, A.; Weinstein, M. I., Selection of the ground state for nonlinear Schrödinger equations, Rev Math Phys, 16, 8, 977-1071 (2004) · Zbl 1111.81313 [41] Staffilani, G., On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math J, 86, 1, 109-142 (1997) · Zbl 0874.35114 [42] Weinstein, M. I., Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J Math Anal, 16, 472-491 (1985) · Zbl 0583.35028 [43] Weinstein, M. I., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm Pure Appl Math, 39, 51-67 (1986) · Zbl 0594.35005
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.