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A survey on asymptotic stability of ground states of nonlinear Schrödinger equations. II. (English) Zbl 1475.35313

Summary: We give short survey on the question of asymptotic stability of ground states of nonlinear Schrödinger equations, focusing primarily on the so called nonlinear Fermi Golden Rule.
For Part I, see [the first author, Quad. Mat. 15, 21–57 (2004; Zbl 1130.35360)].

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 1130.35360
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References:

[1] R. Adami, D. Noja and C. Ortoleva, Orbital and asymptotic stability for standing waves of a nonlinear Schrödinger equation with concentrated nonlinearity in dimension three, J. Math. Phys., 54 (2013), 013501, 33 pp. · Zbl 1322.35122
[2] X. An; A. Soffer, Fermi’s golden rule and \(H^1\) scattering for nonlinear Klein-Gordon equations with metastable states, Discrete Contin. Dyn. Syst., 40, 331-373 (2020) · Zbl 1431.35145
[3] G. Artbazar; K. Yajima, The \(L^p\)-continuity of wave operators for one dimensional Schrödinger operators, J. Math. Sci. Univ. Tokyo, 7, 221-240 (2000) · Zbl 0976.34071
[4] R. Asad and G. Simpson, Embedded eigenvalues and the nonlinear Schrödinger equation, J. Math. Phys., 52 (2011), 033511, 26 pp. · Zbl 1315.35195
[5] D. Bambusi, Asymptotic stability of ground states in some Hamiltonian PDEs with symmetry, Comm. Math. Phys., 320, 499-542 (2013) · Zbl 1267.35140
[6] D. Bambusi; S. Cuccagna, On dispersion of small energy solutions of the nonlinear Klein Gordon equation with a potential, Amer. J. Math., 133, 1421-1468 (2011) · Zbl 1237.35115
[7] D. Bambusi; A. Maspero, Freezing of energy of a soliton in an external potential, Comm. Math. Phys., 344, 155-191 (2016) · Zbl 1342.35320
[8] J. T. Beale, Exact solitary water waves with capillary ripples at infinity, Comm. Pure Appl. Math., 44, 211-257 (1991) · Zbl 0727.76019
[9] M. Beceanu, New estimates for a time-dependent Schrödinger equation, Duke Math. J., 159, 417-477 (2011) · Zbl 1229.35224
[10] A. M. Bloch; P. S. Krishnaprasad; J. E. Marsden; T. S. Ratiu, Dissipation induced instabilities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11, 37-90 (1994) · Zbl 0834.58025
[11] C. Bonanno, Long time dynamics of highly concentrated solitary waves for the nonlinear Schrödinger equation, J. Differential Equations, 258, 717-735 (2015) · Zbl 1304.35632
[12] M. Borghese; R. Jenkins; K. D. T.-R. McLaughlin, Long time asymptotic behavior of the focusing nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35, 887-920 (2018) · Zbl 1390.35020
[13] N. Boussaid; S. Cuccagna, On stability of standing waves of nonlinear Dirac equations, Comm. Partial Differential Equations, 37, 1001-1056 (2012) · Zbl 1251.35098
[14] J. P. Boyd, Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics, Mathematics and its Applications, vol. 442, Generalized solitons and hyperasymptotic perturbation theory, Kluwer Academic Publishers, Dordrecht, 1998. · Zbl 0905.76001
[15] V. S. Buslaev; V. E. Grikurov, Simulation of instability of bright solitons for NLS with saturating nonlinearity, Math. Comput. Simulation, 56, 539-546 (2001) · Zbl 0972.78019
[16] V. S. Buslaev; A. I. Komech; E. A. Kopylova; D. Stuart, On asymptotic stability of solitary waves in Schrödinger quation coupled to nonlinear oscillaton, Commun. Partial Differ. Equ., 33, 669-705 (2008) · Zbl 1185.35247
[17] V. S. Buslaev; G. S. Perel’man, Scattering for the nonlinear Schrödinger equation: States that are close to a soliton, St. Petersburg Math. J., 4, 1111-1142 (1993)
[18] V. S. Buslaev and G. S. Perel’man, On the stability of solitary waves for nonlinear Schrödinger equations, in Nonlinear Evolution Equations, editor N.N. Uraltseva, Transl. Ser. 2, Amer. Math. Soc., Amer. Math. Soc., Providence, 164 (1995), 75-98. · Zbl 0841.35108
[19] V. S. Buslaev; C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 20, 419-475 (2003) · Zbl 1028.35139
[20] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, Courant Lecture Notes, American Mathematical Society, Providence, 2003. · Zbl 1055.35003
[21] S.-M. Chang; S. Gustafson; K. Nakanish; T.-P. Tsai, Spectra of linearized operators for NLS solitary waves, SIAM J. Math. Anal., 39, 1070-1111 (2007/08) · Zbl 1168.35041
[22] A. Comech, Solutions with compact time spectrum to nonlinear Klein-Gordon and Schrödinger equations and the Titchmarsh theorem for partial convolution, Arnold Math. J., 5, 315-338 (2019) · Zbl 1433.37067
[23] A. Comech and S. Cuccagna, On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations, preprint, arXiv: 1801.04079. · Zbl 1460.35321
[24] A. Comech; D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math., 56, 1565-1607 (2003) · Zbl 1072.35165
[25] O. Costin; M. Huang; W. Schlag, On the spectral properties of \(L_{\pm}\) in three dimensions, Nonlinearity, 25, 125-164 (2012) · Zbl 1232.35106
[26] S. Cuccagna, A survey on asymptotic stability of ground states of nonlinear Schrödinger equations, in Dispersive Nonlinear Problems in Mathematical Physics, 21-57, Quad. Mat., 15, Dept. Math., Seconda Univ. Napoli, Caserta, 2004. · Zbl 1130.35360
[27] S. Cuccagna, The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states, Comm. Math. Phys., 305, 279-331 (2011) · Zbl 1222.35183
[28] S. Cuccagna, On asymptotic stability of moving ground states of the nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 366, 2827-2888 (2014) · Zbl 1293.35289
[29] S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure App. Math., 54 (2001), 1110-1145. erratum Comm. Pure Appl. Math., 58 (2005), p. 147. · Zbl 1031.35129
[30] S. Cuccagna, On asymptotic stability of ground states of NLS, Rev. Math. Phys., 15, 877-903 (2003) · Zbl 1084.35089
[31] S. Cuccagna; M. Maeda, On weak interaction between a ground state and a non-trapping potential, J. Differential Eq., 256, 1395-1466 (2014) · Zbl 1285.35107
[32] S. Cuccagna; M. Maeda, On small energy stabilization in the NLS with a trapping potential, Anal. PDE, 8, 1289-1349 (2015) · Zbl 1326.35335
[33] S. Cuccagna; M. Maeda, On weak interaction between a ground state and a trapping potential, Discrete Contin. Dyn. Syst., 35, 3343-3376 (2015) · Zbl 1307.35276
[34] S. Cuccagna; M. Maeda, On orbital instability of spectrally stable vortices of the NLS in the plane, J. Nonlinear Sci., 26, 1851-1894 (2016) · Zbl 1360.35240
[35] S. Cuccagna; M. Maeda, On Nonlinear profile decompositions and scattering for an NLS-ODE model, Int. Math. Res. Not. IMRN, 2020, 5679-5722 (2020) · Zbl 1460.35322
[36] S. Cuccagna; M. Maeda, Long time oscillation of solutions of nonlinear Schrödinger equations near minimal mass ground state, J. Differential Equations, 268, 6416-6480 (2020) · Zbl 1439.35439
[37] S. Cuccagna; M. Maeda, On stability of small solitons of the 1-D NLS with a trapping delta potential, SIAM J. Math. Anal., 51, 4311-4331 (2019) · Zbl 1428.35430
[38] S. Cuccagna; M. Maeda; Tuoc V. Phan, On small energy stabilization in the NLKG with a trapping potential, Nonlinear Anal., 146, 32-58 (2016) · Zbl 1356.35142
[39] S. Cuccagna and M. Maeda, Coordinates at small energy and refined profiles for the Nonlinear Schrödinger Equation, preprint, arXiv: 2004.01366.
[40] S. Cuccagna; T. Mizumachi, On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations, Comm. Math. Phys., 284, 51-77 (2008) · Zbl 1155.35092
[41] S. Cuccagna; D. E. Pelinovsky, The asymptotic stability of solitons in the cubic NLS equation on the line, Appl. Anal., 93, 791-822 (2014) · Zbl 1457.35067
[42] S. Cuccagna; D. Pelinovsky; V. Vougalter, Spectra of positive and negative energies in the linearization of the NLS problem, Comm. Pure Appl. Math., 58, 1-29 (2005) · Zbl 1064.35181
[43] S. Cuccagna; M. Tarulli, On stabilization of small solutions in the nonlinear Dirac equation with a trapping potential, J. Math. Anal. Appl., 436, 1332-1368 (2016) · Zbl 1334.35264
[44] S. Cuccagna; M. Tarulli, On asymptotic stability in energy space of ground states of NLS in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26, 1361-1386 (2009) · Zbl 1171.35470
[45] S. Cuccagna; M. Tarulli, On asymptotic stability of standing waves of discrete Schrödinger equation in \(\Bbb Z\), SIAM J. Math. Anal., 41, 861-885 (2009) · Zbl 1189.35303
[46] K. Datchev; J. Holmer, Fast soliton scattering by attractive delta impurities, Comm. Partial Differential Equations, 34, 1074-1113 (2009) · Zbl 1194.35403
[47] S. De Bièvre, F. Genoud and S. Rota Nodari, Orbital stability: Analysis meets geometry, Nonlinear Optical and Atomic Systems, Lecture Notes in Math., Springer, Cham, 2146 (2015), 147-273. · Zbl 1347.37122
[48] P. Deift; X. Zhou, Perturbation theory for infinite-dimensional integrable systems on the line. A case study, Acta Math., 188, 163-262 (2002) · Zbl 1006.35089
[49] L. Demanet; W. Schlag, Numerical verification of a gap condition for a linearized nonlinear Schrödinger equation, Nonlinearity, 19, 829-852 (2006) · Zbl 1106.35044
[50] Q. Deng; A. Soffer; X. Yao, Soliton-potential interactions for nonlinear Schrödinger equation in \(\Bbb R^3\), SIAM J. Math. Anal., 50, 5243-5292 (2018) · Zbl 1428.35500
[51] T. Duyckaerts; C. Kenig; F. Merle, Classification of radial solutions of the focusing, energy-critical wave equation, Camb. J. Math., 1, 75-144 (2013) · Zbl 1308.35143
[52] V. Fleurov and A. Soffer, Soliton in a well. Dynamics and tunneling, preprint, arXiv: 1305.4279v1.
[53] J. Fröhlich; S. Gustafson; B. L. G. Jonsson; I. M. Sigal, Solitary wave dynamics in an external potential, Comm. Math. Phys., 250, 613-642 (2004) · Zbl 1075.35075
[54] Z. Gang, Perturbation expansion and N-th order fermi golden rule of the nonlinear Schrödinger equations, J. Math. Phys., 48 (2007), p. 053509. · Zbl 1144.81430
[55] Z. Gang; I. M. Sigal, Relaxation of solitons in nonlinear Schrödinger equations with potential, Adv. Math., 216, 443-490 (2007) · Zbl 1126.35065
[56] Z. Gang; I. M. Sigal, Asymptotic stability of nonlinear Schrödinger equations with potential, Rev. Math. Phys., 17, 1143-1207 (2005) · Zbl 1086.82013
[57] Z. Gang; M. I. Weinstein, Dynamics of nonlinear Schrödinger/Gross-Pitaeskii equations; Mass transfer in systems with solitons and degenerate neutral modes, Anal. PDE, 1, 267-322 (2008) · Zbl 1175.35136
[58] Z. Gang; M. I. Weinstein, Equipartition of mass in nonlinear Schrödinger/Gross-Pitaeskii equations, Appl. Math. Res. Express. AMRX, 2011, 123-181 (2011) · Zbl 1228.35224
[59] R. H. Goodman; J. L. Marzuola; M. I. Weinstein, Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger/Gross-Pitaevskii equation, Discrete Contin. Dyn. Syst., 35, 225-246 (2015) · Zbl 1304.35638
[60] M. Grillakis, Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system, Comm. Pure Appl. Math., 43, 299-333 (1990) · Zbl 0731.35010
[61] S. Gustafson; K. Nakanishi; T.-P. Tsai, Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves, Int. Math. Res. Not., 66, 3559-3584 (2004) · Zbl 1072.35167
[62] S. Gustafson; T. V. Phan, Stable directions for degenerate excited states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 43, 1716-1758 (2011) · Zbl 1230.35129
[63] P. Hagerty; A. M. Bloch; M. I. Weinstein, Radiation induced instability, SIAM J. Appl. Math., 64, 484-524 (2003/04) · Zbl 1071.34057
[64] A. Hoffman; J. D. Wright, Nanopteron solutions of diatomic Fermi-Pasta-Ulam-Tsingou lattices with small mass-ratio, Phys. D, 358, 33-59 (2017) · Zbl 1378.35067
[65] J. Holmer; J. Marzuola; M. Zworski, Soliton splitting by external delta potentials, J. Nonlinear Sci., 17, 349-367 (2007) · Zbl 1128.35384
[66] J. Holmer; J. Marzuola; M. Zworski, Fast soliton scattering by delta impurities, Comm. Math. Phys., 274, 187-216 (2007) · Zbl 1126.35068
[67] J. Holmer and M. Zworski, Soliton interaction with slowly varying potentials, Int. Math. Res. Not. IMRN, 2008 (2008), Art. ID rnn026, 36. · Zbl 1147.35084
[68] J. Holmer; M. Zworski, Slow soliton interaction with delta impurities, J. Mod. Dyn., 1, 689-718 (2007) · Zbl 1137.35060
[69] J. S. Howland, On the Weinstein-Aronszajn formula, Arch. Rational Mech. Anal., 39, 323-339 (1970) · Zbl 0225.47013
[70] J. S. Howland, Puiseux series for resonances at an embedded eigenvalue, Pacific J. Math., 55, 157-176 (1974) · Zbl 0312.47010
[71] B. L. G. Jonsson; J. Fröhlich; S. Gustafson; I. M. Sigal, Long time motion of NLS solitary waves in a confining potential, Ann. Henri Poincaré, 7, 621-660 (2006) · Zbl 1100.81019
[72] R. Johnson; X. B. Pan, On an elliptic equation related to the blow-up phenomenon in the nonlinear Schrödinger equation, Proc. Roy. Soc. Edinburgh Sect. A, 123, 763-782 (1993) · Zbl 0788.35041
[73] M. A. Johnson; J. D. Wright, Generalized solitary waves in the gravity-capillary Whitham equation, Stud. Appl. Math., 144, 102-130 (2020) · Zbl 1454.35285
[74] J.-L. Journé; A. Soffer; C. D. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44, 573-604 (1991) · Zbl 0743.35008
[75] T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162, 258-279 (1965/1966) · Zbl 0139.31203
[76] M. Keel; T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120, 955-980 (1998) · Zbl 0922.35028
[77] E. Kirr; P. G. Kevrekidis; D. E. Pelinovsky., Symmetry-breaking bifurcation in Nonlinear Schrödinger equation with symmetric potentials, Comm. Math. Phys., 308, 795-844 (2011) · Zbl 1235.34128
[78] E. W. Kirr; P. G. Kevrekidis; E. Shlizerman; M. I. Weinstein, Symmetry-breaking bifurcation in nonlinear Schrödinger/Gross-Pitaevskii equations, SIAM J. Math. Anal., 40, 566-604 (2008) · Zbl 1157.35479
[79] E. Kirr; Ö. Mizrak, Asymptotic stability of ground states in 3D nonlinear Schrödinger equation including subcritical cases, J. Funct. Anal., 257, 3691-3747 (2009) · Zbl 1187.35238
[80] E. Kirr and V. Natarajan, The global bifurcation picture for ground states in nonlinear Schr ödinger equations, preprint, arXiv: 1811.05716.
[81] E. Kirr; A. Zarnescu, Asymptotic stability of ground states in 2D nonlinear Schr ödinger equation including subcritical cases, J. Differential Equations, 247, 710-735 (2009) · Zbl 1171.35112
[82] A. I. Komech, On attractor of a singular nonlinear U(1)-invariant Klein-Gordon equation, in Progress in Analysis, Vol. I, II (Berlin, 2001), pp. 599-611, World Sci. Publ., River Edge, NJ, 2003. · Zbl 1060.35022
[83] A. Komech; A. Komech, Global attraction to solitary waves for Klein-Gordon equation with mean field interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26, 855-868 (2009) · Zbl 1177.35201
[84] A. Komech; A. Komech, Global attraction to solitary waves for a nonlinear Dirac equation with mean field interaction, SIAM J. Math. Anal., 42, 2944-2964 (2010) · Zbl 1237.35022
[85] A. Komech; A. Komech, On global attraction to solitary waves for the Klein-Gordon field coupled to several nonlinear oscillators, J. Math. Pures Appl., 93, 91-111 (2010) · Zbl 1180.35124
[86] A. Komech; E. Kopylova; D. Stuart, On asymptotic stability of solitons in a nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 11, 1063-1079 (2012) · Zbl 1282.35352
[87] M. Kowalczyk; Y. Martel; C. Muñoz, Kink dynamics in the \(\phi^4\) model: Asymptotic stability for odd perturbations in the energy space, J. Amer. Math. Soc., 30, 769-798 (2017) · Zbl 1387.35419
[88] M. Kowalczyk, Y. Martel and C. Muñoz, Soliton dynamics for the 1D NLKG equation with symmetry and in the absence of internal modes, Jour. of the Europ. Math. Soc., to appear.
[89] M. Kowalczyk, Y. Marte, C. Muñoz and H. Van Den Bosch, A sufficient condition for asymptotic stability of kinks in general \((1+1)\)-scalar field models, preprint, arXiv: 2008.01276.
[90] J. Krieger; W. Schlag, Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension, J. Amer. Math. Soc., 19, 815-920 (2006) · Zbl 1281.35077
[91] C. J. Lustri, Nanoptera and Stokes curves in the 2-periodic Fermi-Pasta-Ulam-Tsingou equation, Phys. D, 402 (2020), 132239, 13 pp. · Zbl 1453.82045
[92] C. J. Lustri; M. A. Porter, Nanoptera in a period-2 Toda chain, SIAM J. Appl. Dyn. Syst., 17, 1182-1212 (2018) · Zbl 1407.34020
[93] Y. Martel; F. Merle, Asymptotic stability of solitons of the gKdV equations with general nonlinearity, Math. Ann., 341, 391-427 (2008) · Zbl 1153.35068
[94] J. L. Marzuola; S. Raynor; G. Simpson, A system of ODEs for a perturbation of a minimal mass soliton, J. Nonlinear Sci., 20, 425-461 (2010) · Zbl 1205.35295
[95] J. L. Marzuola; G. Simpson, Spectral analysis for matrix Hamiltonian operators, Nonlinearity, 24, 389-429 (2011) · Zbl 1213.35371
[96] J. L. Marzuola; M. I. Weinstein, Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations, Discrete Contin. Dyn. Syst., 28, 1505-1554 (2010) · Zbl 1223.35288
[97] S. Masaki; J. Murphy; J. Segata, Stability of small solitary waves for the \(1d\) NLS with an attractive delta potential, Anal. PDE, 13, 1099-1128 (2020) · Zbl 1447.35299
[98] S. Masaki; J. Murphy; J. Segata, Modified scattering for the one-dimensional cubic NLS with a repulsive delta potential, Int. Math. Res. Not. IMRN, 2019, 7577-7603 (2019)
[99] S. Masaki, J. Murphy and J. Segata, Asymptotic stability of solitary waves for the \(1d\) NLS with an attractive delta potential, preprint, arXiv: 2008.11645. · Zbl 1447.35299
[100] F. Merle; P. Raphael, On a sharp lower bound on the blow-up rate for the \(L^2\) critical nonlinear Schrödinger equation, J. Amer. Math. Soc., 19, 37-90 (2006) · Zbl 1075.35077
[101] F. Merle; P. Raphael, Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation, Geom. Funct. Anal., 13, 591-642 (2003) · Zbl 1061.35135
[102] F. Merle; P. Raphael, The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. of Math., 161, 157-222 (2005) · Zbl 1185.35263
[103] T. Mizumachi, Asymptotic stability of small solitons to 1D nonlinear Schr?dinger equations with potential, J. Math. Kyoto Univ., 48, 471-497 (2008) · Zbl 1175.35138
[104] T. Mizumachi, Asymptotic stability of small solitons for 2D nonlinear Schrödinger equations with potential, J. Math. Kyoto Univ., 47, 599-620 (2007) · Zbl 1146.35085
[105] C. Munoz, Sharp inelastic character of slowly varying NLS solitons, preprint, arXiv: 1202.5807.
[106] C. Muñoz, On the soliton dynamics under slowly varying medium, for nonlinear Schrödinger equations, Math. Ann., 353, 867-943 (2012) · Zbl 1291.35264
[107] K. Nakanishi, Global dynamics below excited solitons for the nonlinear Schrödinger equation with a potential, J. Math. Soc. Japan, 69, 1353-1401 (2017) · Zbl 1383.35213
[108] K. Nakanishi; T. V. Phan; T.-P. Tsai, Small solutions of nonlinear Schrödinger equations near first excited states, J. Funct. Anal., 263, 703-781 (2012) · Zbl 1244.35136
[109] I. Naumkin; P. Raphaël, On travelling waves of the non linear Schrödinger equation escaping a potential well, Ann. Henri Poincaré, 21, 1677-1758 (2020) · Zbl 1437.35194
[110] D. E. Pelinovsky; Y. S. Kivshar; V. V. Afanasjev, Internal modes of envelope solitons, Phys. D, 116, 121-142 (1998) · Zbl 0934.35175
[111] G. Perelman, Two soliton collision for nonlinear Schrödinger equations in dimension 1, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28, 357-384 (2011) · Zbl 1217.35176
[112] G. Perelman, A remark on soliton-potential interactions for nonlinear Schrödinger equations, Math. Res. Lett., 16, 477-486 (2009) · Zbl 1172.81012
[113] G. Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, Comm. Partial Differential Equations, 29, 1051-1095 (2004) · Zbl 1067.35113
[114] G. Perelman, On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré, 2, 605-673 (2001) · Zbl 1007.35087
[115] C.-A. Pillet; C. E. Wayne, Invariant manifolds for a class of dispersive, Hamiltonian partial differential equations, J. Differential Equations, 141, 310-326 (1997) · Zbl 0890.35016
[116] I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of N-soliton states of NLS, preprint, arXiv: math/0309114v1. · Zbl 1130.81053
[117] A. Saalmann, Asymptotic stability of \(N\)-solitons in the cubic NLS equation, J. Hyperbolic Differ. Equ., 14, 455-485 (2017) · Zbl 1384.35050
[118] W. Schlag, Stable manifolds for an orbitally unstable NLS, Ann. of Math., 169, 139-227 (2009) · Zbl 1180.35490
[119] I. M. Sigal, Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions, Comm. Math. Phys., 153, 297-320 (1993) · Zbl 0780.35106
[120] A. Soffer; M. I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys., 133, 119-146 (1990) · Zbl 0721.35082
[121] A. Soffer; M. I. Weinstein, Multichannel nonlinear scattering II. The case of anisotropic potentials and data, J. Differential Equations, 98, 376-390 (1992) · Zbl 0795.35073
[122] A. Soffer; M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math., 136, 9-74 (1999) · Zbl 0910.35107
[123] A. Soffer; M. I. Weinstein, Selection of the ground state for nonlinear Schrödinger equations, Rev. Math. Phys., 16, 977-1071 (2004) · Zbl 1111.81313
[124] C. A. Stuart, Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation, Milan J. Math., 76, 329-399 (2008) · Zbl 1179.37101
[125] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Applied Mathematica Sciences vol. 139, 1999, Springer, New York. · Zbl 0867.35094
[126] S. M. Sun, Existence of a generalized solitary wave solution for water with positive Bond number less than \(1/3\), J. Math. Anal. Appl., 156, 471-504 (1991) · Zbl 0725.76028
[127] T.-P. Tsai, Asymptotic dynamics of nonlinear Schrödinger equations with many bound states, J. Differential Equations, 192, 225-282 (2003) · Zbl 1038.35128
[128] T.-P. Tsai; H.-T. Yau, Asymptotic dynamics of nonlinear Schrödinger equations: Resonance dominated and radiation dominated solutions, Comm. Pure Appl. Math., 55, 153-216 (2002) · Zbl 1031.35137
[129] T.-P. Tsai; H.-T. Yau, Relaxation of excited states in nonlinear Schrödinger equations, Int. Math. Res. Not., 2002, 1629-1673 (2002) · Zbl 1011.35120
[130] T.-P. Tsai; H.-T. Yau, Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data, Adv. Theor. Math. Phys., 6, 107-139 (2002) · Zbl 1033.81034
[131] T.-P. Tsai; H.-T. Yau, Stable directions for excited states of nonlinear Schrödinger equations, Comm. Partial Diffierential Equations, 27, 2363-2402 (2002) · Zbl 1021.35113
[132] R. Weder, The \(W_{k, p}\)-continuity of the Schrödinger wave operators on the line, Comm. Math. Phys., 208, 507-520 (1999) · Zbl 0945.34070
[133] R. Weder, \(L^p-L^{\dot p}\) estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential, J. Funct. Anal., 170, 37-68 (2000) · Zbl 0943.34070
[134] M. I. Weinstein, Modulation stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16, 472-491 (1985) · Zbl 0583.35028
[135] K. Yajima, The \(W^{k, p}\)-continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan, 47, 551-581 (1995) · Zbl 0837.35039
[136] K. Yajima, The \(W^{k, p}\)-continuity of wave operators for Schrödinger operators III, J. Math. Sci. Univ. Tokyo, 2, 311-346 (1995) · Zbl 0841.47009
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