## Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension.(English)Zbl 1281.35077

Summary: Standing wave solutions of the one-dimensional nonlinear Schrödinger equation $\partial_t \psi + \partial_{x}^2 \psi =-|\psi|^{2\sigma} \psi$ with $$\sigma>2$$ are well known to be unstable. We show that asymptotic stability can be achieved provided the perturbations of these standing waves are small and chosen to belong to a codimension one Lipschitz surface. Thus, we construct codimension one asymptotically stable manifolds for all supercritical NLS in one dimension. The considerably more difficult $$L^2$$-critical case, for which one wishes to understand the conditional stability of the pseudo-conformal blow-up solutions, is studied in the authors’ companion paper [J. Eur. Math. Soc. (JEMS) 11, No. 1, 1–125 (2009; Zbl 1163.35035)].

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q51 Soliton equations 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems 37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems

Zbl 1163.35035
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### References:

 [1] Galtbayar Artbazar and Kenji Yajima, The \?^{\?}-continuity of wave operators for one dimensional Schrödinger operators, J. Math. Sci. Univ. Tokyo 7 (2000), no. 2, 221 – 240. · Zbl 0976.34071 [2] Peter W. Bates and Christopher K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics reported, Vol. 2, Dynam. Report. Ser. Dynam. Systems Appl., vol. 2, Wiley, Chichester, 1989, pp. 1 – 38. · Zbl 0674.58024 [3] Henri Berestycki and Thierry Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 9, 489 – 492 (French, with English summary). · Zbl 0492.35010 [4] Jean Bourgain and W. Wang, Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 197 – 215 (1998). Dedicated to Ennio De Giorgi. · Zbl 1043.35137 [5] V. S. Buslaev and G. S. Perel$$^{\prime}$$man, Scattering for the nonlinear Schrödinger equation: states that are close to a soliton, Algebra i Analiz 4 (1992), no. 6, 63 – 102 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 6, 1111 – 1142. · Zbl 0853.35112 [6] V. S. Buslaev and G. S. Perel$$^{\prime}$$man, On the stability of solitary waves for nonlinear Schrödinger equations, Nonlinear evolution equations, Amer. Math. Soc. Transl. Ser. 2, vol. 164, Amer. Math. Soc., Providence, RI, 1995, pp. 75 – 98. · Zbl 0841.35108 [7] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), no. 4, 549 – 561. · Zbl 0513.35007 [8] Michael Christ and Alexander Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), no. 2, 409 – 425. · Zbl 0974.47025 [9] Andrew Comech and Dmitry Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math. 56 (2003), no. 11, 1565 – 1607. · Zbl 1072.35165 [10] Scipio Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math. 54 (2001), no. 9, 1110 – 1145. · Zbl 1031.35129 [11] Scipio Cuccagna and Dmitry Pelinovsky, Bifurcations from the endpoints of the essential spectrum in the linearized nonlinear Schrödinger problem, J. Math. Phys. 46 (2005), no. 5, 053520, 15. · Zbl 1110.35082 [12] Scipio Cuccagna, Dmitry Pelinovsky, and Vitali Vougalter, Spectra of positive and negative energies in the linearized NLS problem, Comm. Pure Appl. Math. 58 (2005), no. 1, 1 – 29. · Zbl 1064.35181 [13] Demanet, L., Schlag, W. Numerical verification of a gap condition for a linearized NLS equation, preprint, 2005, to appear in Nonlinearity. · Zbl 1106.35044 [14] Erdogan, M. B., Schlag, W. Dispersive estimates in the presence of a resonances and/or an eigenvalue at zero energy in dimension three: II, preprint, 2005, to appear in Journal d’Analyse. [15] Siegfried Flügge, Practical quantum mechanics, Springer-Verlag, New York-Heidelberg, 1974. Reprinting in one volume of Vols. I, II. · Zbl 0934.81001 [16] J. Fröhlich, S. Gustafson, B. L. G. Jonsson, and I. M. Sigal, Solitary wave dynamics in an external potential, Comm. Math. Phys. 250 (2004), no. 3, 613 – 642. · Zbl 1075.35075 [17] Jürg Fröhlich, Tai-Peng Tsai, and Horng-Tzer Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation, Comm. Math. Phys. 225 (2002), no. 2, 223 – 274. · Zbl 1025.81015 [18] Gang, Z., Sigal, I. M. Asymptotic Stability of Nonlinear Schrödinger Equations with Potential, preprint, 2005, to appear in Reviews in Mathematical Physics. · Zbl 1086.82013 [19] Gang, Z., Sigal, I. M. Relaxation to Trapped Solitons in Nonlinear Schrödinger Equations with Potential, preprint, 2006. [20] F. Gesztesy, C. K. R. T. Jones, Y. Latushkin, and M. Stanislavova, A spectral mapping theorem and invariant manifolds for nonlinear Schrödinger equations, Indiana Univ. Math. J. 49 (2000), no. 1, 221 – 243. · Zbl 0969.35123 [21] Manoussos Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure Appl. Math. 43 (1990), no. 3, 299 – 333. · Zbl 0731.35010 [22] Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), no. 1, 160 – 197. · Zbl 0656.35122 [23] Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal. 94 (1990), no. 2, 308 – 348. · Zbl 0711.58013 [24] M. Goldberg and W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys. 251 (2004), no. 1, 157 – 178. · Zbl 1086.81077 [25] Philip Hartman, Ordinary differential equations, Classics in Applied Mathematics, vol. 38, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Corrected reprint of the second (1982) edition [Birkhäuser, Boston, MA; MR0658490 (83e:34002)]; With a foreword by Peter Bates. · Zbl 1009.34001 [26] P. D. Hislop and I. M. Sigal, Introduction to spectral theory, Applied Mathematical Sciences, vol. 113, Springer-Verlag, New York, 1996. With applications to Schrödinger operators. · Zbl 0855.47002 [27] Hundertmark, D., Lee, Y. R. Exponential decay of eigenfunctions and generalized eigenfunctions of non-selfadjoint matrix Schrödinger operators related to NLS, preprint, 2005. · Zbl 1155.35065 [28] Tosio Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann. 162 (1965/1966), 258 – 279. · Zbl 0139.31203 [29] Krieger, J., Schlag, W. Non-generic blow-up solutions for the critical focusing NLS in 1-d, preprint, 2005. · Zbl 1163.35035 [30] Charles Li and Stephen Wiggins, Invariant manifolds and fibrations for perturbed nonlinear Schrödinger equations, Applied Mathematical Sciences, vol. 128, Springer-Verlag, New York, 1997. · Zbl 0897.35070 [31] F. Merle and P. Raphael, Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation, Geom. Funct. Anal. 13 (2003), no. 3, 591 – 642. · Zbl 1061.35135 [32] Frank Merle and Pierre Raphael, On universality of blow-up profile for \?² critical nonlinear Schrödinger equation, Invent. Math. 156 (2004), no. 3, 565 – 672. · Zbl 1067.35110 [33] Frank Merle and Pierre Raphael, On a sharp lower bound on the blow-up rate for the \?² critical nonlinear Schrödinger equation, J. Amer. Math. Soc. 19 (2006), no. 1, 37 – 90. · Zbl 1075.35077 [34] Minoru Murata, Asymptotic expansions in time for solutions of Schrödinger-type equations, J. Funct. Anal. 49 (1982), no. 1, 10 – 56. · Zbl 0499.35019 [35] Galina Perelman, Some results on the scattering of weakly interacting solitons for nonlinear Schrödinger equations, Spectral theory, microlocal analysis, singular manifolds, Math. Top., vol. 14, Akademie Verlag, Berlin, 1997, pp. 78 – 137. · Zbl 0931.35164 [36] Galina Perelman, On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré 2 (2001), no. 4, 605 – 673. · Zbl 1007.35087 [37] Galina Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, Comm. Partial Differential Equations 29 (2004), no. 7-8, 1051 – 1095. · Zbl 1067.35113 [38] Claude-Alain Pillet and C. Eugene Wayne, Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations, J. Differential Equations 141 (1997), no. 2, 310 – 326. · Zbl 0890.35016 [39] Pierre Raphael, Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation, Math. Ann. 331 (2005), no. 3, 577 – 609. · Zbl 1082.35143 [40] Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. · Zbl 0242.46001 [41] Igor Rodnianski and Wilhelm Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155 (2004), no. 3, 451 – 513. · Zbl 1063.35035 [42] Igor Rodnianski, Wilhelm Schlag, and Avraham Soffer, Dispersive analysis of charge transfer models, Comm. Pure Appl. Math. 58 (2005), no. 2, 149 – 216. · Zbl 1130.81053 [43] Rodnianski, I., Schlag, W., Soffer, A. Asymptotic stability of $$N$$-soliton states of NLS, preprint, 2003. [44] Schlag, W. Stable manifolds for an orbitally unstable NLS. Preprint, 2004, to appear in Annals of Math. [45] Schlag, W. Dispersive estimates for Schrödinger operators: A survey. Preprint, 2004, to appear in “Mathematical Aspects of Nonlinear Dispersive Equations”, Princeton University Press. · Zbl 1143.35001 [46] Jalal Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys. 91 (1983), no. 3, 313 – 327. · Zbl 0539.35067 [47] Jalal Shatah and Walter Strauss, Instability of nonlinear bound states, Comm. Math. Phys. 100 (1985), no. 2, 173 – 190. · Zbl 0603.35007 [48] Hart F. Smith and Christopher D. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. Partial Differential Equations 25 (2000), no. 11-12, 2171 – 2183. · Zbl 0972.35014 [49] A. Soffer and M. I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys. 133 (1990), no. 1, 119 – 146. · Zbl 0721.35082 [50] A. Soffer and M. I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data, J. Differential Equations 98 (1992), no. 2, 376 – 390. · Zbl 0795.35073 [51] Walter A. Strauss, Nonlinear wave equations, CBMS Regional Conference Series in Mathematics, vol. 73, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1989. · Zbl 0714.35003 [52] Catherine Sulem and Pierre-Louis Sulem, The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999. Self-focusing and wave collapse. · Zbl 0928.35157 [53] Tai-Peng Tsai and Horng-Tzer Yau, Stable directions for excited states of nonlinear Schrödinger equations, Comm. Partial Differential Equations 27 (2002), no. 11-12, 2363 – 2402. · Zbl 1021.35113 [54] Ricardo Weder, The \?_{\?,\?}-continuity of the Schrödinger wave operators on the line, Comm. Math. Phys. 208 (1999), no. 2, 507 – 520. · Zbl 0945.34070 [55] Michael I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), no. 3, 472 – 491. · Zbl 0583.35028 [56] Michael I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986), no. 1, 51 – 67. · Zbl 0594.35005
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