Available in electronic format
Available in print format		 Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN: 1088-6834(e) ISSN: 0894-0347(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension

Author(s): J. Krieger; W. Schlag
Journal: J. Amer. Math. Soc. 19 (2006), 815-920.
MSC (2000): Primary 35Q55, 35Q51, 37K40, 37K45
Posted: February 20, 2006
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Standing wave solutions of the one-dimensional nonlinear Schrödinger equation

$\displaystyle i\partial_t \psi + \partial_{x}^2 \psi = -\vert\psi\vert^{2\sigma} \psi $

with $ \sigma>2$ are well known to be unstable. In this paper 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 Non-generic blow-up solutions for the critical focusing NLS in 1-d, preprint, 2005.

References:

1.
Artbazar, G., Yajima, K. The $ L\sp p$-continuity of wave operators for one dimensional Schrödinger operators. J. Math. Sci. Univ. Tokyo 7 (2000), no. 2, 221-240. MR 1768465 (2001f:34166)

2.
Bates, P. W., Jones, C. K. R. T. Invariant manifolds for semilinear partial differential equations. Dynamics reported, Vol. 2, 1-38, Dynam. Report. Ser. Dynam. Systems Appl., 2, Wiley, Chichester, 1989. MR 1000974 (90g:58017)

3.
Berestycki, H., Cazenave, T. 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. MR 0646873 (84f:35120)

4.
Bourgain, J., Wang, W., Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity, Annali Scuola Ecole Normale Cl.(4) 25 (1997), no.1-2, 197-215. MR 1655515 (99m:35219)

5.
Buslaev, V. S., Perelman, G. S. Scattering for the nonlinear Schrödinger equation: States that are close to a soliton. (Russian) Algebra i Analiz 4 (1992), no. 6, 63-102; translation in St. Petersburg Math. J. 4 (1993), no. 6, 1111-1142.MR 1199635 (94b:35256)

6.
Buslaev, V. S., Perelman, G. S. On the stability of solitary waves for nonlinear Schrödinger equations. Nonlinear evolution equations, 75-98, Amer. Math. Soc. Transl. Ser. 2, 164, Amer. Math. Soc., Providence, RI, 1995.MR 1334139 (96e:35157)

7.
Cazenave, T., Lions, P.-L. Orbital stability of standing waves for some nonlinear Schrödinger equations. Comm. Math. Phys. 85 (1982), 549-561.MR 0677997 (84i:81015)

8.
Christ, M., Kiselev, A. Maximal functions associated with filtrations, J. Funct. Anal. 179 (2001), 409-425. MR 1809116 (2001i:47054)

9.
Comech, A., Pelinovsky, D. Purely nonlinear instability of standing waves with minimal energy. Comm. Pure Appl. Math. 56 (2003), no. 11, 1565-1607.MR 1995870 (2005h:37176)

10.
Cuccagna, S. Stabilization of solutions to nonlinear Schrödinger equations. Comm. Pure Appl. Math. 54 (2001), no. 9, 1110-1145.MR 1835384 (2002g:35193)

11.
Cuccagna, S., Pelinovsky, D. Bifurcations from the endpoints of the essential spectrum in the linearized nonlinear Schrödinger problem, J. Math. Phys. 46 (2005), no. 5.MR 2143030 (2005m:35279)

12.
Cuccagna, S., Pelinovsky, D., Vougalter, V. Spectra of positive and negative energies in the linearized NLS problem. Comm. Pure Appl. Math. 58 (2005), no. 1, 1-29. MR 2094265 (2005k:35374)

13.
Demanet, L., Schlag, W. Numerical verification of a gap condition for a linearized NLS equation, preprint, 2005, to appear in Nonlinearity.

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.
Flügge, S. Practical quantum mechanics. Reprinting of Vols. I, II in one volume. Springer-Verlag, New York-Heidelberg, 1974.MR 0366248 (51:2496)

16.
Fröhlich, J., Gustafson, S., Jonsson, B. L. G., Sigal, I. M., Solitary wave dynamics in an external potential. Comm. Math. Phys. 250 (2004), no. 3, 613-642. MR 2094474 (2005h:35320)

17.
Fröhlich, J., Tsai, T. P., Yau, H. T. On the point-particle (Newtonian) limit of the non-linear Hartree equation. Comm. Math. Phys. 225 (2002), no. 2, 223-274.MR 1889225 (2003e:81047)

18.
Gang, Z., Sigal, I. M. Asymptotic Stability of Nonlinear Schrödinger Equations with Potential, preprint, 2005, to appear in Reviews in Mathematical Physics.

19.
Gang, Z., Sigal, I. M. Relaxation to Trapped Solitons in Nonlinear Schrödinger Equations with Potential, preprint, 2006.

20.
Gesztesy, F., Jones, C. K. R. T., Latushkin, Y., Stanislavova, M. A spectral mapping theorem and invariant manifolds for nonlinear Schrödinger equations. Indiana Univ. Math. J. 49 (2000), no. 1, 221-243.MR 1777032 (2001g:37144)

21.
Grillakis, M. Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system. Comm. Pure Appl. Math. 43 (1990), no. 3, 299-333.MR 1040143 (91d:58231)

22.
Grillakis, M., Shatah, J., Strauss, W. Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74 (1987), no. 1, 160-197. MR 0901236 (88g:35169)

23.
Grillakis, M., Shatah, J., Strauss, W. Stability theory of solitary waves in the presence of symmetry. II. J. Funct. Anal. 94 (1990), 308-348. MR 1081647 (92a:35135)

24.
Goldberg, M., Schlag, W. Dispersive estimates for Schrödinger operators in dimensions one and three. Comm. Math. Phys. 251 (2004), no. 1, 157-178. MR 2096737 (2005g:81339)

25.
Hartman, P. Ordinary differential equations. Corrected reprint of the second (1982) edition. Classics in Applied Mathematics, 38. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. MR 1929104 (2003h:34001)

26.
Hislop, P. D., Sigal, I. M. Introduction to spectral theory. With applications to Schrödinger operators. Applied Mathematical Sciences, 113. Springer-Verlag, New York, 1996. MR 1361167 (98h:47003)

27.
Hundertmark, D., Lee, Y. R. Exponential decay of eigenfunctions and generalized eigenfunctions of non-selfadjoint matrix Schrödinger operators related to NLS, preprint, 2005.

28.
Kato, T. Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162 (1965/1966), 258-279. MR 0190801 (32:8211)

29.
Krieger, J., Schlag, W. Non-generic blow-up solutions for the critical focusing NLS in 1-d, preprint, 2005.

30.
Li, C., Wiggins, S. Invariant manifolds and fibrations for perturbed nonlinear Schrödinger equations. Applied Mathematical Sciences, 128. Springer-Verlag, New York, 1997. MR 1475929 (99e:58165)

31.
Merle, F., Raphael, P. Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation. Geom. Funct. Anal. 13 (2003), no. 3, 591-642.MR 1995801 (2005j:35207)

32.
Merle, F., Raphael, P. On universality of blow-up profile for $ L\sp 2$ critical nonlinear Schrödinger equation. Invent. Math. 156 (2004), no. 3, 565-672.MR 2061329 (2006a:35283)

33.
Merle, F., Raphael, P. On a sharp lower bound on the blow-up rate for the $ L\sp 2$ critical nonlinear Schrödinger equation. J. Amer. Math. Soc. 19 (2006), no. 1, 37-90.MR 2169042

34.
Murata, M. Asymptotic expansions in time for solutions of Schrödinger-type equations. J. Funct. Anal. 49 (1) (1982), 10-56. MR 0680855 (85d:35019)

35.
Perelman, G. Some Results on the Scattering of Weakly Interacting Solitons for Nonlinear Schrödinger Equations. In ``Spectral theory, microlocal analysis, singular manifolds", Akad. Verlag (1997), 78-137. MR 1608275 (99h:35203)

36.
Perelman, G. On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. Henri Poincaré 2 (2001), no. 4, 605-673. MR 1852922 (2002m:35205)

37.
Perelman, G. Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations. Comm. Partial Differential Equations 29 (2004), no. 7-8, 1051-1095.MR 2097576 (2005g:35277)

38.
Pillet, C. A., Wayne, C. E. Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations. J. Diff. Eq. 141 (1997), no. 2, 310-326. MR 1488355 (99b:35193)

39.
Raphael, P. 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. MR 2122541 (2006b:35303)

40.
Reed, M., Simon, B. Methods of modern mathematical physics. IV. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 0493421 (58:12429c)

41.
Rodnianski, I., Schlag, W. Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155 (2004), no. 3, 451-513. MR 2038194 (2005h:35295)

42.
Rodnianski, I., Schlag, W., Soffer, A. Dispersive Analysis of Charge Transfer Models, Comm. Pure Appl. Math. 58 (2005), no. 2, 149-216. MR 2094850 (2005i:81181)

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.

46.
Shatah, J. Stable standing waves of nonlinear Klein-Gordon equations. Comm. Math. Phys. 91 (1983), no. 3, 313-327. MR 0723756 (84m:35111)

47.
Shatah, J., Strauss, W. Instability of nonlinear bound states. Comm. Math. Phys. 100 (1985), no. 2, 173-190. MR 0804458 (87b:35159)

48.
Smith, H., Sogge, C. Global Strichartz estimates for nontrapping perturbations of the Laplacian. Comm. Partial Differential Equations 25 (2000), no. 11-12, 2171-2183. MR 1789924 (2001j:35180)

49.
Soffer, A., Weinstein, M. Multichannel nonlinear scattering for nonintegrable equations. Comm. Math. Phys. 133 (1990), 119-146. MR 1071238 (91h:35303)

50.
Soffer, A., Weinstein, M. Multichannel nonlinear scattering, II. The case of anisotropic potentials and data. J. Diff. Eq. 98 (1992), 376-390.MR 1170476 (93i:35137)

51.
Strauss, W. A. Nonlinear wave equations. CBMS Regional Conference Series in Mathematics, 73. American Mathematical Society, Providence, RI, 1989.MR 1032250 (91g:35002)

52.
Sulem, C., Sulem, P.-L. The nonlinear Schrödinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999.MR 1696311 (2000f:35139)

53.
Tsai, T.-P., Yau, H.-T. Stable directions for excited states of nonlinear Schrödinger equations. Comm. Partial Differential Equations 27 (2002), no. 11-12, 2363-2402. MR 1944033 (2004k:35359)

54.
Weder, R. The $ W\sb {k,p}$-continuity of the Schrödinger wave operators on the line. Comm. Math. Phys. 208 (1999), no. 2, 507-520.MR 1729096 (2001c:34178)

55.
Weinstein, Michael I. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16 (1985), no. 3, 472-491. MR 0783974 (86i:35130)

56.
Weinstein, Michael I. Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. 39 (1986), no. 1, 51-67.MR 0820338 (87f:35023)

Similar Articles:

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 35Q55, 35Q51, 37K40, 37K45

Retrieve articles in all Journals with MSC (2000): 35Q55, 35Q51, 37K40, 37K45

Additional Information:

J. Krieger
Affiliation: Department of Mathematics, Harvard University, Science Center, 1 Oxford Street, Cambridge, Massachusetts 02138
Email: jkrieger@math.harvard.edu

W. Schlag
Affiliation: Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago, Illinois 60637
Email: schlag@math.uchicago.edu

DOI: 10.1090/S0894-0347-06-00524-8
PII: S 0894-0347(06)00524-8
Keywords: Critical Schr\"odinger equation, stable manifolds, modulation theory, spectral theory
Received by editor(s): January 13, 2005
Posted: February 20, 2006
Additional Notes: The first author was partially supported by the NSF grant DMS-0401177. He also wishes to thank Caltech, where part of this work was done.
The second author was partially supported by the NSF grant DMS-0300081 and a Sloan fellowship.
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google