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

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- by J. Krieger and W. Schlag PDF
- J. Amer. Math. Soc.
**19**(2006), 815-920 Request permission

## Abstract:

Standing wave solutions of the one-dimensional nonlinear Schrödinger equation \[ i\partial _t \psi + \partial _{x}^2 \psi = -|\psi |^{2\sigma } \psi \] 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

- Galtbayar Artbazar and Kenji Yajima,
*The $L^p$-continuity of wave operators for one dimensional Schrödinger operators*, J. Math. Sci. Univ. Tokyo**7**(2000), no. 2, 221–240. MR**1768465** - 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. MR**1000974** - 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). MR**646873** - 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. MR**1655515** - V. S. Buslaev and G. S. Perel′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. MR**1199635** - V. S. Buslaev and G. S. Perel′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. MR**1334139**, DOI 10.1090/trans2/164/04 - 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. MR**677997**, DOI 10.1007/BF01403504 - Michael Christ and Alexander Kiselev,
*Maximal functions associated to filtrations*, J. Funct. Anal.**179**(2001), no. 2, 409–425. MR**1809116**, DOI 10.1006/jfan.2000.3687 - Andrew Comech and Dmitry Pelinovsky,
*Purely nonlinear instability of standing waves with minimal energy*, Comm. Pure Appl. Math.**56**(2003), no. 11, 1565–1607. MR**1995870**, DOI 10.1002/cpa.10104 - Scipio Cuccagna,
*Stabilization of solutions to nonlinear Schrödinger equations*, Comm. Pure Appl. Math.**54**(2001), no. 9, 1110–1145. MR**1835384**, DOI 10.1002/cpa.1018 - 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. MR**2143030**, DOI 10.1063/1.1901345 - 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. MR**2094265**, DOI 10.1002/cpa.20050
Demanet Demanet, L., Schlag, W. - Siegfried Flügge,
*Practical quantum mechanics*, Springer-Verlag, New York-Heidelberg, 1974. Reprinting in one volume of Vols. I, II. MR**0366248** - 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. MR**2094474**, DOI 10.1007/s00220-004-1128-1 - 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. MR**1889225**, DOI 10.1007/s002200100579
SZ1 Gang, Z., Sigal, I. M. - 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. MR**1777032**, DOI 10.1512/iumj.2000.49.1838 - 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. MR**1040143**, DOI 10.1002/cpa.3160430302 - 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. MR**901236**, DOI 10.1016/0022-1236(87)90044-9 - 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. MR**1081647**, DOI 10.1016/0022-1236(90)90016-E - M. Goldberg and W. Schlag,
*Dispersive estimates for Schrödinger operators in dimensions one and three*, Comm. Math. Phys.**251**(2004), no. 1, 157–178. MR**2096737**, DOI 10.1007/s00220-004-1140-5 - 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. MR**1929104**, DOI 10.1137/1.9780898719222 - 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. MR**1361167**, DOI 10.1007/978-1-4612-0741-2
HL Hundertmark, D., Lee, Y. R. - Tosio Kato,
*Wave operators and similarity for some non-selfadjoint operators*, Math. Ann.**162**(1965/66), 258–279. MR**190801**, DOI 10.1007/BF01360915
KS Krieger, J., Schlag, W. - Charles Li and Stephen Wiggins,
*Invariant manifolds and fibrations for perturbed nonlinear Schrödinger equations*, Applied Mathematical Sciences, vol. 128, Springer-Verlag, New York, 1997. MR**1475929**, DOI 10.1007/978-1-4612-1838-8 - 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. MR**1995801**, DOI 10.1007/s00039-003-0424-9 - Frank Merle and Pierre Raphael,
*On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation*, Invent. Math.**156**(2004), no. 3, 565–672. MR**2061329**, DOI 10.1007/s00222-003-0346-z - Frank Merle and Pierre Raphael,
*On a sharp lower bound on the blow-up rate for the $L^2$ critical nonlinear Schrödinger equation*, J. Amer. Math. Soc.**19**(2006), no. 1, 37–90. MR**2169042**, DOI 10.1090/S0894-0347-05-00499-6 - Minoru Murata,
*Asymptotic expansions in time for solutions of Schrödinger-type equations*, J. Funct. Anal.**49**(1982), no. 1, 10–56. MR**680855**, DOI 10.1016/0022-1236(82)90084-2 - 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. MR**1608275** - Galina Perelman,
*On the formation of singularities in solutions of the critical nonlinear Schrödinger equation*, Ann. Henri Poincaré**2**(2001), no. 4, 605–673. MR**1852922**, DOI 10.1007/PL00001048 - Galina Perelman,
*Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations*, Comm. Partial Differential Equations**29**(2004), no. 7-8, 1051–1095. MR**2097576**, DOI 10.1081/PDE-200033754 - 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. MR**1488355**, DOI 10.1006/jdeq.1997.3345 - 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. MR**2122541**, DOI 10.1007/s00208-004-0596-0 - Michael Reed and Barry Simon,
*Methods of modern mathematical physics. I. Functional analysis*, Academic Press, New York-London, 1972. MR**0493419** - 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. MR**2038194**, DOI 10.1007/s00222-003-0325-4 - Igor Rodnianski, Wilhelm Schlag, and Avraham Soffer,
*Dispersive analysis of charge transfer models*, Comm. Pure Appl. Math.**58**(2005), no. 2, 149–216. MR**2094850**, DOI 10.1002/cpa.20066
RSS2 Rodnianski, I., Schlag, W., Soffer, A. - Jalal Shatah,
*Stable standing waves of nonlinear Klein-Gordon equations*, Comm. Math. Phys.**91**(1983), no. 3, 313–327. MR**723756**, DOI 10.1007/BF01208779 - Jalal Shatah and Walter Strauss,
*Instability of nonlinear bound states*, Comm. Math. Phys.**100**(1985), no. 2, 173–190. MR**804458**, DOI 10.1007/BF01212446 - 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. MR**1789924**, DOI 10.1080/03605300008821581 - A. Soffer and M. I. Weinstein,
*Multichannel nonlinear scattering for nonintegrable equations*, Comm. Math. Phys.**133**(1990), no. 1, 119–146. MR**1071238**, DOI 10.1007/BF02096557 - 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. MR**1170476**, DOI 10.1016/0022-0396(92)90098-8 - 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. MR**1032250** - 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. MR**1696311** - 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. MR**1944033**, DOI 10.1081/PDE-120016161 - Ricardo Weder,
*The $W_{k,p}$-continuity of the Schrödinger wave operators on the line*, Comm. Math. Phys.**208**(1999), no. 2, 507–520. MR**1729096**, DOI 10.1007/s002200050767 - Michael I. Weinstein,
*Modulational stability of ground states of nonlinear Schrödinger equations*, SIAM J. Math. Anal.**16**(1985), no. 3, 472–491. MR**783974**, DOI 10.1137/0516034 - Michael I. Weinstein,
*Lyapunov stability of ground states of nonlinear dispersive evolution equations*, Comm. Pure Appl. Math.**39**(1986), no. 1, 51–67. MR**820338**, DOI 10.1002/cpa.3160390103

*Numerical verification of a gap condition for a linearized NLS equation*, preprint, 2005, to appear in Nonlinearity. ES Erdoğan, 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.

*Asymptotic Stability of Nonlinear Schrödinger Equations with Potential*, preprint, 2005, to appear in Reviews in Mathematical Physics. SZ2 Gang, Z., Sigal, I. M.

*Relaxation to Trapped Solitons in Nonlinear Schrödinger Equations with Potential*, preprint, 2006.

*Exponential decay of eigenfunctions and generalized eigenfunctions of non-selfadjoint matrix Schrödinger operators related to NLS*, preprint, 2005.

*Non-generic blow-up solutions for the critical focusing NLS in 1-d*, preprint, 2005.

*Asymptotic stability of $N$-soliton states of NLS*, preprint, 2003. Sch1 Schlag, W.

*Stable manifolds for an orbitally unstable NLS.*Preprint, 2004, to appear in Annals of Math. Sch2 Schlag, W.

*Dispersive estimates for Schrödinger operators: A survey.*Preprint, 2004, to appear in “Mathematical Aspects of Nonlinear Dispersive Equations”, Princeton University Press.

## Additional Information

**J. Krieger**- Affiliation: Department of Mathematics, Harvard University, Science Center, 1 Oxford Street, Cambridge, Massachusetts 02138
- MR Author ID: 688045
- Email: jkrieger@math.harvard.edu
**W. Schlag**- Affiliation: Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago, Illinois 60637
- MR Author ID: 313635
- Email: schlag@math.uchicago.edu
- Received by editor(s): January 13, 2005
- Published electronically: 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 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**19**(2006), 815-920 - MSC (2000): Primary 35Q55, 35Q51, 37K40, 37K45
- DOI: https://doi.org/10.1090/S0894-0347-06-00524-8
- MathSciNet review: 2219305