Kyoto Journal of Mathematics

On the one-dimensional cubic nonlinear Schrödinger equation below L2

Tadahiro Oh and Catherine Sulem

Full-text: Open access


In this paper, we review several recent results concerning well-posedness of the one-dimensional, cubic nonlinear Schrödinger equation (NLS) on the real line R and on the circle T for solutions below the L2-threshold. We point out common results for NLS on R and the so-called Wick-ordered NLS (WNLS) on T, suggesting that WNLS may be an appropriate model for the study of solutions below L2(T). In particular, in contrast with a recent result of Molinet, who proved that the solution map for the periodic cubic NLS equation is not weakly continuous from L2(T) to the space of distributions, we show that this is not the case for WNLS.

Article information

Kyoto J. Math., Volume 52, Number 1 (2012), 99-115.

First available in Project Euclid: 19 February 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]


Oh, Tadahiro; Sulem, Catherine. On the one-dimensional cubic nonlinear Schrödinger equation below $L^{2}$. Kyoto J. Math. 52 (2012), no. 1, 99--115. doi:10.1215/21562261-1503772.

Export citation


  • [1] M. Ablowitz, D. Kaup, D. A. Newell, and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974), 249–315.
  • [2] M. Ablowitz and Y. Ma, The periodic cubic Schrödinger equation, Stud. Appl. Math. 65 (1981), 113–158.
  • [3] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I: Schrödinger equations, Geom. Funct. Anal. 3 (1993), 107–156.
  • [4] J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys. 176 (1996), 421–445.
  • [5] J. Bourgain, Periodic Korteweg-de Vries equation with measures as initial data, Selecta Math. (N.S.) 3 (1997), 115–159.
  • [6] J. Bourgain, Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity, Internat. Math. Res. Notices 1998, no. 5, 253–283.
  • [7] J. Bourgain, “Nonlinear Schrödinger equations” in Hyperbolic Equations and Frequency Interactions (Park City, Ut., 1995), IAS/Park City Math. Ser. 5, Amer. Math. Soc., Providence, 1999, 3–157.
  • [8] N. Burq, P. Gérard, and N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on Sd, Math. Res. Lett. 9 (2002), 323–335.
  • [9] R. Carles, E. Dumas, and C. Sparber, Multiphase weakly nonlinear geometric optics for Schrödinger equations, SIAM J. Math. Anal. 42 (2010), 489–518.
  • [10] M. Christ, “Power series solution of a nonlinear Schrödinger equation” in Mathematical Aspects of Nonlinear Dispersive Equations, Ann. of Math. Stud. 163, Princeton Univ. Press, Princeton, N.J., 2007, 131–155.
  • [11] M. Christ, Nonuniqueness of weak solutions of the nonlinear Schroedinger equation, preprint, arXiv:math/0503366v1 [math.AP]
  • [12] M. Christ, J. Colliander, and T. Tao, Asymptotics, frequency modulation, and low-regularity ill-posedness of canonical defocusing equations, Amer. J. Math. 125 (2003), 1235–1293.
  • [13] M. Christ, J. Colliander, T. Tao, A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order, J. Funct. Anal. 254 (2008), 368–395.
  • [14] M. Christ, J. Colliander, T. Tao, Instability of the periodic nonlinear Schrödinger equation, preprint, arXiv:math/0311227v1 [math.AP]
  • [15] M. Christ, J. Holmer, and D. Tataru, A priori bounds and instability for the nonlinear Schrödinger equation, workshop lecture at “Nonlinear waves and dispersion,” Institut Henri Poincaré, Paris, 2009.
  • [16] J. Colliander and T. Oh, Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L2, Duke Math. J. 161 (2012), 367–414.
  • [17] S. Cui and C. Kenig, Weak continuity of dynamical systems for the KdV and mKdV equations, Differential Integral Equations 23 (2010), 1001–1022.
  • [18] S. Cui and C. Kenig, Weak continuity of the flow map for the Benjamin-Ono equation on the line, J. Fourier Anal. Appl. 16 (2010), 1021–1052.
  • [19] O. Goubet and L. Molinet, Global attractor for weakly damped nonlinear Schrödinger equations in L2(ℝ), Nonlinear Anal. 71 (2009), 317–320.
  • [21] A. Grünrock, Bi- and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS, Int. Math. Res. Not. 2005, no. 41, 2525–2558.
  • [22] A. Grünrock and S. Herr, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal. 39 (2008), 1890–1920.
  • [23] S. Gutiérrez, J. Rivas, and L. Vega, Formation of singularities and self-similar vortex motion under the localized induction approximation, Comm. Partial Differential Equations 28 (2003), 927–968.
  • [24] H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech. 51 (1972), 477–485.
  • [25] S. Janson, Gaussian Hilbert Spaces, Cambridge Tracts in Math. 129, Cambridge Univ. Press, Cambridge, 1997.
  • [26] T. Kato, On nonlinear Schrödinger equations, II: Hs-solutions and unconditional well-posedness, J. Anal. Math. 67 (1995), 281–306.
  • [27] C. Kenig and Y. Martel, Asymptotic stability of solitons for the Benjamin-Ono equation, Rev. Mat. Iberoam. 25 (2009), 909–970.
  • [28] C. Kenig, G. Ponce, and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001), 617–633.
  • [29] H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. 2007, no. 16, art. ID rnm05.
  • [30] H. Kuo, Introduction to Stochastic Integration, Universitext, Springer, New York, 2006.
  • [31] M. Ledoux and M. Talagrand, Probability in Banach Spaces: Isoperimetry and Processes, reprint of the 1991 ed., Classics Math., Springer, Berlin, 2011.
  • [32] Y. Martel and F. Merle, A Liouville theorem for the critical generalized Korteweg-de Vries equation, J. Math. Pures Appl. 79 (2000), 339–425.
  • [33] Y. Martel and F. Merle, Blow up in finite time and dynamics of blow up solutions for the L2-critical generalized KdV equation, J. Amer. Math. Soc. 15 (2002), 617–664.
  • [34] L. Molinet, On ill-posedness for the one-dimensional periodic cubic Schrödinger equation, Math. Res. Lett. 16 (2009), 111–120.
  • [36] C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Appl. Math. Sci. 139, Springer, New York, 1999.
  • [37] Y. Tsutsumi, L2-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), 115–125.
  • [38] N. Tzvetkov, Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation, Probab. Theory Related Fields 146 (2010), 481–514.
  • [39] A. Vargas and L. Vega, Global wellposedness for 1D non-linear Schrödinger equation for data with an infinite L2 norm, J. Math. Pures Appl. (9) 80 (2001), 1029–1044.
  • [40] P. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Math. 1756, Springer, Berlin, 2001.
  • [41] A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189–201.