The Nonlinear Schrödinger Equation on Z and R with Bounded Initial Data: Examples and Conjectures

Abstract

We study the nonlinear Schrödinger equation (NLS) with bounded initial data which does not vanish at infinity. Examples include periodic, quasi-periodic and random initial data. On the lattice we prove that solutions are polynomially bounded in time for any bounded data. In the continuum, local existence is proved for real analytic data by a Newton iteration scheme. Global existence for NLS with a regularized nonlinearity follows by analyzing a local energy norm.

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References

  1. 1.

    Binder, I., Damanik, D., Goldstein, M., Lukic, M.: Almost periodicity in time of solutions of the KdV equation. Duke Math. J. 167(14), 2633–2678 (2018)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II. The KdV-equation. Geom. Funct. Anal. 3(3), 209–262 (1993)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Bourgain, J.: Periodic nonlinear Schrödinger equation and invariant measures. Commun. Math. Phys. 166(1), 1–26 (1994)

    ADS  MATH  Google Scholar 

  4. 4.

    Bourgain, J.: Invariant measures for the \(2\)D-defocusing nonlinear Schrödinger equation. Commun. Math. Phys. 176(2), 421–445 (1996)

    ADS  MATH  Google Scholar 

  5. 5.

    Bourgain, J.: Invariant measures for NLS in infinite volume. Commun. Math. Phys. 210(3), 605–620 (2000)

    ADS  MathSciNet  MATH  Google Scholar 

  6. 6.

    Carlen, E.A., Fröhlich, J., Lebowitz, J.: Exponential relaxation to equilibrium for a one-dimensional focusing non-linear schrödinger equation with noise. Commun. Math. Phys. 342(1), 303–332 (2016)

    ADS  MATH  Google Scholar 

  7. 7.

    Chaichenets, L., Hundertmark, D., Kunstmann, P., Pattakos, N.: On the existence of global solutions of the one-dimensional cubic NLS for initial data in the modulation space \(M_{p, q}({\mathbb{R}})\). J. Differ. Equ. 263(8), 4429–4441 (2017)

    ADS  MATH  Google Scholar 

  8. 8.

    Chatterjee, S., Kirkpatrick, K.: Probabilistic methods for discrete nonlinear schrödinger equations. Commun. Pure Appl. Math. 65(5), 727–757 (2012)

    MATH  Google Scholar 

  9. 9.

    Damanik, D., Goldstein, M.: On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data. J. Am. Math. Soc. 29(3), 825–856 (2016)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Greene, R.E., Jacobowitz, H.: Analytic isometric embeddings. Ann. Math. 2(93), 189–204 (1971)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Hörmander, Lars: On the Nash-Moser implicit function theorem. Ann. Acad. Sci. Fenn. Ser. A I Math., 10, 255–259 (1985)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Kappeler, T., Schaad, B., Topalov, P.: Scattering-like phenomena of the periodic defocusing NLS equation. Math. Res. Lett. 24(3), 803–826 (2017)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Lebowitz, J.L., Mounaix, P., Wang, W.-M.: Approach to equilibrium for the stochastic nls. Commun. Math. Phys. 321(1), 69–84 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

  14. 14.

    Lebowitz, J.L., Rose, H.A., Speer, E.R.: Statistical mechanics of the nonlinear Schrödinger equation. J. Stat. Phys. 50(3–4), 657–687 (1988)

    ADS  MATH  Google Scholar 

  15. 15.

    Lukkarinen, J., Spohn, H.: Weakly nonlinear schrödinger equation with random initial data. Invent. Math. 183(1), 79–188 (2011)

    ADS  MathSciNet  MATH  Google Scholar 

  16. 16.

    Moser, J.: A new technique for the construction of solutions of nonlinear differential equations. Proc. Nat. Acad. Sci. USA 47, 1824–1831 (1961)

    ADS  MathSciNet  MATH  Google Scholar 

  17. 17.

    Mendl, C.B., Spohn, H.: Low temperature dynamics of the one-dimensional discrete nonlinear schrödinger equation. J. Stat. Mech. 2015(8), P08028 (2015)

    Google Scholar 

  18. 18.

    Tadahiro, O.: Global existence for the defocusing nonlinear schrödinger equations with limit periodic initial data. Commun. Pure Appl. Anal. 14(4), 1563–1580 (2015)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Tadahiro, O.: On nonlinear Schrödinger equations with almost periodic initial data. SIAM J. Math. Anal. 47(2), 1253–1270 (2015)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Turaev, D., Zelik, S.: Analytical proof of space-time chaos in ginzburg-landau equations. Dyn. Syst. 28(4), 1713–1751 (2010)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Wang, W.-M.: Infinite energy quasi-periodic solutions to nonlinear schrödinger equations on \({\mathbb{R}}\). arXiv preprint arXiv:1908.11627 (2019)

Download references

Acknowledgements

The authors thank J. Bourgain, P. Deift, J. Lebowitz and W. Schlag for helpful discussions. The first author gratefully acknowledges the support of NSF Grants DMS-1500424 and DMS-1764358 while writing this paper. He also gratefully acknowledges the support by the von Neumann fellowship at the Institute for Advanced Study while writing this paper. The second author is supported in part by NSF grant DMS-160074.

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Correspondence to Thomas Spencer.

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(Dedicated to Joel Lebowitz with admiration for his inspiring leadership.)

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Communicated by Ivan Corwin.

Appendix: Linear Schrödinger Time Evolution on \({\mathbb {Z}}\)

Appendix: Linear Schrödinger Time Evolution on \({\mathbb {Z}}\)

Let \( \psi (t,n) = e^{it \Delta }\psi _0(n)\) where \(\psi _0 = \sum _j a_j \delta _j\) and \(\Delta \) is the finite difference Laplacian.

Lemma A

We can choose \(a_n,\,|a_j| \le 1\) so that \(|\psi (t,0)| \ge \delta t^{1/2}, \delta >0\).

Proof

The fundamental solution to the lattice Schrödinger equation can be expressed in terms of the integral

$$\begin{aligned} F_n(t) = (2\pi )^{-1}\int _0^{2\pi }e^{it\cos (\theta )+i n\theta } d\theta , \,\, n\in {\mathbb {Z}} \,, \end{aligned}$$

which is closely related to the Bessel function. This integral has two saddle points \(\theta _s, \, \pi - \theta _s\) where \(\sin (\theta _s) = n/t,\,\, \cos (\theta _s) = [1-(n/t)^2]^{1/2}\). By classical stationary phase, if \(|n| \le t/2\)

$$\begin{aligned} F_n(t) \approx [t\cos (\theta _s)]^{-1/2}\, \cos (\phi (t,n)(1 +O(1/t)) \end{aligned}$$

when n is even and

$$\begin{aligned} F(t,n)=i[t\cos (\theta _s)]^{-1/2}\sin (\phi (t,n)(1+O(1/t)), \end{aligned}$$

when n is odd. Here

$$\begin{aligned} \phi (t,n)= \pi /4+t\cos (\theta _s)+n \theta _s\,. \end{aligned}$$

Since \(|\psi (t,0)|= |\sum _n a_n F_n(t)|\) we will choose \(a_n\) so that the sum equals

$$\begin{aligned} \sum _{ |n|\le t/2}|F(t,n)| \ge \delta t^{1/2}\,. \end{aligned}$$

To prove this lower bound note that if \(|\cos (\phi (t,n))|\) is small when n is even then \(|\sin (\phi (t,n+1))|\ge 1/4\) since \(|\phi (t,n+1)- \phi (t,n)| \approx |\theta _s|= \arcsin n/t \le \delta t^{1/2}\). \(\square \)

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Dodson, B., Soffer, A. & Spencer, T. The Nonlinear Schrödinger Equation on Z and R with Bounded Initial Data: Examples and Conjectures. J Stat Phys 180, 910–934 (2020). https://doi.org/10.1007/s10955-020-02552-w

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Keywords

  • Propagation speed
  • Well-posedness
  • Local conservation laws
  • Newton iteration