A KAM Theorem for Two Dimensional Completely Resonant Reversible Schrödinger Systems


In this paper, we prove an abstract KAM (Kolmogorov–Arnold–Moser) theorem for infinite dimensional reversible Schrödinger systems. Using this KAM theorem together with partial Birkhoff normal form method, we obtain the existence of quasi-periodic solutions for a class of completely resonant reversible coupled nonlinear Schrödinger systems on two dimensional torus.

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  1. 1.

    The norm of vector valued function \(G:D_\rho (r,s)\times {\mathcal {O}}\rightarrow {\mathbb {C}}^b\), \(b<\infty ,\) is defined as \(\Vert G\Vert _{D_\rho (r,s)\times {\mathcal {O}}}=\mathop {\sum }\nolimits ^b_{a=1}\Vert G_a\Vert _{D_\rho (r,s)\times {\mathcal {O}}}\).


  1. 1.

    Berti, M., Bolle, P.: Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential. Nonlinearity 25(9), 2579–2613 (2012)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Berti, M., Bolle, P.: Quasi-periodic solutions with Sobolev regularity of NLS on \(\mathbb{T}^d\) with a multiplicative potential. J. Eur. Math. Soc. (JEMS) 15(1), 229–286 (2013)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Berti, M., Corsi, L., Procesi, M.: An abstract Nash–Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifolds. Commun. Math. Phys. 334(3), 1413–1454 (2015)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bourgain, J.: Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE. Int. Math. Res. Not. 11, 475ff (1994)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Ann. Math. (2) 148(2), 363–439 (1998)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bourgain, J.: Green’s Function Estimates for Lattice Schrödinger Operators and Applications. Annals of Mathematics Studies, vol. 158. Princeton University Press, Princeton (2005)

    Google Scholar 

  7. 7.

    Bourgain, J., Wang, W.: Quasi-periodic solutions of nonlinear random Schrödinger equations. J. Eur. Math. Soc. (JEMS) 10(1), 1–45 (2008)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Chierchia, L., You, J.: KAM tori for 1D nonlinear wave equations with periodic boundary conditions. Commun. Math. Phys. 211(2), 497–525 (2000)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Eliasson, L., Grébert, B., Kuksin, S.: KAM for the nonlinear beam equation. Geom. Funct. Anal. 26(6), 1588–1715 (2016)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Eliasson, L., Kuksin, S.: KAM for the nonlinear Schrödinger equation. Ann. Math. (2) 172(1), 371–435 (2010)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Ge, C., Geng, J., Lou, Z.: KAM tori for reversible derivative beam equations on \(\mathbb{T}^2\). Math. Z. (2020). https://doi.org/10.1007/s00209-020-02575-9

    Article  Google Scholar 

  12. 12.

    Geng, J., Xu, X., You, J.: An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation. Adv. Math. 226(6), 5361–5402 (2011)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Geng, J., You, J.: KAM tori for higher dimensional beam equations with constant potentials. Nonlinearity 19(10), 2405–2423 (2006)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Geng, J., You, J.: A KAM theorem for higher dimensional nonlinear Schrödinger equations. J. Dyn. Differ. Equ. 25(2), 451–476 (2013)

    Article  Google Scholar 

  15. 15.

    Geng, J., Xue, S.: Invariant tori for two-dimensional nonlinear Schrödinger equations with large forcing terms. J. Math. Phys. 60, 052703 (2019)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Grébert, B., Vilaça Da Rocha, V.: Stable and unstable time quasi periodic solutions for a system of coupled NLS equations. Nonlinearity 31, 4776 (2018)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Kuksin, S.: Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum. Funktsional. Anal. i Prilozhen. 21(3), 22–37, 95 (1987)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Kuksin, S., Pöschel, J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Ann. Math. (2) 143(1), 149–179 (1996)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Newell, A., Moloney, J.: Nonlinear Optics. Advanced Topics in the Interdisciplinary Mathematical Sciences. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA (1992)

    Google Scholar 

  20. 20.

    Pöschel, J.: On elliptic lower-dimensional tori in Hamiltonian systems. Math. Z. 202(4), 559–608 (1989)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Pöschel, J.: A KAM-theorem for some nonlinear partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23(1), 119–148 (1996)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Procesi, C., Procesi, M.: A KAM algorithm for the resonant non-linear Schrödinger equation. Adv. Math. 272, 399–470 (2015)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Procesi, M., Xu, X.: Quasi-Töplitz functions in KAM theorem. SIAM J. Math. Anal. 45(4), 2148–2181 (2013)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Sun, Y., Lou, Z., Geng, J.: A KAM theorem for higher dimensional reversible nonlinear Schrödinger equations. Preprint (2017)

  25. 25.

    Wang, W.: Energy supercritical nonlinear Schrödinger equations: quasiperiodic solutions. Duke Math. J. 165(6), 1129–1192 (2016)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Wang, W.: Quasi-Periodic Solutions for Nonlinear Klein–Gordon Equations. arXiv:1609.00309 (2017)

  27. 27.

    Wayne, C.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Commun. Math. Phys. 127(3), 479–528 (1990)

    MathSciNet  Article  Google Scholar 

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We would like to thank anonymous referee for helping to improve this paper. The research was supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11971012). Z. Lou was supported by NSFC (Grant No. 11901291) and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20190395). Y. Sun was supported by the China Scholarship Council (CSC) (Grant No. 202006190134).

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Correspondence to Jiansheng Geng.

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Suppose vector field \(X(\theta , I, z, {\bar{z}})\) is defined on \(D_\rho (r,s)=\{y=(\theta , I, z, {\bar{z}}):|Im \theta |<r,\, |I|<s,\, \Vert z\Vert _{\rho }<s,\, \Vert {\bar{z}}\Vert _{\rho }<s\}.\)

Definition 6.1

(Reversible vector field) Suppose S is an involution map: \(S^2=id.\) Vector field X is called reversible with respect to S (or S-reversible), if

$$\begin{aligned} DS\cdot X=- X\circ S, \end{aligned}$$


$$\begin{aligned} (DS(y))X(y)=-X(S(y)),\,y\in D_\rho (r,s), \end{aligned}$$

where DS is the tangent map of S.

Definition 6.2

Suppose S is an involution map: \(S^2=id.\) Vector field X is called invariant with respect to S (or S-invariant), if

$$\begin{aligned} DS\cdot X=X\circ S. \end{aligned}$$

Definition 6.3

A transformation \(\Phi \) is called invariant with respect to above involution S (or S-invariant), if \(\Phi \circ S=S\circ \Phi .\)

Lemma 6.1

  1. (1)

    If X and Y are both S-reversible (or S-invariant), then [XY] is S-invariant.

  2. (2)

    If X is S-reversible, Y is S-invariant and the transformation \(\Phi \) is S-invariant, then [XY] and \(\Phi ^*{X}\) are both S-reversible. In particular, the flow \(\phi _{Y} ^t\) of Y are S-invariant, thus \((\phi _{Y} ^t)^*{X}\) is S-reversible.

Lemma 6.2

(Cauchy’s inequality, [14]) Let \(0<\delta <r.\) For an analytic function \(f(\theta , I, z, {\bar{z}})\) on \(D_\rho (r,s),\)

$$\begin{aligned}&\left\| \frac{\partial f}{\partial \theta _b}\Vert _{D_\rho (r-\delta ,s)}\le \frac{c}{\delta }\right\| f\Vert _{D_\rho (r,s)},\\&\left\| \frac{\partial f}{\partial I_b}\right\| _{D_\rho (r,s/2)}\le \frac{c}{s}\Vert f\Vert _{D_\rho (r,s)}, \end{aligned}$$


$$\begin{aligned} \left\| \frac{\partial f}{\partial z^\sigma _i}\right\| _{D_\rho (r,s/2)}\le \frac{c}{s}\Vert f\Vert _{D_\rho (r,s)}\mathrm {e}^{\rho |i|},\,\sigma =\pm . \end{aligned}$$

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Geng, J., Lou, Z. & Sun, Y. A KAM Theorem for Two Dimensional Completely Resonant Reversible Schrödinger Systems. J Dyn Diff Equat (2021). https://doi.org/10.1007/s10884-021-09941-z

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  • KAM
  • NLS
  • Quasi-periodic solution
  • Reversible vector field
  • Birkhoff normal form

Mathematics Subject Classification

  • 37K55
  • 35B15