Taiwanese Journal of Mathematics

Quasi-periodic Solutions for Nonlinear Schrödinger Equations with Legendre Potential

Guanghua Shi and Dongfeng Yan

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In this paper, the nonlinear Schrödinger equations with Legendre potential $\mathbf{i} u_{t} - u_{xx} + V_L(x)u + mu + \sec x \cdot |u|^2 u = 0$ subject to certain boundary conditions is considered, where $V_L(x) = -\frac{1}{2} - \frac{1}{4} \tan^2 x$, $x \in (-\pi/2,\pi/2)$. It is proved that for each given positive constant $m \gt 0$, the above equation admits lots of quasi-periodic solutions with two frequencies. The proof is based on a partial Birkhoff normal form technique and an infinite-dimensional Kolmogorov-Arnold-Moser theory.

Article information

Taiwanese J. Math., Advance publication (2019), 17 pages.

First available in Project Euclid: 5 August 2019

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Digital Object Identifier

Primary: 37K55: Perturbations, KAM for infinite-dimensional systems 37C55: Periodic and quasiperiodic flows and diffeomorphisms

Kolmogorov-Arnold-Moser theory quasi-periodic solutions singular differential operator


Shi, Guanghua; Yan, Dongfeng. Quasi-periodic Solutions for Nonlinear Schrödinger Equations with Legendre Potential. Taiwanese J. Math., advance publication, 5 August 2019. doi:10.11650/tjm/190707. https://projecteuclid.org/euclid.twjm/1564970424

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  • P. Baldi, M. Berti, E. Haus and R. Montalto, Time quasi-periodic gravity water waves in finite depth, Invent. Math. 214 (2018), no. 2, 739–911.
  • P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann. 359 (2014), no. 1-2, 471–536.
  • M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equation, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 2, 301–373.
  • J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Internat. Math. Res. Notices 1994 (1994), no. 11, 475–497.
  • ––––, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math. (2) 148 (1998), no. 2, 363–439.
  • ––––, On invariant tori of full dimension for 1D periodic NLS, J. Funct. Anal. 229 (2005), no. 1, 62–94.
  • C. Cao and X. Yuan, Quasi-periodic solutions for perturbed generalized nonlinear vibrating string equation with singularities, Discrete Contin. Dyn. Syst. 37 (2017), no. 4, 1867–1901.
  • L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys. 211 (2000), no. 2, 497–525.
  • R. Feola, F. Giuliani and M. Procesi, Reducible KAM tori for Degasperis-Procesi equation, arXiv:1812.08498.
  • R. Feola and M. Procesi, Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differential Equations 259 (2015), no. 7, 3389–3447.
  • M. Gao and J. Liu, Quasi-periodic solutions for 1D wave equation with higher order nonlinearity, J. Differential Equations 252 (2012), no. 2, 1466–1493.
  • T. Kappeler and J. Pöschel, KdV & KAM, Springer-Verlag, Berlin, 2003.
  • S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl. 21 (1987), no. 3, 192–205.
  • ––––, Perturbation of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Math. USSR-Izv. 32 (1989), no. 1, 39–62.
  • ––––, Nearly Integrable Infinite-dimensional Hamiltonian Systems, Lecture Notes in Mathematics 1556, Springer-Verlag, Berlin, 1993.
  • ––––, A KAM-theorem for equations of the Korteweg-de Vries type, Rev. Math. Math. Phys. 10 (1998), no. 3, 1–64.
  • ––––, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications 19, Oxford University Press, Oxford, 2000.
  • S. B. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math. (2) 143 (1996), no. 1, 149–179.
  • J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large-variable coefficient, Comm. Pure Appl. Math. 63 (2010), no. 9, 1145–1172.
  • ––––, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Comm. Math. Phys. 307 (2011), no. 3, 629–673.
  • ––––, KAM for the derivative nonlinear Schrödinger equation with periodic boundary conditions, J. Differential Equations 256 (2014), no. 4, 1627–1652.
  • L. Mi, Quasi-periodic solutions of derivative nonlinear Schrödinger equations with a given potential, J. Math. Anal. Appl. 390 (2012), no. 1, 335–354.
  • J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), no. 1, 119–148.
  • ––––, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv. 71 (1996), no. 2, 269–296.
  • C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127 (1990), no. 3, 479–528.
  • X. Yuan, Quasi-periodic solutions of nonlinear wave equations with a prescribed potential, Discrete Contin. Dyn. Syst. 16 (2006), no. 3, 615–634.
  • ––––, Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations 230 (2006), no. 1, 213–274.
  • J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations, Nonlinearity 24 (2011), no. 4, 1189–1228.