KAM Tori for a Two Dimensional Beam Equation with aQuintic Nonlinear Term and Quasi-periodic Forcing

• Min Zhang ^[1] ; Jie Rui ^[1] ; Yan Li ^[1] ; Jian Zhang ^[1]
  1. [1] China University of Petroleum (East China)
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This work studies a two-dimensional beam equation with a quintic nonlinear term andquasi-periodic forcingutt+2u+εφ(t)h(u)=0,x∈T2,t∈Rwith periodic boundary conditions, whereεis a small positive parameter;φ(t)is a realanalytic quasi-periodic function intwith frequency vectorη=(η1,η2...,ηn∗)⊂[,2]n∗for a given positive integern∗and some constant>0; andhis a realanalytic function of the formh(u)=c1u+c5u5+∑ˆi≥6cˆiuˆi,c1,c5=0.Firstly, the linear part of Hamiltonian system corresponding to the equation is trans-formed to constant coefficients by a linear quasi-periodic change of variables. Then,a symplectic transformation is used to convert the Hamiltonian system into an angle-dependent block-diagonal normal form, which can be achieved by selecting theappropriate tangential sites. Finally, it is obtained that a Whitney smooth family ofsmall-amplitude quasi-periodic solutions for the equation by developing an abstractKAM (Kolmogorov–Arnold–Moser) theorem for infinite dimensional Hamiltonian systems.

• Referencias bibliográficas
  • 1. Arnold, V.I.: Mathematical methods in classical mechanics. Nauka, (1974), English transl. Springer,(1978)
  • 2. Bambusi, D., Graffi, S.: Time Quasi-periodic unbounded perturbations of Schrödinger operators andKAM methods. Commun. Math. Phys.219, 465–480...
  • 3. Eliasson, L.H., Kuksin, S.B.: KAM for the nonlinear Schrödinger equation. Ann. Math.172, 371–435(2010)
  • 4. Eliasson, L.H., Grebert, B., Kuksin, S.B.: KAM for the nonlinear beam equation. Geom. Funct. Anal.26, 1588–1715 (2016)
  • 5. Eliasson, L.H., Kuksin, S.B.: On reducibility of Schrödinger equations with quasiperiodic in timepotentials. Commun. Math. Phys.286, 125–135...
  • 6. Eliasson, L.H.: Reducibility for Linear Quasi-periodic Differential Equations. Winter School, St Eti-enne deTinée (2011)
  • 7. Geng, J., You, J.: A KAM theorem for Hamiltonian partial differential equations in higher dimensionalspaces. Commun. Math. Phys.262, 343–372...
  • 8. Geng, J., You, J.: KAM tori for higher dimensional beam equations with constant potentials. Nonlin-earity19, 2405–2423 (2006)
  • 9. Geng, J., Xu, X., You, J.: An infinite dimensional KAM theorem and its application to the two dimen-sional cubic Schrödinger equation....
  • 10. Geng, J., Xue, S.: Reducible KAM tori for two-dimensional quintic Schrödinger equations. Sci. SinicaMath.51, 457–498 (2021)
  • 11. Geng, J., Zhou, S.: An infinite dimensional KAM theorem with application to two dimensional com-pletely resonant beam equation. J. Math....
  • 12. Grébert, B., Paturel, E.: On reducibility of quantum harmonic oscillator onRdwith quasiperiodic intime potential. Annales de la faculté...
  • 13. Haus, E., Procesi, M.: Growth of sobolev norms for the quintic NLS onT2. Anal. PDE8, 883–922(2015)
  • 14. Kolmogorov, A.N.: On the conservation of conditionally periodic motions for a small Chang in Hamil-ton’s function. Dokl. Akad. Nauk USSR98,...
  • 15. Kuksin, S.B.: Nearly integrable infinite-dimensional Hamiltonian systems. Volume 1556 of LectureNotes in Math. Springer, New York (1993)
  • 16. Moser, J.: Convergent series expansions for quasi-periodic motions. Math. Ann.169, 136–176 (1967)
  • 17. Pöschel, J.: A KAM-theorem for some nonlinear PDEs. Ann. Sc. Norm. Super. Pisa Cl. Sci. IV Ser.23(15), 119–148 (1996)
  • 18. Procesi, C., Procesi, M.: A KAM algorithm for the resonant nonlinear Schrödinger equation. Adv.Math.272, 399–470 (2015)
  • 19. Procesi, M.: A normal form for beam and non-local nonlinear Schrödinger equations. J. Phys. A43,434028 (2010)
  • 20. Wayne, C.E.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory.Commun. Math. Phys.127, 479–528 (1990)
  • 21. Xu,J.,You,J.:Persistenceoflower-dimensionaltoriunderthefirstMelnikovsnon-resonancecondition.J. Math. Pures Appl.80, 1045–1067 (2001)
  • 22. Yuan, X.: Quasi-periodic solutions of nonlinear Schrödinger equations of higher dimension. J. Differ.Equ.195, 230–242 (2003)
  • 23. Zhang, M.: Quasi-periodic solutions of two dimensional Schrödinger equations with Quasi-periodicforcing. Nonlinear Anal.135, 1–34 (2016)
  • 24. Zhang, M., Wang, Y., Li, Y.: Reducibility and quasi-periodic solutions for a two dimensional beamequation with quasi-periodic in time...
  • 25. Zhang, M., Si, J.: Construction of quasi-periodic solutions for the quintic Schrödinger equation on thetwo-dimensional torusT2.Trans.Amer.Math.Soc.374,...