# American Institute of Mathematical Sciences

February  2014, 34(2): 689-707. doi: 10.3934/dcds.2014.34.689

## Invariant Tori for Benjamin-Ono Equation with Unbounded quasi-periodically forced Perturbation

 1 Department of Mathematics and Information Science, Binzhou University, Binzhou Shandong 256600, China 2 School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received  August 2012 Revised  May 2013 Published  August 2013

In this paper, we consider the non-autonomous Benjamin-Ono equation $$u_t+\mathscr{H}u_{xx}- uu_x- (F(\omega t,x,u))_x=0$$ under periodic boundary conditions. Using an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field and partial Birkhoff normal form, we will prove that there exists a Cantorian branch of KAM tori and thus many time quasi-periodic solutions for the above equation.
Citation: Lufang Mi, Kangkang Zhang. Invariant Tori for Benjamin-Ono Equation with Unbounded quasi-periodically forced Perturbation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 689-707. doi: 10.3934/dcds.2014.34.689
##### References:
 [1] M. J. Ablowitz and A. S. Fokas, The inverse scattering transform for the Benjamin-Ono equation-a pivot for multidimensional problems, Stud. Appl. Math., 68 (1983), 1-10. [2] D. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods, Comm. Math. Phys., 219 (2001), 465-480. doi: 10.1007/s002200100426. [3] M. Berti and M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations, Comm. Partial Differential Equations, 31 (2006), 959-985. doi: 10.1080/03605300500358129. [4] Zh. Burgeĭn, Recent progress on quasi-periodic lattice Schrödinger operators and Hamiltonian PDEs, Russian Math. Surveys, 59 (2004), 231-246. doi: 10.1070/RM2004v059n02ABEH000716. [5] M. Gao and J. Liu, Quasi-periodic solutions for derivative nonlinear Schrödinger equation, Discrete Contin. Dyn. Syst., 32 (2012), 2101-2123. doi: 10.3934/dcds.2012.32.2101. [6] L. Jiao and Y. Wang, The construction of quasi-periocic solutions of quasi-periodic forced Schrödinger equation, Commun. Pure Appl. Anal., 8 (2009), 1585-1606. doi: 10.3934/cpaa.2009.8.1585. [7] T. Kappeler and J. Pöschel, "KdV & KAM," Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas, 3rd Series, A Series of Modern Surveys in Mathematics], 45, Springer-Verlag, Berlin, 2003. [8] S. Klainerman, Long-time behaviour of solutions to nonliear wave equations, in "Proceedings of International Congress of Mathematicians, Vol. 1, 2" (Warsaw, 1983), PWN, Warsaw, (1984), 1209-1215. [9] S. B. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. Math. (2), 143 (1996), 149-179. doi: 10.2307/2118656. [10] S. B. Kuksin, On small-denominators equations with large variable coefficients, Z. Angew. Math. Phys., 48 (1997), 262-271. doi: 10.1007/PL00001476. [11] S. B. Kuksin, "Analysis of Hamiltonian PDEs," Oxford Lecture Series in Mathematics and its Applications, 19, Oxford University Press, Oxford, 2000. [12] S. B. Kuksin, Fifteen years of KAM in PDE, in "Geometry, Topology, and Mathematical Physics" (eds. V. M. Buchstaber and I. M. Krichever), Amer. Math. Soc. Transl. Ser. 2, 212 Amer. Math. Soc., Providence, RI, (2004), 237-258. [13] P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964), 611-613. doi: 10.1063/1.1704154. [14] J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large-variable coefficient, Commun. Pure Appl. Math., 63 (2010), 1145-1172. doi: 10.1002/cpa.20314. [15] J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Commun. Math. Phys., 307 (2011), 629-673. doi: 10.1007/s00220-011-1353-3. [16] L. Mi, Quasi-periodic Solutions of derivative nonlinear Schrödinger equations with a given potential, J. Math. Anal. Appl., 390 (2012), 335-354. doi: 10.1016/j.jmaa.2012.01.046. [17] L. Molinet, Global well-posedness in the energy space for the Benjamin-Ono equation on the circle, Math. Ann., 337 (2007), 353-383. doi: 10.1007/s00208-006-0038-2. [18] L. Molinet, Global well-posedness in $L^2$ for the periodic Benjamin-Ono equation, Amer. J. Math., 130 (2008), 635-683. doi: 10.1353/ajm.0.0001. [19] J. Pöschel, Quasi-periodic solutions for a nonlinear wave equations, Comment. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420. [20] J. Si, Quasi-periodic solutions of a non-autonomous wave equations with quasi-periodic forcing, J. Differential Equations, 252 (2012), 5274-5360. doi: 10.1016/j.jde.2012.01.034. [21] Y. Wang, Quasi-periodic solutions of a quasi-periodically forced nonlinear beam equation, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 2682-2700. doi: 10.1016/j.cnsns.2011.10.022. [22] J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differtial equations, Nonliearity, 24 (2011), 1189-1228. doi: 10.1088/0951-7715/24/4/010.

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##### References:
 [1] M. J. Ablowitz and A. S. Fokas, The inverse scattering transform for the Benjamin-Ono equation-a pivot for multidimensional problems, Stud. Appl. Math., 68 (1983), 1-10. [2] D. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods, Comm. Math. Phys., 219 (2001), 465-480. doi: 10.1007/s002200100426. [3] M. Berti and M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations, Comm. Partial Differential Equations, 31 (2006), 959-985. doi: 10.1080/03605300500358129. [4] Zh. Burgeĭn, Recent progress on quasi-periodic lattice Schrödinger operators and Hamiltonian PDEs, Russian Math. Surveys, 59 (2004), 231-246. doi: 10.1070/RM2004v059n02ABEH000716. [5] M. Gao and J. Liu, Quasi-periodic solutions for derivative nonlinear Schrödinger equation, Discrete Contin. Dyn. Syst., 32 (2012), 2101-2123. doi: 10.3934/dcds.2012.32.2101. [6] L. Jiao and Y. Wang, The construction of quasi-periocic solutions of quasi-periodic forced Schrödinger equation, Commun. Pure Appl. Anal., 8 (2009), 1585-1606. doi: 10.3934/cpaa.2009.8.1585. [7] T. Kappeler and J. Pöschel, "KdV & KAM," Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas, 3rd Series, A Series of Modern Surveys in Mathematics], 45, Springer-Verlag, Berlin, 2003. [8] S. Klainerman, Long-time behaviour of solutions to nonliear wave equations, in "Proceedings of International Congress of Mathematicians, Vol. 1, 2" (Warsaw, 1983), PWN, Warsaw, (1984), 1209-1215. [9] S. B. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. Math. (2), 143 (1996), 149-179. doi: 10.2307/2118656. [10] S. B. Kuksin, On small-denominators equations with large variable coefficients, Z. Angew. Math. Phys., 48 (1997), 262-271. doi: 10.1007/PL00001476. [11] S. B. Kuksin, "Analysis of Hamiltonian PDEs," Oxford Lecture Series in Mathematics and its Applications, 19, Oxford University Press, Oxford, 2000. [12] S. B. Kuksin, Fifteen years of KAM in PDE, in "Geometry, Topology, and Mathematical Physics" (eds. V. M. Buchstaber and I. M. Krichever), Amer. Math. Soc. Transl. Ser. 2, 212 Amer. Math. Soc., Providence, RI, (2004), 237-258. [13] P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964), 611-613. doi: 10.1063/1.1704154. [14] J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large-variable coefficient, Commun. Pure Appl. Math., 63 (2010), 1145-1172. doi: 10.1002/cpa.20314. [15] J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Commun. Math. Phys., 307 (2011), 629-673. doi: 10.1007/s00220-011-1353-3. [16] L. Mi, Quasi-periodic Solutions of derivative nonlinear Schrödinger equations with a given potential, J. Math. Anal. Appl., 390 (2012), 335-354. doi: 10.1016/j.jmaa.2012.01.046. [17] L. Molinet, Global well-posedness in the energy space for the Benjamin-Ono equation on the circle, Math. Ann., 337 (2007), 353-383. doi: 10.1007/s00208-006-0038-2. [18] L. Molinet, Global well-posedness in $L^2$ for the periodic Benjamin-Ono equation, Amer. J. Math., 130 (2008), 635-683. doi: 10.1353/ajm.0.0001. [19] J. Pöschel, Quasi-periodic solutions for a nonlinear wave equations, Comment. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420. [20] J. Si, Quasi-periodic solutions of a non-autonomous wave equations with quasi-periodic forcing, J. Differential Equations, 252 (2012), 5274-5360. doi: 10.1016/j.jde.2012.01.034. [21] Y. Wang, Quasi-periodic solutions of a quasi-periodically forced nonlinear beam equation, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 2682-2700. doi: 10.1016/j.cnsns.2011.10.022. [22] J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differtial equations, Nonliearity, 24 (2011), 1189-1228. doi: 10.1088/0951-7715/24/4/010.
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