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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

Lufang Mi 1, and  Kangkang Zhang 2,

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.
Keywords: normal form., KAM theory, Benjamin-Ono equation, Quasi-periodic solution.
Mathematics Subject Classification: Primary: 37K55, 35Q55, Secondary: 35Q4.
Citation: Lufang Mi, Kangkang Zhang. Invariant Tori for Benjamin-Ono Equation with Unbounded quasi-periodically forced Perturbation. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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Zh. Burgeĭn, Recent progress on quasi-periodic lattice Schrödinger operators and Hamiltonian PDEs, Russian Math. Surveys, 59 (2004), 231-246. doi: 10.1070/RM2004v059n02ABEH000716.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. B. Kuksin, On small-denominators equations with large variable coefficients, Z. Angew. Math. Phys., 48 (1997), 262-271. doi: 10.1007/PL00001476.  Google Scholar

[11]

S. B. Kuksin, "Analysis of Hamiltonian PDEs," Oxford Lecture Series in Mathematics and its Applications, 19, Oxford University Press, Oxford, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. Pöschel, Quasi-periodic solutions for a nonlinear wave equations, Comment. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. B. Kuksin, On small-denominators equations with large variable coefficients, Z. Angew. Math. Phys., 48 (1997), 262-271. doi: 10.1007/PL00001476.  Google Scholar

[11]

S. B. Kuksin, "Analysis of Hamiltonian PDEs," Oxford Lecture Series in Mathematics and its Applications, 19, Oxford University Press, Oxford, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. Pöschel, Quasi-periodic solutions for a nonlinear wave equations, Comment. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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