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May  2015, 14(3): 941-957. doi: 10.3934/cpaa.2015.14.941

KAM Tori for generalized Benjamin-Ono equation

Dongfeng Yan 1,

1. 

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan, 450001, China

Received  July 2014 Revised  December 2014 Published  March 2015

In this paper, we investigate one-dimensional generalized Benjamin-Ono equation, \begin{eqnarray} u_t+\mathcal{H}u_{xx}+u^{4}u_x=0,x\in\mathbb{T}, \end{eqnarray} and prove the existence of quasi-periodic solutions with two frequencies. The proof is based on partial Birkhoff normal form and an unbounded KAM theorem developed by Liu-Yuan[Commun.Math.Phys.307(2011)629-673].
Keywords: quasi-periodic solutions, KAM tori, unbounded KAM theorem., Birkhoff normal form.
Mathematics Subject Classification: Primary: O175.14, O175.2.
Citation: Dongfeng Yan. KAM Tori for generalized Benjamin-Ono equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 941-957. doi: 10.3934/cpaa.2015.14.941
References:
[1]

P. Baldi, Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type,, \emph{Ann. Inst. H. Poincar Anal. Non Linaire}, 30 (2013), 33.  doi: 10.1016/j.anihpc.2012.06.001.  Google Scholar

[2]

P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation,, \emph{Math. Ann.}, 359 (2014), 471.  doi: 10.1007/s00208-013-1001-7.  Google Scholar

[3]

P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear KdV,, \emph{C. R. Math. Acad. Sci. Paris}, 352 (2014), 603.  doi: 10.1016/j.crma.2014.04.012.  Google Scholar

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T. B. Benjamin, Internal waves of permanent form in fluids of great depth,, \emph{J. Fluid Mech.}, 29 (1967), 559.   Google Scholar

[5]

M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equations,, \emph{Arch. Ration. Mech. Anal.}, 212 (2014), 905.  doi: 10.1007/s00205-014-0726-0.  Google Scholar

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J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and application to nonlinear pde,, \emph{Int. Math. Res. Notices}, 11 (1994), 475.  doi: 10.1155/S1073792894000516.  Google Scholar

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J. Bourgain, On Melnikov's persistence problem,, \emph{Math. Res. Lett.}, 4 (1997), 445.  doi: 10.4310/MRL.1997.v4.n4.a1.  Google Scholar

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J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation,, \emph{Ann. Math.}, 148 (1998), 363.  doi: 10.2307/121001.  Google Scholar

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J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications,, Annals of mathematics studies, (2005).   Google Scholar

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J. Bourgain, On invariant tori of full dimension for 1D periodic NLS,, \emph{J. Funct. Anal.}, 229 (2005), 62.  doi: 10.1016/j.jfa.2004.10.019.  Google Scholar

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L. Chierchia and J. You, KAM tori for 1D nonlinear wave equation with periodic boundary conditions,, \emph{Commun. Math. Phys.}, 211 (2000), 497.  doi: 10.1007/s002200050824.  Google Scholar

[12]

G. Iooss, P. I. Plotnikov and J. F. Toland, Standing waves on an infinitely deep perfect fluid under gravity,, \emph{Arch. Ration. Mech. Anal.}, 177 (2005), 367.  doi: 10.1007/s00205-005-0381-6.  Google Scholar

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J. R. Iorio, On the Cauchy problem for the Benjamin-Ono equation,, \emph{Comm. Partial Differential equations}, 11 (1986), 1031.  doi: 10.1080/03605308608820456.  Google Scholar

[14]

T. Kappler and J. Pöschel, KdV $&$ KAM,, Springer-Verlag, (2003).  doi: 10.1007/978-3-662-08054-2.  Google Scholar

[15]

C. E. Kenig, G. Ponce and L. Vega, On the generalized Benjamin-Ono equations,, \emph{Trans. Amer. Math. Soc.}, 342 (1994), 155.  doi: 10.2307/2154688.  Google Scholar

[16]

S. B. Kuksin, Hamiltonian perturbation of infinite-dimensional linear system with an imaginary spectrum,, \emph{Funkt. Anal. Prilozh.}, 21 (1987), 22.   Google Scholar

[17]

S. B. Kuksin, Perturbation of quasiperiodic solutions of infinite-dimensional Hamiltonian systems,, \emph{Izv. Akad. Nauk SSSR, 52 (1989), 41.   Google Scholar

[18]

S. B. Kuksin, Nearly Integrable Infinite-dimensional Hamiltonian Systems,, Springer-Verlag, (1993).   Google Scholar

[19]

S. B. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation,, \emph{Ann. of Math.}, 143 (1996), 149.  doi: 10.2307/2118656.  Google Scholar

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S. B. Kuksin, On small denominators equations with large variable coefficients,, \emph{J. Appl. Math. Phys.}, 48 (1997), 262.  doi: 10.1007/PL00001476.  Google Scholar

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S. B. Kuksin, A KAM theorem for equations of the Korteweg-de Vries type,, \emph{Rev. Math-Math Phys.}, 10 (1998), 1.   Google Scholar

[22]

S. B. Kuksin, Analysis of Hamiltonian PDEs,, Oxford Univ. Press, (2000).   Google Scholar

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J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large variable coefficient,, \emph{Commun. Pure Appl. Math.}, 63 (2010), 1145.  doi: 10.1002/cpa.20314.  Google Scholar

[24]

J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations,, \emph{Commun. Math. Phys.}, 307 (2011), 629.  doi: 10.1007/s00220-011-1353-3.  Google Scholar

[25]

J. Liu and X. Yuan, KAM for the derivative nonliear Schrödinger equation with periodic boundary conditions,, \emph{Journal of Differential Equations}, 256 (2014), 1627.  doi: 10.1016/j.jde.2013.11.007.  Google Scholar

[26]

L. Mi, Quasi-periodic solutions of derivative nonlinear Schrödinger equations with a given potential,, \emph{Journal of Mathematical Analysis and Applications}, 390 (2012), 335.  doi: 10.1016/j.jmaa.2012.01.046.  Google Scholar

[27]

L. Mi and K. Zhang, Invariant tori for Benjamin-Ono equation with unbounded quasi-periodically forced perturbation,, \emph{Discrete and Continuous Dynamical Systems-Series A}, 34 (2014), 689.  doi: 10.3934/dcds.2014.34.689.  Google Scholar

[28]

L. Molinet and F. Ribaud, Well-posedness results for the generalized Benjamin-Ono equation with small initial data,, \emph{J. Math. Pures Appl.}, 83 (2004), 277.  doi: 10.1016/j.matpur.2003.11.005.  Google Scholar

[29]

H. Ono, Algebraic solitary waves in stratified fluids,, \emph{Journal of the Physical Society of Japan}, 39 (1975), 1082.   Google Scholar

[30]

J. Pöschel, A KAM theorem for some nonlinear PDEs,, \emph{Ann. Scuola Norm. Sup. Pisacl. Sci.}, 23 (1996), 119.   Google Scholar

[31]

J. Pöschel, Quasi-periodic solutions for nonlinear wave equations,, \emph{Comm. Math. Helv.}, 71 (1996), 269.  doi: 10.1007/BF02566420.  Google Scholar

[32]

T. Tao, Global well-posedness of the Benjamin-Ono equation in H1(R),, \emph{J. Hyperbolic Differ. Equ.}, 1 (2004), 27.  doi: 10.1142/S0219891604000032.  Google Scholar

[33]

C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equation via KAM theory,, \emph{Commun. Math. Phys.}, 127 (1990), 479.   Google Scholar

[34]

X. Yuan and K. Zhang, A reduction theorem for time dependent Schrödinger operator with finite differentiable unbounded perturbation,, \emph{J. Math. Phys.}, 54 (2013).  doi: 10.1063/1.4803852.  Google Scholar

[35]

J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations,, \emph{Nonlinearity}, 24 (2011), 1189.  doi: 10.1088/0951-7715/24/4/010.  Google Scholar

show all references

References:
[1]

P. Baldi, Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type,, \emph{Ann. Inst. H. Poincar Anal. Non Linaire}, 30 (2013), 33.  doi: 10.1016/j.anihpc.2012.06.001.  Google Scholar

[2]

P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation,, \emph{Math. Ann.}, 359 (2014), 471.  doi: 10.1007/s00208-013-1001-7.  Google Scholar

[3]

P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear KdV,, \emph{C. R. Math. Acad. Sci. Paris}, 352 (2014), 603.  doi: 10.1016/j.crma.2014.04.012.  Google Scholar

[4]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth,, \emph{J. Fluid Mech.}, 29 (1967), 559.   Google Scholar

[5]

M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equations,, \emph{Arch. Ration. Mech. Anal.}, 212 (2014), 905.  doi: 10.1007/s00205-014-0726-0.  Google Scholar

[6]

J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and application to nonlinear pde,, \emph{Int. Math. Res. Notices}, 11 (1994), 475.  doi: 10.1155/S1073792894000516.  Google Scholar

[7]

J. Bourgain, On Melnikov's persistence problem,, \emph{Math. Res. Lett.}, 4 (1997), 445.  doi: 10.4310/MRL.1997.v4.n4.a1.  Google Scholar

[8]

J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation,, \emph{Ann. Math.}, 148 (1998), 363.  doi: 10.2307/121001.  Google Scholar

[9]

J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications,, Annals of mathematics studies, (2005).   Google Scholar

[10]

J. Bourgain, On invariant tori of full dimension for 1D periodic NLS,, \emph{J. Funct. Anal.}, 229 (2005), 62.  doi: 10.1016/j.jfa.2004.10.019.  Google Scholar

[11]

L. Chierchia and J. You, KAM tori for 1D nonlinear wave equation with periodic boundary conditions,, \emph{Commun. Math. Phys.}, 211 (2000), 497.  doi: 10.1007/s002200050824.  Google Scholar

[12]

G. Iooss, P. I. Plotnikov and J. F. Toland, Standing waves on an infinitely deep perfect fluid under gravity,, \emph{Arch. Ration. Mech. Anal.}, 177 (2005), 367.  doi: 10.1007/s00205-005-0381-6.  Google Scholar

[13]

J. R. Iorio, On the Cauchy problem for the Benjamin-Ono equation,, \emph{Comm. Partial Differential equations}, 11 (1986), 1031.  doi: 10.1080/03605308608820456.  Google Scholar

[14]

T. Kappler and J. Pöschel, KdV $&$ KAM,, Springer-Verlag, (2003).  doi: 10.1007/978-3-662-08054-2.  Google Scholar

[15]

C. E. Kenig, G. Ponce and L. Vega, On the generalized Benjamin-Ono equations,, \emph{Trans. Amer. Math. Soc.}, 342 (1994), 155.  doi: 10.2307/2154688.  Google Scholar

[16]

S. B. Kuksin, Hamiltonian perturbation of infinite-dimensional linear system with an imaginary spectrum,, \emph{Funkt. Anal. Prilozh.}, 21 (1987), 22.   Google Scholar

[17]

S. B. Kuksin, Perturbation of quasiperiodic solutions of infinite-dimensional Hamiltonian systems,, \emph{Izv. Akad. Nauk SSSR, 52 (1989), 41.   Google Scholar

[18]

S. B. Kuksin, Nearly Integrable Infinite-dimensional Hamiltonian Systems,, Springer-Verlag, (1993).   Google Scholar

[19]

S. B. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation,, \emph{Ann. of Math.}, 143 (1996), 149.  doi: 10.2307/2118656.  Google Scholar

[20]

S. B. Kuksin, On small denominators equations with large variable coefficients,, \emph{J. Appl. Math. Phys.}, 48 (1997), 262.  doi: 10.1007/PL00001476.  Google Scholar

[21]

S. B. Kuksin, A KAM theorem for equations of the Korteweg-de Vries type,, \emph{Rev. Math-Math Phys.}, 10 (1998), 1.   Google Scholar

[22]

S. B. Kuksin, Analysis of Hamiltonian PDEs,, Oxford Univ. Press, (2000).   Google Scholar

[23]

J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large variable coefficient,, \emph{Commun. Pure Appl. Math.}, 63 (2010), 1145.  doi: 10.1002/cpa.20314.  Google Scholar

[24]

J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations,, \emph{Commun. Math. Phys.}, 307 (2011), 629.  doi: 10.1007/s00220-011-1353-3.  Google Scholar

[25]

J. Liu and X. Yuan, KAM for the derivative nonliear Schrödinger equation with periodic boundary conditions,, \emph{Journal of Differential Equations}, 256 (2014), 1627.  doi: 10.1016/j.jde.2013.11.007.  Google Scholar

[26]

L. Mi, Quasi-periodic solutions of derivative nonlinear Schrödinger equations with a given potential,, \emph{Journal of Mathematical Analysis and Applications}, 390 (2012), 335.  doi: 10.1016/j.jmaa.2012.01.046.  Google Scholar

[27]

L. Mi and K. Zhang, Invariant tori for Benjamin-Ono equation with unbounded quasi-periodically forced perturbation,, \emph{Discrete and Continuous Dynamical Systems-Series A}, 34 (2014), 689.  doi: 10.3934/dcds.2014.34.689.  Google Scholar

[28]

L. Molinet and F. Ribaud, Well-posedness results for the generalized Benjamin-Ono equation with small initial data,, \emph{J. Math. Pures Appl.}, 83 (2004), 277.  doi: 10.1016/j.matpur.2003.11.005.  Google Scholar

[29]

H. Ono, Algebraic solitary waves in stratified fluids,, \emph{Journal of the Physical Society of Japan}, 39 (1975), 1082.   Google Scholar

[30]

J. Pöschel, A KAM theorem for some nonlinear PDEs,, \emph{Ann. Scuola Norm. Sup. Pisacl. Sci.}, 23 (1996), 119.   Google Scholar

[31]

J. Pöschel, Quasi-periodic solutions for nonlinear wave equations,, \emph{Comm. Math. Helv.}, 71 (1996), 269.  doi: 10.1007/BF02566420.  Google Scholar

[32]

T. Tao, Global well-posedness of the Benjamin-Ono equation in H1(R),, \emph{J. Hyperbolic Differ. Equ.}, 1 (2004), 27.  doi: 10.1142/S0219891604000032.  Google Scholar

[33]

C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equation via KAM theory,, \emph{Commun. Math. Phys.}, 127 (1990), 479.   Google Scholar

[34]

X. Yuan and K. Zhang, A reduction theorem for time dependent Schrödinger operator with finite differentiable unbounded perturbation,, \emph{J. Math. Phys.}, 54 (2013).  doi: 10.1063/1.4803852.  Google Scholar

[35]

J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations,, \emph{Nonlinearity}, 24 (2011), 1189.  doi: 10.1088/0951-7715/24/4/010.  Google Scholar

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