# American Institute of Mathematical Sciences

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

## KAM Tori for generalized Benjamin-Ono equation

 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].
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 [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

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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