KAM Tori for generalized Benjamin-Ono equation

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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

Abstract
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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 and 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, Ann. Inst. H. Poincar Anal. Non Linaire, 30 (2013), 33-77. doi: 10.1016/j.anihpc.2012.06.001.
[2]	P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., 359 (2014), 471-536. doi: 10.1007/s00208-013-1001-7.
[3]	P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear KdV, C. R. Math. Acad. Sci. Paris, 352 (2014), 603-607. doi: 10.1016/j.crma.2014.04.012.
[4]	T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592.
[5]	M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equations, Arch. Ration. Mech. Anal., 212 (2014), 905-955. doi: 10.1007/s00205-014-0726-0.
[6]	J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and application to nonlinear pde, Int. Math. Res. Notices, 11 (1994), 475-497. doi: 10.1155/S1073792894000516.
[7]	J. Bourgain, On Melnikov's persistence problem, Math. Res. Lett., 4 (1997), 445-458. doi: 10.4310/MRL.1997.v4.n4.a1.
[8]	J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. Math., 148 (1998), 363-439. doi: 10.2307/121001.
[9]	J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of mathematics studies, Princeton University Press, 2005.
[10]	J. Bourgain, On invariant tori of full dimension for 1D periodic NLS, J. Funct. Anal., 229 (2005), 62-94. doi: 10.1016/j.jfa.2004.10.019.
[11]	L. Chierchia and J. You, KAM tori for 1D nonlinear wave equation with periodic boundary conditions, Commun. Math. Phys., 211 (2000), 497-525. doi: 10.1007/s002200050824.
[12]	G. Iooss, P. I. Plotnikov and J. F. Toland, Standing waves on an infinitely deep perfect fluid under gravity, Arch. Ration. Mech. Anal., 177 (2005), 367-478. doi: 10.1007/s00205-005-0381-6.
[13]	J. R. Iorio, On the Cauchy problem for the Benjamin-Ono equation, Comm. Partial Differential equations, 11 (1986), 1031-1081. doi: 10.1080/03605308608820456.
[14]	T. Kappler and J. Pöschel, KdV $&$ KAM, Springer-Verlag,Berlin,Heidelberg, 2003. doi: 10.1007/978-3-662-08054-2.
[15]	C. E. Kenig, G. Ponce and L. Vega, On the generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 342 (1994), 155-172. doi: 10.2307/2154688.
[16]	S. B. Kuksin, Hamiltonian perturbation of infinite-dimensional linear system with an imaginary spectrum, Funkt. Anal. Prilozh., 21 (1987), 22-37. [English translation in Funct. Anal. Appl., 21(1987), 192-205.]
[17]	S. B. Kuksin, Perturbation of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Izv. Akad. Nauk SSSR, ser. Mat., 52 (1989), 41-63. [English translation in Math. USSR Izv., 32 (1989), 39-62.]
[18]	S. B. Kuksin, Nearly Integrable Infinite-dimensional Hamiltonian Systems, Springer-Verlag, Berlin, 1993.
[19]	S. B. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math., 143 (1996), 149-179. doi: 10.2307/2118656.
[20]	S. B. Kuksin, On small denominators equations with large variable coefficients, J. Appl. Math. Phys., 48(1997), 262-271. doi: 10.1007/PL00001476.
[21]	S. B. Kuksin, A KAM theorem for equations of the Korteweg-de Vries type, Rev. Math-Math Phys., 10 (1998), 1-64.
[22]	S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Univ. Press,Oxford, 2000.
[23]	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.
[24]	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.
[25]	J. Liu and X. Yuan, KAM for the derivative nonliear Schrödinger equation with periodic boundary conditions, Journal of Differential Equations, 256 (2014), 1627-1652. doi: 10.1016/j.jde.2013.11.007.
[26]	L. Mi, Quasi-periodic solutions of derivative nonlinear Schrödinger equations with a given potential, Journal of Mathematical Analysis and Applications, 390 (2012), 335-354. doi: 10.1016/j.jmaa.2012.01.046.
[27]	L. Mi and K. Zhang, Invariant tori for Benjamin-Ono equation with unbounded quasi-periodically forced perturbation, Discrete and Continuous Dynamical Systems-Series A, 34 (2014), 689-707. doi: 10.3934/dcds.2014.34.689.
[28]	L. Molinet and F. Ribaud, Well-posedness results for the generalized Benjamin-Ono equation with small initial data, J. Math. Pures Appl., 83 (2004), 277-311. doi: 10.1016/j.matpur.2003.11.005.
[29]	H. Ono, Algebraic solitary waves in stratified fluids, Journal of the Physical Society of Japan, 39 (1975), 1082-1091.
[30]	J. Pöschel, A KAM theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisacl. Sci., 23 (1996), 119-148.
[31]	J. Pöschel, Quasi-periodic solutions for nonlinear wave equations, Comm. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420.
[32]	T. Tao, Global well-posedness of the Benjamin-Ono equation in H1(R), J. Hyperbolic Differ. Equ., 1 (2004), 27-49. doi: 10.1142/S0219891604000032.
[33]	C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equation via KAM theory, Commun. Math. Phys., 127 (1990), 479-528.
[34]	X. Yuan and K. Zhang, A reduction theorem for time dependent Schrödinger operator with finite differentiable unbounded perturbation, J. Math. Phys., 54 (2013), 052701. doi: 10.1063/1.4803852.
[35]	J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations, Nonlinearity, 24 (2011), 1189-1228. doi: 10.1088/0951-7715/24/4/010.

show all references

References:

[1]	P. Baldi, Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type, Ann. Inst. H. Poincar Anal. Non Linaire, 30 (2013), 33-77. doi: 10.1016/j.anihpc.2012.06.001.
[2]	P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., 359 (2014), 471-536. doi: 10.1007/s00208-013-1001-7.
[3]	P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear KdV, C. R. Math. Acad. Sci. Paris, 352 (2014), 603-607. doi: 10.1016/j.crma.2014.04.012.
[4]	T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592.
[5]	M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equations, Arch. Ration. Mech. Anal., 212 (2014), 905-955. doi: 10.1007/s00205-014-0726-0.
[6]	J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and application to nonlinear pde, Int. Math. Res. Notices, 11 (1994), 475-497. doi: 10.1155/S1073792894000516.
[7]	J. Bourgain, On Melnikov's persistence problem, Math. Res. Lett., 4 (1997), 445-458. doi: 10.4310/MRL.1997.v4.n4.a1.
[8]	J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. Math., 148 (1998), 363-439. doi: 10.2307/121001.
[9]	J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of mathematics studies, Princeton University Press, 2005.
[10]	J. Bourgain, On invariant tori of full dimension for 1D periodic NLS, J. Funct. Anal., 229 (2005), 62-94. doi: 10.1016/j.jfa.2004.10.019.
[11]	L. Chierchia and J. You, KAM tori for 1D nonlinear wave equation with periodic boundary conditions, Commun. Math. Phys., 211 (2000), 497-525. doi: 10.1007/s002200050824.
[12]	G. Iooss, P. I. Plotnikov and J. F. Toland, Standing waves on an infinitely deep perfect fluid under gravity, Arch. Ration. Mech. Anal., 177 (2005), 367-478. doi: 10.1007/s00205-005-0381-6.
[13]	J. R. Iorio, On the Cauchy problem for the Benjamin-Ono equation, Comm. Partial Differential equations, 11 (1986), 1031-1081. doi: 10.1080/03605308608820456.
[14]	T. Kappler and J. Pöschel, KdV $&$ KAM, Springer-Verlag,Berlin,Heidelberg, 2003. doi: 10.1007/978-3-662-08054-2.
[15]	C. E. Kenig, G. Ponce and L. Vega, On the generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 342 (1994), 155-172. doi: 10.2307/2154688.
[16]	S. B. Kuksin, Hamiltonian perturbation of infinite-dimensional linear system with an imaginary spectrum, Funkt. Anal. Prilozh., 21 (1987), 22-37. [English translation in Funct. Anal. Appl., 21(1987), 192-205.]
[17]	S. B. Kuksin, Perturbation of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Izv. Akad. Nauk SSSR, ser. Mat., 52 (1989), 41-63. [English translation in Math. USSR Izv., 32 (1989), 39-62.]
[18]	S. B. Kuksin, Nearly Integrable Infinite-dimensional Hamiltonian Systems, Springer-Verlag, Berlin, 1993.
[19]	S. B. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math., 143 (1996), 149-179. doi: 10.2307/2118656.
[20]	S. B. Kuksin, On small denominators equations with large variable coefficients, J. Appl. Math. Phys., 48(1997), 262-271. doi: 10.1007/PL00001476.
[21]	S. B. Kuksin, A KAM theorem for equations of the Korteweg-de Vries type, Rev. Math-Math Phys., 10 (1998), 1-64.
[22]	S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Univ. Press,Oxford, 2000.
[23]	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.
[24]	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.
[25]	J. Liu and X. Yuan, KAM for the derivative nonliear Schrödinger equation with periodic boundary conditions, Journal of Differential Equations, 256 (2014), 1627-1652. doi: 10.1016/j.jde.2013.11.007.
[26]	L. Mi, Quasi-periodic solutions of derivative nonlinear Schrödinger equations with a given potential, Journal of Mathematical Analysis and Applications, 390 (2012), 335-354. doi: 10.1016/j.jmaa.2012.01.046.
[27]	L. Mi and K. Zhang, Invariant tori for Benjamin-Ono equation with unbounded quasi-periodically forced perturbation, Discrete and Continuous Dynamical Systems-Series A, 34 (2014), 689-707. doi: 10.3934/dcds.2014.34.689.
[28]	L. Molinet and F. Ribaud, Well-posedness results for the generalized Benjamin-Ono equation with small initial data, J. Math. Pures Appl., 83 (2004), 277-311. doi: 10.1016/j.matpur.2003.11.005.
[29]	H. Ono, Algebraic solitary waves in stratified fluids, Journal of the Physical Society of Japan, 39 (1975), 1082-1091.
[30]	J. Pöschel, A KAM theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisacl. Sci., 23 (1996), 119-148.
[31]	J. Pöschel, Quasi-periodic solutions for nonlinear wave equations, Comm. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420.
[32]	T. Tao, Global well-posedness of the Benjamin-Ono equation in H1(R), J. Hyperbolic Differ. Equ., 1 (2004), 27-49. doi: 10.1142/S0219891604000032.
[33]	C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equation via KAM theory, Commun. Math. Phys., 127 (1990), 479-528.
[34]	X. Yuan and K. Zhang, A reduction theorem for time dependent Schrödinger operator with finite differentiable unbounded perturbation, J. Math. Phys., 54 (2013), 052701. doi: 10.1063/1.4803852.
[35]	J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations, Nonlinearity, 24 (2011), 1189-1228. doi: 10.1088/0951-7715/24/4/010.

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