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September  2016, 15(5): 1857-1869. doi: 10.3934/cpaa.2016019

Smooth quasi-periodic solutions for the perturbed mKdV equation

Siqi Xu 1, and  Dongfeng Yan 1,

1. 

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

Received  November 2015 Revised  March 2016 Published  July 2016

This paper aims to study the time quasi-periodic solutions for one dimensional modified KdV (mKdV, for short) equation with perturbation \begin{eqnarray} u_t=-u_{x x x}-6 u^{2}u_x+perturbation ,x\in \mathbb{T}. \end{eqnarray} We show that, for any $n \in \mathbb{N}$ and a subset of $\mathbb{Z} \backslash \{0\}$ like $\{j_1 < j_2 < \cdots < j_n\}$, this equation admits a large amount of smooth n-dimensional invariant tori, along which exists a quantity of smooth quasi-periodic solutions. The proof is based on partial Birkhoff normal form and an unbounded KAM theorem established by Liu-Yuan in [Commun. Math. Phys., 307 (2011), 629-673].
Keywords: quasi-periodic solutions, partial normal form., Unbounded KAM theorem.
Mathematics Subject Classification: O175.14, O175.2.
Citation: Siqi Xu, Dongfeng Yan. Smooth quasi-periodic solutions for the perturbed mKdV equation. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1857-1869. doi: 10.3934/cpaa.2016019
References:
[1]

W. Ames, Nonlinear Partial Differential Equations,, New York: Academic Press, (1967).   Google Scholar

[2]

P. Baldi, Periodic solutions of forced Kirchhoff equations,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sci.}, 8 (2009), 117.   Google Scholar

[3]

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

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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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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, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations,, \emph{Geometric and Functional Analysis}, 3 (1993), 209.  doi: 10.1007/BF01895688.  Google Scholar

[8]

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), 45.  doi: 10.1155/S1073792894000516.  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,, \emph{Annals of Mathematics Studies}, 158 (2005).  doi: 10.1515/9781400837144.  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

[13]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness results for periodic and non-periodic KdV and modified KdV on R and T,, \emph{J. Amer. Math. Soc.}, 16 (2003), 705.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

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R. Feola and M. Procesi, Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations,, \emph{Journal of Differential Equations}, 259 (2015), 3389.  doi: 10.1016/j.jde.2015.04.025.  Google Scholar

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

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T. S. Komatsu and S. I. Sasa, Kink soliton characterizing traffic congestion,, \emph{Phys. Rev. E}, 52 (1995), 5574.   Google Scholar

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

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

[20]

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

[21]

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.} (ZAMP), 48 (1997), 262.  doi: 10.1007/PL00001476.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

S. Matsutani and H. Tsuru, Reflectionless quantum wire,, \emph{Journal of the Physical Society of Japan}, 60 (1991), 3640.  doi: 10.1143/JPSJ.60.3640.  Google Scholar

[29]

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

[30]

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

[31]

R. M. Miura, Korteweg-de Vries equation and generalizations, I. A remarkable explicit nonlinear transformation,, \emph{J. Math. Phys.}, 9 (1968), 1202.   Google Scholar

[32]

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

[33]

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

[34]

Y. Shi and J. Xu, KAM tori for defocusing modified KDV equation,, \emph{Journal of Geometry and Physics}, 90 (2015), 1.  doi: 10.1016/j.geomphys.2014.12.009.  Google Scholar

[35]

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

[36]

D. Yan, KAM tori for generalized Benjamin-Ono equation,, \emph{Communications on Pure $&$ Applied Analysis}, 14 (2015), 941.  doi: 10.3934/cpaa.2015.14.941.  Google Scholar

[37]

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

[38]

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

[39]

V. Ziegler, J. Dinkel, C. Setzer and K. E. Lonngren, On the propagation of nonlinear solitary waves in a distributed Schottky barrier diode transmission line,, \emph{Chaos, 12 (2001), 1719.   Google Scholar

show all references

References:
[1]

W. Ames, Nonlinear Partial Differential Equations,, New York: Academic Press, (1967).   Google Scholar

[2]

P. Baldi, Periodic solutions of forced Kirchhoff equations,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sci.}, 8 (2009), 117.   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations,, \emph{Geometric and Functional Analysis}, 3 (1993), 209.  doi: 10.1007/BF01895688.  Google Scholar

[8]

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), 45.  doi: 10.1155/S1073792894000516.  Google Scholar

[9]

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

[10]

J. Bourgain, Green's function estimates for lattice Schrödinger operators and applications,, \emph{Annals of Mathematics Studies}, 158 (2005).  doi: 10.1515/9781400837144.  Google Scholar

[11]

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

[12]

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

[13]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness results for periodic and non-periodic KdV and modified KdV on R and T,, \emph{J. Amer. Math. Soc.}, 16 (2003), 705.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[14]

R. Feola and M. Procesi, Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations,, \emph{Journal of Differential Equations}, 259 (2015), 3389.  doi: 10.1016/j.jde.2015.04.025.  Google Scholar

[15]

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

[16]

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

[17]

T. S. Komatsu and S. I. Sasa, Kink soliton characterizing traffic congestion,, \emph{Phys. Rev. E}, 52 (1995), 5574.   Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

S. Matsutani and H. Tsuru, Reflectionless quantum wire,, \emph{Journal of the Physical Society of Japan}, 60 (1991), 3640.  doi: 10.1143/JPSJ.60.3640.  Google Scholar

[29]

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

[30]

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

[31]

R. M. Miura, Korteweg-de Vries equation and generalizations, I. A remarkable explicit nonlinear transformation,, \emph{J. Math. Phys.}, 9 (1968), 1202.   Google Scholar

[32]

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

[33]

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

[34]

Y. Shi and J. Xu, KAM tori for defocusing modified KDV equation,, \emph{Journal of Geometry and Physics}, 90 (2015), 1.  doi: 10.1016/j.geomphys.2014.12.009.  Google Scholar

[35]

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

[36]

D. Yan, KAM tori for generalized Benjamin-Ono equation,, \emph{Communications on Pure $&$ Applied Analysis}, 14 (2015), 941.  doi: 10.3934/cpaa.2015.14.941.  Google Scholar

[37]

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

[38]

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

[39]

V. Ziegler, J. Dinkel, C. Setzer and K. E. Lonngren, On the propagation of nonlinear solitary waves in a distributed Schottky barrier diode transmission line,, \emph{Chaos, 12 (2001), 1719.   Google Scholar

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