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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 and 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.

[2]

P. Baldi, Periodic solutions of forced Kirchhoff equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 117-141.

[3]

P. Baldi, Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type, Ann. Inst. H. Poincare Anal. Non Linaire, 30 (2013), 33-77. doi: 10.1016/j.anihpc.2012.06.001.

[4]

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.

[5]

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.

[6]

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.

[7]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geometric and Functional Analysis, 3 (1993), 209-262. doi: 10.1007/BF01895688.

[8]

J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and application to nonlinear pde, Int. Math. Res. Notices, 11 (1994), 45-497. doi: 10.1155/S1073792894000516.

[9]

J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. Math., 148 (1998), 363-439. doi: 10.2307/121001.

[10]

J. Bourgain, Green's function estimates for lattice Schrödinger operators and applications, Annals of Mathematics Studies, 158, Princeton University Press, 2005. doi: 10.1515/9781400837144.

[11]

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.

[12]

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.

[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, J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1.

[14]

R. Feola and M. Procesi, Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, Journal of Differential Equations, 259 (2015), 3389-3447. doi: 10.1016/j.jde.2015.04.025.

[15]

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.

[16]

T. Kappler and J. Pöschel, KdV & KAM, Springer-Verlag, Berlin, Heidelberg, 2003. doi: 10.1007/978-3-662-08054-2.

[17]

T. S. Komatsu and S. I. Sasa, Kink soliton characterizing traffic congestion, Phys. Rev. E, 52 (1995), 5574-5582.

[18]

S. B. Kuksin, Hamiltonian perturbation of infinite-dimensional linear system with an imaginary spectrum, Funkt. Anal. Prilozh., 21, 22-37. [English translation in Funct. Anal. Appl., 21 (1987), 192-205.]

[19]

S. B. Kuksin, Perturbation of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Izv. Akad. Nauk SSSR, ser. Mat., 52, 41-63. [English translation in Math. USSR Izv., 32 (1989), 39-62.]

[20]

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

[21]

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.

[22]

S. B. Kuksin, On small denominators equations with large variable coefficients, J. Appl. Math. Phys. (ZAMP), 48 (1997), 262-271. doi: 10.1007/PL00001476.

[23]

S. B. Kuksin, A KAM theorem for equations of the Korteweg-de Vries type, Rev. Math-Math Phys., 10 (1998), 1-64.

[24]

S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford University Press, Oxford, 2000.

[25]

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.

[26]

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.

[27]

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.

[28]

S. Matsutani and H. Tsuru, Reflectionless quantum wire, Journal of the Physical Society of Japan, 60 (1991), 3640-3644. doi: 10.1143/JPSJ.60.3640.

[29]

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.

[30]

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.

[31]

R. M. Miura, Korteweg-de Vries equation and generalizations, I. A remarkable explicit nonlinear transformation, J. Math. Phys., 9 (1968), 1202-1204.

[32]

J. Pöschel, A KAM theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisacl. Sci., 23 (1996), 119-148.

[33]

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

[34]

Y. Shi and J. Xu, KAM tori for defocusing modified KDV equation, Journal of Geometry and Physics, 90 (2015), 1-10. doi: 10.1016/j.geomphys.2014.12.009.

[35]

C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equation via KAM theory, Commun. Math. Phys., 127 (1990), 479-528.

[36]

D. Yan, KAM tori for generalized Benjamin-Ono equation, Communications on Pure $&$ Applied Analysis, 14 (2015), 941-957. doi: 10.3934/cpaa.2015.14.941.

[37]

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.

[38]

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.

[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, Chaos, Solitons $&$ Fractals, 12 (2001), 1719-1728.

show all references

References:
[1]

W. Ames, Nonlinear Partial Differential Equations, New York: Academic Press, 1967.

[2]

P. Baldi, Periodic solutions of forced Kirchhoff equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 117-141.

[3]

P. Baldi, Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type, Ann. Inst. H. Poincare Anal. Non Linaire, 30 (2013), 33-77. doi: 10.1016/j.anihpc.2012.06.001.

[4]

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.

[5]

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.

[6]

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.

[7]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geometric and Functional Analysis, 3 (1993), 209-262. doi: 10.1007/BF01895688.

[8]

J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and application to nonlinear pde, Int. Math. Res. Notices, 11 (1994), 45-497. doi: 10.1155/S1073792894000516.

[9]

J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. Math., 148 (1998), 363-439. doi: 10.2307/121001.

[10]

J. Bourgain, Green's function estimates for lattice Schrödinger operators and applications, Annals of Mathematics Studies, 158, Princeton University Press, 2005. doi: 10.1515/9781400837144.

[11]

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.

[12]

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.

[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, J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1.

[14]

R. Feola and M. Procesi, Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, Journal of Differential Equations, 259 (2015), 3389-3447. doi: 10.1016/j.jde.2015.04.025.

[15]

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.

[16]

T. Kappler and J. Pöschel, KdV & KAM, Springer-Verlag, Berlin, Heidelberg, 2003. doi: 10.1007/978-3-662-08054-2.

[17]

T. S. Komatsu and S. I. Sasa, Kink soliton characterizing traffic congestion, Phys. Rev. E, 52 (1995), 5574-5582.

[18]

S. B. Kuksin, Hamiltonian perturbation of infinite-dimensional linear system with an imaginary spectrum, Funkt. Anal. Prilozh., 21, 22-37. [English translation in Funct. Anal. Appl., 21 (1987), 192-205.]

[19]

S. B. Kuksin, Perturbation of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Izv. Akad. Nauk SSSR, ser. Mat., 52, 41-63. [English translation in Math. USSR Izv., 32 (1989), 39-62.]

[20]

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

[21]

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.

[22]

S. B. Kuksin, On small denominators equations with large variable coefficients, J. Appl. Math. Phys. (ZAMP), 48 (1997), 262-271. doi: 10.1007/PL00001476.

[23]

S. B. Kuksin, A KAM theorem for equations of the Korteweg-de Vries type, Rev. Math-Math Phys., 10 (1998), 1-64.

[24]

S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford University Press, Oxford, 2000.

[25]

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.

[26]

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.

[27]

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.

[28]

S. Matsutani and H. Tsuru, Reflectionless quantum wire, Journal of the Physical Society of Japan, 60 (1991), 3640-3644. doi: 10.1143/JPSJ.60.3640.

[29]

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.

[30]

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.

[31]

R. M. Miura, Korteweg-de Vries equation and generalizations, I. A remarkable explicit nonlinear transformation, J. Math. Phys., 9 (1968), 1202-1204.

[32]

J. Pöschel, A KAM theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisacl. Sci., 23 (1996), 119-148.

[33]

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

[34]

Y. Shi and J. Xu, KAM tori for defocusing modified KDV equation, Journal of Geometry and Physics, 90 (2015), 1-10. doi: 10.1016/j.geomphys.2014.12.009.

[35]

C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equation via KAM theory, Commun. Math. Phys., 127 (1990), 479-528.

[36]

D. Yan, KAM tori for generalized Benjamin-Ono equation, Communications on Pure $&$ Applied Analysis, 14 (2015), 941-957. doi: 10.3934/cpaa.2015.14.941.

[37]

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.

[38]

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.

[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, Chaos, Solitons $&$ Fractals, 12 (2001), 1719-1728.

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