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

Smooth quasi-periodic solutions for the perturbed mKdV equation

 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].
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 [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

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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