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Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four
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 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.

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##### 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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