# American Institute of Mathematical Sciences

January  2015, 35(1): 173-204. doi: 10.3934/dcds.2015.35.173

## Small-divisor equation with large-variable coefficient and periodic solutions of DNLS equations

 1 School of Science, Shanghai Second Polytechnic University, Shanghai 201209, China

Received  January 2014 Revised  April 2014 Published  August 2014

In this paper, we establish an estimate for the solutions of a small-divisor equation with large variable coefficient. Then by formulating an infinite-dimensional KAM theorem which allows for multiple normal frequencies and unbounded perturbations, we prove that there are many periodic solutions for the derivative nonlinear Schrödinger equations subject to small Hamiltonian perturbations.
Citation: Meina Gao. Small-divisor equation with large-variable coefficient and periodic solutions of DNLS equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 173-204. doi: 10.3934/dcds.2015.35.173
##### References:
 [1] P. Baldi, Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type,, Ann. Inst. H. Poincare (C) Anal. Non Linaire, 30 (2013), 33. doi: 10.1016/j.anihpc.2012.06.001. Google Scholar [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. doi: 10.1007/s00208-013-1001-7. Google Scholar [3] D. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods,, Comm. Math. Phys., 219 (2001), 465. doi: 10.1007/s002200100426. Google Scholar [4] M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equation,, Ann. Sci. Ec. Norm. Super., 46 (2013), 301. Google Scholar [5] N. N. Bogolyubov, Yu. A. Mitropolskii and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics,, Springer-Verlag, (1976). Google Scholar [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. doi: 10.1155/S1073792894000516. Google Scholar [7] J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation,, Ann. Math., 148 (1998), 363. doi: 10.2307/121001. Google Scholar [8] J. Bourgain, Harmoinc Analysis and Partial Differential Equations,, University of Chicago Press, (1999). Google Scholar [9] L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions,, Commun. Math. Phys., 211 (2000), 497. doi: 10.1007/s002200050824. Google Scholar [10] W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equation,, Commun. Pure Appl. Math., 46 (1993), 1409. doi: 10.1002/cpa.3160461102. Google Scholar [11] L. Du and X. Yuan, Invariant tori of nonlinear Schrödinger equations with a given potential,, Dynamics of PDE, 3 (2006), 331. Google Scholar [12] L. H. Eliasson and S. B. Kuksin, KAM for the nonlinear Schrödinger equation,, Ann. Math., 172 (2010), 371. doi: 10.4007/annals.2010.172.371. Google Scholar [13] M. Gao and J. Zhang, Small-divisor equation of higher order with large variable coefficient,, Acta. Mathematica Sinica, 27 (2011), 2005. doi: 10.1007/s10114-011-0064-1. Google Scholar [14] T. Kappeler and J. Pöschel, KdV&KAM,, Springer-Verlag, (2003). doi: 10.1007/978-3-662-08054-2. Google Scholar [15] T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (1980). doi: 10.1007/978-3-642-66282-9. Google Scholar [16] S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum,, Funkt. Anal. Prilozh., 21 (1987), 22. Google Scholar [17] S. B. Kuksin, Perturbations of quasiperiodic solutions of infinite-dimensional Hamiltonian systems,, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 41. Google Scholar [18] S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems,, Lecture Notes in Math., (1556). Google Scholar [19] S. B. Kuksin, Analysis of Hamiltonian PDEs,, Oxford University Press, (2000). Google Scholar [20] S. B. Kuksin and J. Pöschel, Invariant cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation,, Ann. of Math. (2), 143 (1996), 149. doi: 10.2307/2118656. Google Scholar [21] P. Lancaster, Theory of Matrices,, Academic Press LTD, (1969). Google Scholar [22] J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large variable coefficient,, Comm. Pure Appl. Math., 63 (2010), 1145. doi: 10.1002/cpa.20314. Google Scholar [23] J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations,, Commun. Math. Phys., 307 (2011), 629. doi: 10.1007/s00220-011-1353-3. Google Scholar [24] J. Liu and X. Yuan, KAM for the derivative nonlinear Schrödinger equation with periodic boundary conditions,, J. Differential Equations, 256 (2014), 1627. doi: 10.1016/j.jde.2013.11.007. Google Scholar [25] C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory,, Commun. Math. Phys., 127 (1990), 479. doi: 10.1007/BF02104499. Google Scholar [26] J. Pöschel, A KAM-theorem for some nonlinear partial differential equations,, Ann. Sc. Norm. Sup. Pisa., 23 (1996), 119. Google Scholar [27] J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation,, Comment. Math. Helv., 71 (1996), 269. doi: 10.1007/BF02566420. Google Scholar [28] J. You, Perturbations of lower dimensional tori for hamiltonian systems,, J. Differential Equations, 152 (1999), 1. doi: 10.1006/jdeq.1998.3515. Google Scholar [29] X. Yuan, A KAM theorem with appllications to partial differential equations of higher dimension,, Commun. Math. Phys., 275 (2007), 97. doi: 10.1007/s00220-007-0287-2. Google Scholar [30] J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations,, Nonlinearity, 24 (2011), 1198. doi: 10.1088/0951-7715/24/4/010. Google Scholar [31] W. P. Ziemer, Weakly Differentiable Functions,, Graduate Texts in Math., (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar

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##### References:
 [1] P. Baldi, Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type,, Ann. Inst. H. Poincare (C) Anal. Non Linaire, 30 (2013), 33. doi: 10.1016/j.anihpc.2012.06.001. Google Scholar [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. doi: 10.1007/s00208-013-1001-7. Google Scholar [3] D. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods,, Comm. Math. Phys., 219 (2001), 465. doi: 10.1007/s002200100426. Google Scholar [4] M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equation,, Ann. Sci. Ec. Norm. Super., 46 (2013), 301. Google Scholar [5] N. N. Bogolyubov, Yu. A. Mitropolskii and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics,, Springer-Verlag, (1976). Google Scholar [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. doi: 10.1155/S1073792894000516. Google Scholar [7] J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation,, Ann. Math., 148 (1998), 363. doi: 10.2307/121001. Google Scholar [8] J. Bourgain, Harmoinc Analysis and Partial Differential Equations,, University of Chicago Press, (1999). Google Scholar [9] L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions,, Commun. Math. Phys., 211 (2000), 497. doi: 10.1007/s002200050824. Google Scholar [10] W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equation,, Commun. Pure Appl. Math., 46 (1993), 1409. doi: 10.1002/cpa.3160461102. Google Scholar [11] L. Du and X. Yuan, Invariant tori of nonlinear Schrödinger equations with a given potential,, Dynamics of PDE, 3 (2006), 331. Google Scholar [12] L. H. Eliasson and S. B. Kuksin, KAM for the nonlinear Schrödinger equation,, Ann. Math., 172 (2010), 371. doi: 10.4007/annals.2010.172.371. Google Scholar [13] M. Gao and J. Zhang, Small-divisor equation of higher order with large variable coefficient,, Acta. Mathematica Sinica, 27 (2011), 2005. doi: 10.1007/s10114-011-0064-1. Google Scholar [14] T. Kappeler and J. Pöschel, KdV&KAM,, Springer-Verlag, (2003). doi: 10.1007/978-3-662-08054-2. Google Scholar [15] T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (1980). doi: 10.1007/978-3-642-66282-9. Google Scholar [16] S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum,, Funkt. Anal. Prilozh., 21 (1987), 22. Google Scholar [17] S. B. Kuksin, Perturbations of quasiperiodic solutions of infinite-dimensional Hamiltonian systems,, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 41. Google Scholar [18] S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems,, Lecture Notes in Math., (1556). Google Scholar [19] S. B. Kuksin, Analysis of Hamiltonian PDEs,, Oxford University Press, (2000). Google Scholar [20] S. B. Kuksin and J. Pöschel, Invariant cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation,, Ann. of Math. (2), 143 (1996), 149. doi: 10.2307/2118656. Google Scholar [21] P. Lancaster, Theory of Matrices,, Academic Press LTD, (1969). Google Scholar [22] J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large variable coefficient,, Comm. Pure Appl. Math., 63 (2010), 1145. doi: 10.1002/cpa.20314. Google Scholar [23] J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations,, Commun. Math. Phys., 307 (2011), 629. doi: 10.1007/s00220-011-1353-3. Google Scholar [24] J. Liu and X. Yuan, KAM for the derivative nonlinear Schrödinger equation with periodic boundary conditions,, J. Differential Equations, 256 (2014), 1627. doi: 10.1016/j.jde.2013.11.007. Google Scholar [25] C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory,, Commun. Math. Phys., 127 (1990), 479. doi: 10.1007/BF02104499. Google Scholar [26] J. Pöschel, A KAM-theorem for some nonlinear partial differential equations,, Ann. Sc. Norm. Sup. Pisa., 23 (1996), 119. Google Scholar [27] J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation,, Comment. Math. Helv., 71 (1996), 269. doi: 10.1007/BF02566420. Google Scholar [28] J. You, Perturbations of lower dimensional tori for hamiltonian systems,, J. Differential Equations, 152 (1999), 1. doi: 10.1006/jdeq.1998.3515. Google Scholar [29] X. Yuan, A KAM theorem with appllications to partial differential equations of higher dimension,, Commun. Math. Phys., 275 (2007), 97. doi: 10.1007/s00220-007-0287-2. Google Scholar [30] J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations,, Nonlinearity, 24 (2011), 1198. doi: 10.1088/0951-7715/24/4/010. Google Scholar [31] W. P. Ziemer, Weakly Differentiable Functions,, Graduate Texts in Math., (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar
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