# 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 and Continuous Dynamical Systems, 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), 33-77. doi: 10.1016/j.anihpc.2012.06.001. [2] P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., 359 (2014), 471-536. doi: 10.1007/s00208-013-1001-7. [3] D. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods, Comm. Math. Phys., 219 (2001), 465-480. doi: 10.1007/s002200100426. [4] M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equation, Ann. Sci. Ec. Norm. Super., 46 (2013), 301-373. [5] N. N. Bogolyubov, Yu. A. Mitropolskii and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-Verlag, New York, 1976. [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), 475-497. doi: 10.1155/S1073792894000516. [7] J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. Math., 148 (1998), 363-439. doi: 10.2307/121001. [8] J. Bourgain, Harmoinc Analysis and Partial Differential Equations, University of Chicago Press, 1999. [9] L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Commun. Math. Phys., 211 (2000), 497-525. doi: 10.1007/s002200050824. [10] W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equation, Commun. Pure Appl. Math., 46 (1993), 1409-1498. doi: 10.1002/cpa.3160461102. [11] L. Du and X. Yuan, Invariant tori of nonlinear Schrödinger equations with a given potential, Dynamics of PDE, 3 (2006), 331-346. [12] L. H. Eliasson and S. B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. Math., 172 (2010), 371-435. doi: 10.4007/annals.2010.172.371. [13] M. Gao and J. Zhang, Small-divisor equation of higher order with large variable coefficient, Acta. Mathematica Sinica, English Series, 27 (2011), 2005-2032. doi: 10.1007/s10114-011-0064-1. [14] T. Kappeler and J. Pöschel, KdV&KAM, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2. [15] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin Heidelberg New York, 1980. doi: 10.1007/978-3-642-66282-9. [16] S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funkt. Anal. Prilozh., 21 (1987), 22-37; English translation in Funct. Anal. Appl., 21 (1987), 192-205. [17] S. B. Kuksin, Perturbations of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 41-63; English translation in Math. USSR Izv., 32 (1989), 39-62. [18] S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Math., 1556, Springer-Verlag, New York, 1993. [19] S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford University Press, Oxford, 2000. [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), 149-179. doi: 10.2307/2118656. [21] P. Lancaster, Theory of Matrices, Academic Press LTD, New York and London, 1969. [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), 1145-1172. doi: 10.1002/cpa.20314. [23] 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. [24] J. Liu and X. Yuan, KAM for the derivative nonlinear Schrödinger equation with periodic boundary conditions, J. Differential Equations, 256 (2014), 1627-1652. doi: 10.1016/j.jde.2013.11.007. [25] C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys., 127 (1990), 479-528. doi: 10.1007/BF02104499. [26] J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Sc. Norm. Sup. Pisa., 23 (1996), 119-148. [27] J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420. [28] J. You, Perturbations of lower dimensional tori for hamiltonian systems, J. Differential Equations, 152 (1999), 1-29. doi: 10.1006/jdeq.1998.3515. [29] X. Yuan, A KAM theorem with appllications to partial differential equations of higher dimension, Commun. Math. Phys., 275 (2007), 97-137. doi: 10.1007/s00220-007-0287-2. [30] J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations, Nonlinearity, 24 (2011), 1198-2118. doi: 10.1088/0951-7715/24/4/010. [31] W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Math., 120, Springer-Verlag, Berlin Heidelberg New York, 1989. doi: 10.1007/978-1-4612-1015-3.

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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), 33-77. doi: 10.1016/j.anihpc.2012.06.001. [2] P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., 359 (2014), 471-536. doi: 10.1007/s00208-013-1001-7. [3] D. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods, Comm. Math. Phys., 219 (2001), 465-480. doi: 10.1007/s002200100426. [4] M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equation, Ann. Sci. Ec. Norm. Super., 46 (2013), 301-373. [5] N. N. Bogolyubov, Yu. A. Mitropolskii and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-Verlag, New York, 1976. [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), 475-497. doi: 10.1155/S1073792894000516. [7] J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. Math., 148 (1998), 363-439. doi: 10.2307/121001. [8] J. Bourgain, Harmoinc Analysis and Partial Differential Equations, University of Chicago Press, 1999. [9] L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Commun. Math. Phys., 211 (2000), 497-525. doi: 10.1007/s002200050824. [10] W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equation, Commun. Pure Appl. Math., 46 (1993), 1409-1498. doi: 10.1002/cpa.3160461102. [11] L. Du and X. Yuan, Invariant tori of nonlinear Schrödinger equations with a given potential, Dynamics of PDE, 3 (2006), 331-346. [12] L. H. Eliasson and S. B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. Math., 172 (2010), 371-435. doi: 10.4007/annals.2010.172.371. [13] M. Gao and J. Zhang, Small-divisor equation of higher order with large variable coefficient, Acta. Mathematica Sinica, English Series, 27 (2011), 2005-2032. doi: 10.1007/s10114-011-0064-1. [14] T. Kappeler and J. Pöschel, KdV&KAM, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2. [15] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin Heidelberg New York, 1980. doi: 10.1007/978-3-642-66282-9. [16] S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funkt. Anal. Prilozh., 21 (1987), 22-37; English translation in Funct. Anal. Appl., 21 (1987), 192-205. [17] S. B. Kuksin, Perturbations of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 41-63; English translation in Math. USSR Izv., 32 (1989), 39-62. [18] S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Math., 1556, Springer-Verlag, New York, 1993. [19] S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford University Press, Oxford, 2000. [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), 149-179. doi: 10.2307/2118656. [21] P. Lancaster, Theory of Matrices, Academic Press LTD, New York and London, 1969. [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), 1145-1172. doi: 10.1002/cpa.20314. [23] 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. [24] J. Liu and X. Yuan, KAM for the derivative nonlinear Schrödinger equation with periodic boundary conditions, J. Differential Equations, 256 (2014), 1627-1652. doi: 10.1016/j.jde.2013.11.007. [25] C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys., 127 (1990), 479-528. doi: 10.1007/BF02104499. [26] J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Sc. Norm. Sup. Pisa., 23 (1996), 119-148. [27] J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420. [28] J. You, Perturbations of lower dimensional tori for hamiltonian systems, J. Differential Equations, 152 (1999), 1-29. doi: 10.1006/jdeq.1998.3515. [29] X. Yuan, A KAM theorem with appllications to partial differential equations of higher dimension, Commun. Math. Phys., 275 (2007), 97-137. doi: 10.1007/s00220-007-0287-2. [30] J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations, Nonlinearity, 24 (2011), 1198-2118. doi: 10.1088/0951-7715/24/4/010. [31] W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Math., 120, Springer-Verlag, Berlin Heidelberg New York, 1989. doi: 10.1007/978-1-4612-1015-3.
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