# American Institute of Mathematical Sciences

January  2017, 16(1): 25-68. doi: 10.3934/cpaa.2017002

## Invariant tori of a nonlinear Schrödinger equation with quasi-periodically unbounded perturbations

 1 School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, P. R. China 2 School of Mathematics, Shandong University, Jinan 250100, P. R. China

Received  June 2015 Revised  August 2016 Published  November 2016

Fund Project: The first author is supported by the National Natural Science Foundation of China (Grant 11171185, 11571201) and the Research Foundation for Doctor Programme of Henan Polytechnic University (Grant B2016-58). the second author is supported by the National Natural Science Foundation of China (Grant 11171185, 11571201).

This paper is concerned with the derivative nonlinear Schrödinger equation with quasi-periodic forcing under periodic boundary conditions
 ${\rm{i}} u_t+u_{xx}+{\rm{i}} (B+\epsilon g(\beta t))(f(|u|^2)u)_x=0, \ \ x\in \mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}.$
Assume that the frequency vector $\beta$ is co-linear with a fixed Diophantine vector $\bar{\beta}\in \mathbb{R}.{m}$, that is, $\beta=\lambda \bar{\beta}$, $\lambda \in [1/2, 3/2]$. We show that above equation possesses a Cantorian branch of invariant $n$--tori and exists many smooth quasi-periodic solutions with $(m+n)$ non-resonance frequencies $(\lambda\bar{\beta}, \omega_{\ast})$. The proof is based on a Kolmogorov--Arnold--Moser (KAM) iterative procedure for quasi-periodically unbounded vector fields and partial Birkhoff normal form.
Citation: Jie Liu, Jianguo Si. Invariant tori of a nonlinear Schrödinger equation with quasi-periodically unbounded perturbations. Communications on Pure and Applied Analysis, 2017, 16 (1) : 25-68. doi: 10.3934/cpaa.2017002
##### References:
 [1] P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbation of Airy equation, Math. Ann., 359 (2014), 471-536. doi: 10.1007/s00208-013-1001-7. [2] M. Berti and L. Biasco, Branching of Cantor manifolds of elliptic tori and applications to PDEs, Comm. Math. Phys., 305 (2011), 741-796. doi: 10.1007/s00220-011-1264-3. [3] M. Berti and M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations, Comm. Partial Differential Eqautions, 31 (2006), 959-985. doi: 10.1080/03605300500358129. [4] J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math., 148 (1998), 363-439. doi: 10.2307/121001. [5] J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mathematics Studies, 158, Princeton University Press, Princeton, 2005. doi: 10.1515/9781400837144. [6] L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys., 211 (2000), 497-525. doi: 10.1007/s002200050824. [7] W. Craig and C. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409-1498. doi: 10.1002/cpa.3160461102. [8] H. L. Eliasson and S. B. Kuksin, KAM for the non-linear Schrödinger equation, Ann. of Math., 172 (2010), 371-435. doi: 10.4007/annals.2010.172.371. [9] R. Feola and M. Procesi, Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differential Equations, 259 (2015), 3389-3447. doi: 10.1016/j.jde.2015.04.025. [10] R. Feola, KAM for quasi-linear forced Hamiltonian NLS, preprint, arXiv: 1602.01341. [11] J. Geng and J. You, A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Comm. Math. Phys., 262 (2006), 343-372. doi: 10.1007/s00220-005-1497-0. [12] J. Geng and Y. Yi, Quasi-periodic solutions in a nonlinear Schrödinger equation, J. Differential Equations, 233 (2007), 512-542. doi: 10.1016/j.jde.2006.07.027. [13] J. Geng, Invariant tori of full dimension for a nonlinear Schrödinger equation, J. Differential Equations, 252 (2012), 1-34. doi: 10.1016/j.jde.2011.09.006. [14] J. Geng and J. Wu, Real analytic quasi-periodic solutions for the derivative nonlinear Schrödinger equations, J. Math. Phys., 53 (2012), 102702. doi: 10.1063/1.4754822. [15] B. Grébert and L. Thomann, KAM for the quantum harmonic oscillator, Comm. Math. Phys., 307 (2011), 383-427. doi: 10.1007/s00220-011-1327-5. [16] S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, (Russian) Funktsional. Anal. iPrilozhen., 21 (1987), 22-37; English translation in: Funct. Anal. Appl., 21 (1987), 192-205. [17] S. B. Kuksin, Perturbations of conditionally periodic solutions of infinite-dimensional Hamiltonian systems, (Russian) 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 Mathematics, 1556, Springer-Verlag, Berlin, 1993. doi: 10.1007/BFb0092243. [19] S. B. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math., 143 (1996), 147-179. [20] S. B. Kuksin, On small-denominators equations with large variable coefficients, Z. Angew. Math. Phys., 48 (1997), 262-271. doi: 10.1007/PL00001476. [21] S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications, 19, Oxford University Press, Oxford, 2000. [22] T. Kappeler and J. Pöschel, KdV and KAM, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2. [23] L. Jiao and Y. Wang, The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation, Commun. Pure Appl. Anal., 8 (2009), 1585-1606. doi: 10.3934/cpaa.2009.8.1585. [24] J. Liu and J. Si, Invariant tori for a derivative nonlinear Schrödinger equation with quasiperiodic forcing, J Math. Phys. , 56 (2015), 032702. doi: 10.1063/1.4916287. [25] J. Liu, Periodic and quasi-periodic solutions of a derivative nonlinear Schrödinger equation, Appl. Anal., 95 (2016), 801-825. doi: 10.1080/00036811.2015.1032942. [26] 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. [27] J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Comm. Math. Phys., 307 (2011), 629-673. doi: 10.1007/s00220-011-1353-3. [28] 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. [29] L. Mi and K. Zhang, Invariant tori for Benjamin-Ono equation with unbounded quasiperiodically forced perturbation, Discrete Contin. Dyn. Syst., 34 (2014), 689-707. doi: 10.3934/dcds.2014.34.689. [30] J. Pöschel, A KAM theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa. Cl. Sci., 23 (1996), 119-148. [31] J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420. [32] J. Rui and J. Si, Quasi-periodic solutions for quasi-periodically forced nonlinear Schrödinger equations with quasi-periodic inhomogeneous terms, Phys. D, 286/287 (2014), 1-31. doi: 10.1016/j.physd.2014.07.005. [33] M. B. Sevryuk, The reversible context 2 in KAM theory: the first steps, Regul. Chaotic Dyn., 16 (2011), 24-38. doi: 10.1134/S1560354710520035. [34] J. Si, Quasi-periodic solutions of a non-autonomous wave equations with quasi-periodic forcing, J. Differential Equations, 252 (2012), 5274-5360. doi: 10.1016/j.jde.2012.01.034. [35] Y. Wang and J. Si, A result on quasi-periodic solutions of a nonlinear beam equation with a quasi-periodic forcing term, Z. Angew. Math. Phys., 63 (2012), 189-190. doi: 10.1007/s00033-011-0172-x. [36] C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys., 127 (1990), 479-528. [37] X. Yuan, Quasi-periodic solutions of nonlinear Schrödinger equations of higher dimension, J. Differential Equations, 195 (2003), 230-242. doi: 10.1016/S0022-0396(03)00095-0. [38] X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations, 230 (2006), 213-274. doi: 10.1016/j.jde.2005.12.012. [39] M. Zhang and J. Si, Quasi-periodic solutions of nonlinear wave equations with quasi-periodic forcing, Phys. D, 238 (2009), 2185-2215. doi: 10.1016/j.physd.2009.09.003.

show all references

##### References:
 [1] P. Baldi, M. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbation of Airy equation, Math. Ann., 359 (2014), 471-536. doi: 10.1007/s00208-013-1001-7. [2] M. Berti and L. Biasco, Branching of Cantor manifolds of elliptic tori and applications to PDEs, Comm. Math. Phys., 305 (2011), 741-796. doi: 10.1007/s00220-011-1264-3. [3] M. Berti and M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations, Comm. Partial Differential Eqautions, 31 (2006), 959-985. doi: 10.1080/03605300500358129. [4] J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math., 148 (1998), 363-439. doi: 10.2307/121001. [5] J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mathematics Studies, 158, Princeton University Press, Princeton, 2005. doi: 10.1515/9781400837144. [6] L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys., 211 (2000), 497-525. doi: 10.1007/s002200050824. [7] W. Craig and C. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409-1498. doi: 10.1002/cpa.3160461102. [8] H. L. Eliasson and S. B. Kuksin, KAM for the non-linear Schrödinger equation, Ann. of Math., 172 (2010), 371-435. doi: 10.4007/annals.2010.172.371. [9] R. Feola and M. Procesi, Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differential Equations, 259 (2015), 3389-3447. doi: 10.1016/j.jde.2015.04.025. [10] R. Feola, KAM for quasi-linear forced Hamiltonian NLS, preprint, arXiv: 1602.01341. [11] J. Geng and J. You, A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Comm. Math. Phys., 262 (2006), 343-372. doi: 10.1007/s00220-005-1497-0. [12] J. Geng and Y. Yi, Quasi-periodic solutions in a nonlinear Schrödinger equation, J. Differential Equations, 233 (2007), 512-542. doi: 10.1016/j.jde.2006.07.027. [13] J. Geng, Invariant tori of full dimension for a nonlinear Schrödinger equation, J. Differential Equations, 252 (2012), 1-34. doi: 10.1016/j.jde.2011.09.006. [14] J. Geng and J. Wu, Real analytic quasi-periodic solutions for the derivative nonlinear Schrödinger equations, J. Math. Phys., 53 (2012), 102702. doi: 10.1063/1.4754822. [15] B. Grébert and L. Thomann, KAM for the quantum harmonic oscillator, Comm. Math. Phys., 307 (2011), 383-427. doi: 10.1007/s00220-011-1327-5. [16] S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, (Russian) Funktsional. Anal. iPrilozhen., 21 (1987), 22-37; English translation in: Funct. Anal. Appl., 21 (1987), 192-205. [17] S. B. Kuksin, Perturbations of conditionally periodic solutions of infinite-dimensional Hamiltonian systems, (Russian) 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 Mathematics, 1556, Springer-Verlag, Berlin, 1993. doi: 10.1007/BFb0092243. [19] S. B. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math., 143 (1996), 147-179. [20] S. B. Kuksin, On small-denominators equations with large variable coefficients, Z. Angew. Math. Phys., 48 (1997), 262-271. doi: 10.1007/PL00001476. [21] S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications, 19, Oxford University Press, Oxford, 2000. [22] T. Kappeler and J. Pöschel, KdV and KAM, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2. [23] L. Jiao and Y. Wang, The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation, Commun. Pure Appl. Anal., 8 (2009), 1585-1606. doi: 10.3934/cpaa.2009.8.1585. [24] J. Liu and J. Si, Invariant tori for a derivative nonlinear Schrödinger equation with quasiperiodic forcing, J Math. Phys. , 56 (2015), 032702. doi: 10.1063/1.4916287. [25] J. Liu, Periodic and quasi-periodic solutions of a derivative nonlinear Schrödinger equation, Appl. Anal., 95 (2016), 801-825. doi: 10.1080/00036811.2015.1032942. [26] 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. [27] J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Comm. Math. Phys., 307 (2011), 629-673. doi: 10.1007/s00220-011-1353-3. [28] 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. [29] L. Mi and K. Zhang, Invariant tori for Benjamin-Ono equation with unbounded quasiperiodically forced perturbation, Discrete Contin. Dyn. Syst., 34 (2014), 689-707. doi: 10.3934/dcds.2014.34.689. [30] J. Pöschel, A KAM theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa. Cl. Sci., 23 (1996), 119-148. [31] J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420. [32] J. Rui and J. Si, Quasi-periodic solutions for quasi-periodically forced nonlinear Schrödinger equations with quasi-periodic inhomogeneous terms, Phys. D, 286/287 (2014), 1-31. doi: 10.1016/j.physd.2014.07.005. [33] M. B. Sevryuk, The reversible context 2 in KAM theory: the first steps, Regul. Chaotic Dyn., 16 (2011), 24-38. doi: 10.1134/S1560354710520035. [34] J. Si, Quasi-periodic solutions of a non-autonomous wave equations with quasi-periodic forcing, J. Differential Equations, 252 (2012), 5274-5360. doi: 10.1016/j.jde.2012.01.034. [35] Y. Wang and J. Si, A result on quasi-periodic solutions of a nonlinear beam equation with a quasi-periodic forcing term, Z. Angew. Math. Phys., 63 (2012), 189-190. doi: 10.1007/s00033-011-0172-x. [36] C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys., 127 (1990), 479-528. [37] X. Yuan, Quasi-periodic solutions of nonlinear Schrödinger equations of higher dimension, J. Differential Equations, 195 (2003), 230-242. doi: 10.1016/S0022-0396(03)00095-0. [38] X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations, 230 (2006), 213-274. doi: 10.1016/j.jde.2005.12.012. [39] M. Zhang and J. Si, Quasi-periodic solutions of nonlinear wave equations with quasi-periodic forcing, Phys. D, 238 (2009), 2185-2215. doi: 10.1016/j.physd.2009.09.003.
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