# American Institute of Mathematical Sciences

June  2012, 32(6): 2101-2123. doi: 10.3934/dcds.2012.32.2101

## Quasi-periodic solutions for derivative nonlinear Schrödinger equation

 1 School of Science, Shanghai Second Polytechnic University, Shanghai 201209, China 2 School of Mathematical Sciences, Fudan University, Shanghai 200433

Received  February 2011 Revised  June 2011 Published  February 2012

In this paper, we discuss the existence of time quasi-periodic solutions for the derivative nonlinear Schrödinger equation $$\label{1.1}\mathbf{i} u_t+u_{xx}+\mathbf{i} f(x,u,\bar{u})u_x+g(x,u,\bar{u})=0$$ subject to Dirichlet boundary conditions. Using an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field and Birkhoff normal form, we will prove that there exist a Cantorian branch of KAM tori and thus many time quasi-periodic solutions for the above equation.
Citation: Meina Gao, Jianjun Liu. Quasi-periodic solutions for derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101
##### References:
 [1] D. Bambusi, Birkhoff normal form for some nonlinear PDEs,, Comm. Math. Phys., 234 (2003), 253. doi: 10.1007/s00220-002-0774-4. [2] 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. [3] D. Bambusi and B. Grébert, Birkhoff normal form for partial differential equations with tame modulus,, Duke Math. J., 135 (2006), 507. [4] D. Bambusi and S. Paleari, Families of periodic solutions of resonant PDEs, J. Nonlinear Sci., 11 (2001), 69. doi: 10.1007/s003320010010. [5] M. Berti and P. Bolle, Cantor families of periodic solutions for completely resonant nonlinear wave equations,, Duke Math. J., 134 (2006), 359. doi: 10.1215/S0012-7094-06-13424-5. [6] J. Bourgain, "Global Solutions of Nonlinear Schrödinger Equations,", American Mathematical Society, (1999). [7] J. Bourgain, Periodic solutions of nonlinear wave equations, Harmonic analysis and partial differential equations,, in, (1999), 69. [8] H. Chihara, Local existence for semilinear Schrödinger equations,, Math. Japon., 42 (1995), 35. [9] G. Gentile and M. Procesi, Periodic solutions for a class of nonlinear partial differential equations in higher dimension,, Comm. Math. Phys., 289 (2009), 863. doi: 10.1007/s00220-009-0817-1. [10] B. Grébert, R. Imekraz and E. Paturel, Normal forms for semilinear quantum harmonic oscillators,, Comm. Math. Phys., 291 (2009), 763. doi: 10.1007/s00220-009-0800-x. [11] E. Faou, B. Grébert and E. Paturel, Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. I. Finite-dimensional discretization,, Numer. Math., 114 (2010), 429. doi: 10.1007/s00211-009-0258-y. [12] E. Faou, B. Grébert and E. Paturel, Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. II. Abstract splitting,, Numer. Math., 114 (2010), 459. doi: 10.1007/s00211-009-0257-z. [13] N. Hayashi and T. Ozawa, Remarks on nonlinear Schrödinger equations in one space dimension,, Differential Integral Equations, 7 (1994), 453. [14] T. Kappeler and J. Pöschel, "KdV&KAM,", Springer-Verlag, (2003). [15] C. Kenig, G. Ponce and L. Vega, On the IVP for the nonlinear Schrödinger equations,, in, (1995), 353. [16] C. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 255. [17] C. Kenig, G. Ponce and L. Vega, On the initial value problem for the Ishimori system,, Ann. Henri Poincaré, 1 (2000), 341. doi: 10.1007/PL00001008. [18] C. Kenig, G. Ponce and L. Vega, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations,, Invent. Math., 134 (1998), 489. doi: 10.1007/s002220050272. [19] S. Klainerman, Long-time behaviour of solutions to nonlinear wave equations,, in, 1, 2 (1984), 1209. [20] S. B. Kuksin, On small-denominators equations with large variable coefficients,, Z. Angew. Math. Phys., 48 (1997), 262. doi: 10.1007/PL00001476. [21] S. B. Kuksin, "Analysis of Hamiltonian PDEs,", Oxford University Press, (2000). [22] 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. [23] P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations,, J. Mathematical Phys., 5 (1964), 611. doi: 10.1063/1.1704154. [24] 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. [25] 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. [26] J. Pöschel, Quasi-periodic solutions for a nonlinear wave equations,, Comment. Math. Helv., 71 (1996), 269. doi: 10.1007/BF02566420. [27] X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations,, J. Differential Equations, 203 (2006), 213. doi: 10.1016/j.jde.2005.12.012. [28] J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations,, Nonlinearity, 24 (2011), 1198.

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##### References:
 [1] D. Bambusi, Birkhoff normal form for some nonlinear PDEs,, Comm. Math. Phys., 234 (2003), 253. doi: 10.1007/s00220-002-0774-4. [2] 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. [3] D. Bambusi and B. Grébert, Birkhoff normal form for partial differential equations with tame modulus,, Duke Math. J., 135 (2006), 507. [4] D. Bambusi and S. Paleari, Families of periodic solutions of resonant PDEs, J. Nonlinear Sci., 11 (2001), 69. doi: 10.1007/s003320010010. [5] M. Berti and P. Bolle, Cantor families of periodic solutions for completely resonant nonlinear wave equations,, Duke Math. J., 134 (2006), 359. doi: 10.1215/S0012-7094-06-13424-5. [6] J. Bourgain, "Global Solutions of Nonlinear Schrödinger Equations,", American Mathematical Society, (1999). [7] J. Bourgain, Periodic solutions of nonlinear wave equations, Harmonic analysis and partial differential equations,, in, (1999), 69. [8] H. Chihara, Local existence for semilinear Schrödinger equations,, Math. Japon., 42 (1995), 35. [9] G. Gentile and M. Procesi, Periodic solutions for a class of nonlinear partial differential equations in higher dimension,, Comm. Math. Phys., 289 (2009), 863. doi: 10.1007/s00220-009-0817-1. [10] B. Grébert, R. Imekraz and E. Paturel, Normal forms for semilinear quantum harmonic oscillators,, Comm. Math. Phys., 291 (2009), 763. doi: 10.1007/s00220-009-0800-x. [11] E. Faou, B. Grébert and E. Paturel, Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. I. Finite-dimensional discretization,, Numer. Math., 114 (2010), 429. doi: 10.1007/s00211-009-0258-y. [12] E. Faou, B. Grébert and E. Paturel, Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. II. Abstract splitting,, Numer. Math., 114 (2010), 459. doi: 10.1007/s00211-009-0257-z. [13] N. Hayashi and T. Ozawa, Remarks on nonlinear Schrödinger equations in one space dimension,, Differential Integral Equations, 7 (1994), 453. [14] T. Kappeler and J. Pöschel, "KdV&KAM,", Springer-Verlag, (2003). [15] C. Kenig, G. Ponce and L. Vega, On the IVP for the nonlinear Schrödinger equations,, in, (1995), 353. [16] C. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 255. [17] C. Kenig, G. Ponce and L. Vega, On the initial value problem for the Ishimori system,, Ann. Henri Poincaré, 1 (2000), 341. doi: 10.1007/PL00001008. [18] C. Kenig, G. Ponce and L. Vega, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations,, Invent. Math., 134 (1998), 489. doi: 10.1007/s002220050272. [19] S. Klainerman, Long-time behaviour of solutions to nonlinear wave equations,, in, 1, 2 (1984), 1209. [20] S. B. Kuksin, On small-denominators equations with large variable coefficients,, Z. Angew. Math. Phys., 48 (1997), 262. doi: 10.1007/PL00001476. [21] S. B. Kuksin, "Analysis of Hamiltonian PDEs,", Oxford University Press, (2000). [22] 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. [23] P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations,, J. Mathematical Phys., 5 (1964), 611. doi: 10.1063/1.1704154. [24] 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. [25] 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. [26] J. Pöschel, Quasi-periodic solutions for a nonlinear wave equations,, Comment. Math. Helv., 71 (1996), 269. doi: 10.1007/BF02566420. [27] X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations,, J. Differential Equations, 203 (2006), 213. doi: 10.1016/j.jde.2005.12.012. [28] J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations,, Nonlinearity, 24 (2011), 1198.
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