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Article Contents

# Quasi-periodic solutions for derivative nonlinear Schrödinger equation

• 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.
Mathematics Subject Classification: Primary: 37K55; Secondary: 35Q55.

 Citation:

•  [1] D. Bambusi, Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys., 234 (2003), 253-285.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), 465-480.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), 507-567. [4] D. Bambusi and S. Paleari, Families of periodic solutions of resonant PDEs J. Nonlinear Sci., 11 (2001), 69-87.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), 359-419.doi: 10.1215/S0012-7094-06-13424-5. [6] J. Bourgain, "Global Solutions of Nonlinear Schrödinger Equations," American Mathematical Society, Providence, Rhode Island, 1999. [7] J. Bourgain, Periodic solutions of nonlinear wave equations, Harmonic analysis and partial differential equations, in "Chicago Lectures in Math.," Univ. Chicago Press, Chicago, IL, (1999), 69-97. [8] H. Chihara, Local existence for semilinear Schrödinger equations, Math. Japon., 42 (1995), 35-51. [9] G. Gentile and M. Procesi, Periodic solutions for a class of nonlinear partial differential equations in higher dimension, Comm. Math. Phys., 289 (2009), 863-906.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), 763-798.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), 429-458.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), 459-490.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), 453-461. [14] T. Kappeler and J. Pöschel, "KdV&KAM," Springer-Verlag, Berlin, 2003. [15] C. Kenig, G. Ponce and L. Vega, On the IVP for the nonlinear Schrödinger equations, in "Harmonic Analysis and operator theory (Caracas, 1994)", Contemporary Math., 189, Amer. Math. Soc., Providence, RI, (1995), 353-367. [16] C. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 255-288. [17] C. Kenig, G. Ponce and L. Vega, On the initial value problem for the Ishimori system, Ann. Henri Poincaré, 1 (2000), 341-384.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), 489-545.doi: 10.1007/s002220050272. [19] S. Klainerman, Long-time behaviour of solutions to nonlinear wave equations, in "Proceedings of International Congress of Mathematics," 1, 2 (Warsaw, 1983), PWN, Warsaw, (1984), 1209-1215. [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 University Press, Oxford, 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), 149-179.doi: 10.2307/2118656. [23] P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Mathematical Phys., 5 (1964), 611-613.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), 1145-1172. [25] 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. [26] J. Pöschel, Quasi-periodic solutions for a nonlinear wave equations, Comment. Math. Helv., 71 (1996), 269-296.doi: 10.1007/BF02566420. [27] X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations, 203 (2006), 213-274.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), 1198-2118.