Quasi-periodic solutions for derivative nonlinear  Schrödinger equation

Meina Gao; Jianjun Liu

doi:10.3934/dcds.2012.32.2101

Article Contents

2012, Volume 32, Issue 6: 2101-2123. Doi: 10.3934/dcds.2012.32.2101

This issue Previous Article A direct proof of the Tonelli's partial regularity result Next Article Generalized Stokes system in Orlicz spaces

Quasi-periodic solutions for derivative nonlinear Schrödinger equation

Meina Gao^1, and
Jianjun Liu^2,

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

Received: January 31, 2011

Revised: May 31, 2011

Published: May 2012

Abstract Related Papers Cited by

Abstract

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.

Keywords:

KAM theory,
Derivative nonlinear Schrödinger equation,
Normal form.

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

Citation:

Related Papers

Cited by

References

[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.