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.  Google Scholar

[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.  Google Scholar

[3]

D. Bambusi and B. Grébert, Birkhoff normal form for partial differential equations with tame modulus,, Duke Math. J., 135 (2006), 507.   Google Scholar

[4]

D. Bambusi and S. Paleari, Families of periodic solutions of resonant PDEs, J. Nonlinear Sci., 11 (2001), 69.  doi: 10.1007/s003320010010.  Google Scholar

[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.  Google Scholar

[6]

J. Bourgain, "Global Solutions of Nonlinear Schrödinger Equations,", American Mathematical Society, (1999).   Google Scholar

[7]

J. Bourgain, Periodic solutions of nonlinear wave equations, Harmonic analysis and partial differential equations,, in, (1999), 69.   Google Scholar

[8]

H. Chihara, Local existence for semilinear Schrödinger equations,, Math. Japon., 42 (1995), 35.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

N. Hayashi and T. Ozawa, Remarks on nonlinear Schrödinger equations in one space dimension,, Differential Integral Equations, 7 (1994), 453.   Google Scholar

[14]

T. Kappeler and J. Pöschel, "KdV&KAM,", Springer-Verlag, (2003).   Google Scholar

[15]

C. Kenig, G. Ponce and L. Vega, On the IVP for the nonlinear Schrödinger equations,, in, (1995), 353.   Google Scholar

[16]

C. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 255.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. Klainerman, Long-time behaviour of solutions to nonlinear wave equations,, in, 1, 2 (1984), 1209.   Google Scholar

[20]

S. B. Kuksin, On small-denominators equations with large variable coefficients,, Z. Angew. Math. Phys., 48 (1997), 262.  doi: 10.1007/PL00001476.  Google Scholar

[21]

S. B. Kuksin, "Analysis of Hamiltonian PDEs,", Oxford University Press, (2000).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

J. Pöschel, Quasi-periodic solutions for a nonlinear wave equations,, Comment. Math. Helv., 71 (1996), 269.  doi: 10.1007/BF02566420.  Google Scholar

[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.  Google Scholar

[28]

J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations,, Nonlinearity, 24 (2011), 1198.   Google Scholar

show all references

References:
[1]

D. Bambusi, Birkhoff normal form for some nonlinear PDEs,, Comm. Math. Phys., 234 (2003), 253.  doi: 10.1007/s00220-002-0774-4.  Google Scholar

[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.  Google Scholar

[3]

D. Bambusi and B. Grébert, Birkhoff normal form for partial differential equations with tame modulus,, Duke Math. J., 135 (2006), 507.   Google Scholar

[4]

D. Bambusi and S. Paleari, Families of periodic solutions of resonant PDEs, J. Nonlinear Sci., 11 (2001), 69.  doi: 10.1007/s003320010010.  Google Scholar

[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.  Google Scholar

[6]

J. Bourgain, "Global Solutions of Nonlinear Schrödinger Equations,", American Mathematical Society, (1999).   Google Scholar

[7]

J. Bourgain, Periodic solutions of nonlinear wave equations, Harmonic analysis and partial differential equations,, in, (1999), 69.   Google Scholar

[8]

H. Chihara, Local existence for semilinear Schrödinger equations,, Math. Japon., 42 (1995), 35.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

N. Hayashi and T. Ozawa, Remarks on nonlinear Schrödinger equations in one space dimension,, Differential Integral Equations, 7 (1994), 453.   Google Scholar

[14]

T. Kappeler and J. Pöschel, "KdV&KAM,", Springer-Verlag, (2003).   Google Scholar

[15]

C. Kenig, G. Ponce and L. Vega, On the IVP for the nonlinear Schrödinger equations,, in, (1995), 353.   Google Scholar

[16]

C. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 255.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. Klainerman, Long-time behaviour of solutions to nonlinear wave equations,, in, 1, 2 (1984), 1209.   Google Scholar

[20]

S. B. Kuksin, On small-denominators equations with large variable coefficients,, Z. Angew. Math. Phys., 48 (1997), 262.  doi: 10.1007/PL00001476.  Google Scholar

[21]

S. B. Kuksin, "Analysis of Hamiltonian PDEs,", Oxford University Press, (2000).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

J. Pöschel, Quasi-periodic solutions for a nonlinear wave equations,, Comment. Math. Helv., 71 (1996), 269.  doi: 10.1007/BF02566420.  Google Scholar

[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.  Google Scholar

[28]

J. Zhang, M. Gao and X. Yuan, KAM tori for reversible partial differential equations,, Nonlinearity, 24 (2011), 1198.   Google Scholar

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