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 and Continuous Dynamical Systems, 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), 253-285. 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.

show all references

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.

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