American Institute of Mathematical Sciences

Advanced
December  2010, 3(4): 579-600. doi: 10.3934/dcdss.2010.3.579

Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $|u|^{2p}u$

Hongzi Cong 1, , Jianjun Liu 1, and  Xiaoping Yuan 1,

1. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China, China, China

Received  March 2009 Revised  June 2010 Published  August 2010

In this paper we prove that there is a Cantorian branch of 2-dimensional KAM invariant tori for the complex Ginzburg-Landau equation with the nonlinearity $|u|^{2p}u,\ p\geq1$.
Keywords: partial normal form., Invariant tori, KAM theory.
Mathematics Subject Classification: Primary: 37K55; Secondary: 35Q56, 35K5.
Citation: Hongzi Cong, Jianjun Liu, Xiaoping Yuan. Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $|u|^{2p}u$. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 579-600. doi: 10.3934/dcdss.2010.3.579
References:
[1]

K. W. Chung and X. Yuan, Periodic and quasi-periodic solutions for the complex Ginzburg-Landau equation,, Nonlinearity, 21 (2008), 435.  doi: 10.1088/0951-7715/21/3/004.  Google Scholar

[2]

H. Cong, J. Liu and X. Yuan, Quasiperiodic solutions for the cubic complex Ginzburg-Landau equation,, J. Math. Physics, 50 (2009).  doi: 10.1063/1.3157213.  Google Scholar

[3]

C. D. Levermore and M. Oliver, The complex Ginzburg-Landau equation as a model problem,, in, 31 ().   Google Scholar

[4]

Zh. Liang, Quasi-periodic solutions for $1D$ Schrödinger equation with the nonlinearity $|u|^{2p}u$,, J. Differential Equations, 244 (2008), 2185.  doi: 10.1016/j.jde.2008.02.015.  Google Scholar

[5]

B. P. Luce, Homoclinic explosions in the complex Ginzburg-Landau equation,, Physica D, 84 (1995), 553.  doi: 10.1016/0167-2789(95)00047-8.  Google Scholar

[6]

S. C. Mancas and S. R. Choudhury, Bifurcations of plane wave (CW) solutions in the complex cubic-quintic Ginzburg-Landau equation,, Math. Comput. Simul., 74 (2007), 266.  doi: 10.1016/j.matcom.2006.10.009.  Google Scholar

[7]

G. Cruz-Pacheco, C. D. Levermore and B. P. Luce, Complex Ginzburg-Landau equations as perturbations of nonlinear Schrödinger equations: A Melnikov approach,, Physica D, 197 (2004), 269.  doi: 10.1016/j.physd.2004.07.012.  Google Scholar

[8]

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

[9]

P. Takáč, Invariant $2$-tori in the time-dependent Ginzburg-Landau equation,, Nonlinearity, 5 (1992), 289.  doi: 10.1088/0951-7715/5/2/002.  Google Scholar

[10]

C. Valls, Quasiperiodic solutions for dissipative Boussinesq systems,, Comm. Math. Phys., 265 (2006), 305.  doi: 10.1007/s00220-006-0026-0.  Google Scholar

[11]

X. Yuan, Quasi-periodic solutions of nonlinear Schrödinger equations of higher dimension,, J. Differential Equations, 195 (2003), 230.  doi: 10.1016/S0022-0396(03)00095-0.  Google Scholar

[12]

X. Yuan, A KAM theorem with applications to partial differential equations of higher dimensions,, Comm. Math. Phys., 275 (2007), 97.  doi: 10.1007/s00220-007-0287-2.  Google Scholar

show all references

References:
[1]

K. W. Chung and X. Yuan, Periodic and quasi-periodic solutions for the complex Ginzburg-Landau equation,, Nonlinearity, 21 (2008), 435.  doi: 10.1088/0951-7715/21/3/004.  Google Scholar

[2]

H. Cong, J. Liu and X. Yuan, Quasiperiodic solutions for the cubic complex Ginzburg-Landau equation,, J. Math. Physics, 50 (2009).  doi: 10.1063/1.3157213.  Google Scholar

[3]

C. D. Levermore and M. Oliver, The complex Ginzburg-Landau equation as a model problem,, in, 31 ().   Google Scholar

[4]

Zh. Liang, Quasi-periodic solutions for $1D$ Schrödinger equation with the nonlinearity $|u|^{2p}u$,, J. Differential Equations, 244 (2008), 2185.  doi: 10.1016/j.jde.2008.02.015.  Google Scholar

[5]

B. P. Luce, Homoclinic explosions in the complex Ginzburg-Landau equation,, Physica D, 84 (1995), 553.  doi: 10.1016/0167-2789(95)00047-8.  Google Scholar

[6]

S. C. Mancas and S. R. Choudhury, Bifurcations of plane wave (CW) solutions in the complex cubic-quintic Ginzburg-Landau equation,, Math. Comput. Simul., 74 (2007), 266.  doi: 10.1016/j.matcom.2006.10.009.  Google Scholar

[7]

G. Cruz-Pacheco, C. D. Levermore and B. P. Luce, Complex Ginzburg-Landau equations as perturbations of nonlinear Schrödinger equations: A Melnikov approach,, Physica D, 197 (2004), 269.  doi: 10.1016/j.physd.2004.07.012.  Google Scholar

[8]

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

[9]

P. Takáč, Invariant $2$-tori in the time-dependent Ginzburg-Landau equation,, Nonlinearity, 5 (1992), 289.  doi: 10.1088/0951-7715/5/2/002.  Google Scholar

[10]

C. Valls, Quasiperiodic solutions for dissipative Boussinesq systems,, Comm. Math. Phys., 265 (2006), 305.  doi: 10.1007/s00220-006-0026-0.  Google Scholar

[11]

X. Yuan, Quasi-periodic solutions of nonlinear Schrödinger equations of higher dimension,, J. Differential Equations, 195 (2003), 230.  doi: 10.1016/S0022-0396(03)00095-0.  Google Scholar

[12]

X. Yuan, A KAM theorem with applications to partial differential equations of higher dimensions,, Comm. Math. Phys., 275 (2007), 97.  doi: 10.1007/s00220-007-0287-2.  Google Scholar

[1]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444
[2]

Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 16: 331-348. doi: 10.3934/jmd.2020012
[3]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073
[4]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264
[5]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451
[6]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047
[7]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458
[8]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051
[9]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences