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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

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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

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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

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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

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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

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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

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