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

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

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 $|u|^{2p}u,\ p\geq1$.
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

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