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

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Properly-degenerate KAM theory (following V. I. Arnold)

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

Abstract
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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 and 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), 435-451. doi: 10.1088/0951-7715/21/3/004.
[2]	H. Cong, J. Liu and X. Yuan, Quasiperiodic solutions for the cubic complex Ginzburg-Landau equation, J. Math. Physics, 50 (2009), 063516. doi: 10.1063/1.3157213.
[3]	C. D. Levermore and M. Oliver, The complex Ginzburg-Landau equation as a model problem,, in, 31 ().
[4]	Zh. Liang, Quasi-periodic solutions for $1D$ Schrödinger equation with the nonlinearity $\|u\|^{2p}u$, J. Differential Equations, 244 (2008), 2185-2225. doi: 10.1016/j.jde.2008.02.015.
[5]	B. P. Luce, Homoclinic explosions in the complex Ginzburg-Landau equation, Physica D, 84 (1995), 553-581. doi: 10.1016/0167-2789(95)00047-8.
[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), 266-280. doi: 10.1016/j.matcom.2006.10.009.
[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), 269-285. doi: 10.1016/j.physd.2004.07.012.
[8]	J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420.
[9]	P. Takáč, Invariant $2$-tori in the time-dependent Ginzburg-Landau equation, Nonlinearity, 5 (1992), 289-321. doi: 10.1088/0951-7715/5/2/002.
[10]	C. Valls, Quasiperiodic solutions for dissipative Boussinesq systems, Comm. Math. Phys., 265 (2006), 305-331. doi: 10.1007/s00220-006-0026-0.
[11]	X. Yuan, Quasi-periodic solutions of nonlinear Schrödinger equations of higher dimension, J. Differential Equations, 195 (2003), 230-242. doi: 10.1016/S0022-0396(03)00095-0.
[12]	X. Yuan, A KAM theorem with applications to partial differential equations of higher dimensions, Comm. Math. Phys., 275 (2007), 97-137. doi: 10.1007/s00220-007-0287-2.

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), 435-451. doi: 10.1088/0951-7715/21/3/004.
[2]	H. Cong, J. Liu and X. Yuan, Quasiperiodic solutions for the cubic complex Ginzburg-Landau equation, J. Math. Physics, 50 (2009), 063516. doi: 10.1063/1.3157213.
[3]	C. D. Levermore and M. Oliver, The complex Ginzburg-Landau equation as a model problem,, in, 31 ().
[4]	Zh. Liang, Quasi-periodic solutions for $1D$ Schrödinger equation with the nonlinearity $\|u\|^{2p}u$, J. Differential Equations, 244 (2008), 2185-2225. doi: 10.1016/j.jde.2008.02.015.
[5]	B. P. Luce, Homoclinic explosions in the complex Ginzburg-Landau equation, Physica D, 84 (1995), 553-581. doi: 10.1016/0167-2789(95)00047-8.
[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), 266-280. doi: 10.1016/j.matcom.2006.10.009.
[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), 269-285. doi: 10.1016/j.physd.2004.07.012.
[8]	J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420.
[9]	P. Takáč, Invariant $2$-tori in the time-dependent Ginzburg-Landau equation, Nonlinearity, 5 (1992), 289-321. doi: 10.1088/0951-7715/5/2/002.
[10]	C. Valls, Quasiperiodic solutions for dissipative Boussinesq systems, Comm. Math. Phys., 265 (2006), 305-331. doi: 10.1007/s00220-006-0026-0.
[11]	X. Yuan, Quasi-periodic solutions of nonlinear Schrödinger equations of higher dimension, J. Differential Equations, 195 (2003), 230-242. doi: 10.1016/S0022-0396(03)00095-0.
[12]	X. Yuan, A KAM theorem with applications to partial differential equations of higher dimensions, Comm. Math. Phys., 275 (2007), 97-137. doi: 10.1007/s00220-007-0287-2.

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