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

Previous Article
Properly-degenerate KAM theory (following V. I. Arnold)
DCDS-S Home
This Issue
Next Article
Convergence radius in the Poincaré-Siegel problem

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
Related Papers

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]	Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643
[2]	Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2141-2160. doi: 10.3934/dcds.2017092
[3]	Dongfeng Yan. KAM Tori for generalized Benjamin-Ono equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 941-957. doi: 10.3934/cpaa.2015.14.941
[4]	Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363
[5]	Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[6]	Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57
[7]	Virginie De Witte, Willy Govaerts. Numerical computation of normal form coefficients of bifurcations of odes in MATLAB. Conference Publications, 2011, 2011 (Special) : 362-372. doi: 10.3934/proc.2011.2011.362
[8]	Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561
[9]	John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170
[10]	Shengqing Hu, Bin Liu. Degenerate lower dimensional invariant tori in reversible system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3735-3763. doi: 10.3934/dcds.2018162
[11]	Hsuan-Wen Su. Finding invariant tori with Poincare's map. Communications on Pure & Applied Analysis, 2008, 7 (2) : 433-443. doi: 10.3934/cpaa.2008.7.433
[12]	Ugo Locatelli, Antonio Giorgilli. Invariant tori in the Sun--Jupiter--Saturn system. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 377-398. doi: 10.3934/dcdsb.2007.7.377
[13]	Fuzhong Cong, Yong Li. Invariant hyperbolic tori for Hamiltonian systems with degeneracy. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 371-382. doi: 10.3934/dcds.1997.3.371
[14]	Hans Koch. On the renormalization of Hamiltonian flows, and critical invariant tori. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 633-646. doi: 10.3934/dcds.2002.8.633
[15]	Xiaocai Wang. Non-floquet invariant tori in reversible systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3439-3457. doi: 10.3934/dcds.2018147
[16]	Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413
[17]	Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069
[18]	Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080
[19]	Ali Hamidoǧlu. On general form of the Tanh method and its application to nonlinear partial differential equations. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 175-181. doi: 10.3934/naco.2016007
[20]	C. Chandre. Renormalization for cubic frequency invariant tori in Hamiltonian systems with two degrees of freedom. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 457-465. doi: 10.3934/dcdsb.2002.2.457

2018 Impact Factor: 0.545

American Institute of Mathematical Sciences