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