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Traveling wave solutions and its stability for 3D Ginzburg-Landau type equation
1. | School of Mathematics and Systems Science & LMIB, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China |
2. | Department of Mechanical Engineering, Zhejiang University, Hangzhou 310027, China |
3. | School of Mathematics and Systems Science, Beijing University of Aeronautics-Astronautics, Beijing 100191, China, China, China |
References:
[1] |
J. M. Ghidaglia and B. Héron, Dimension of the attractor associated to the Ginzburg-Landau equation,, Phys. D, 28 (1987), 282.
doi: 10.1016/0167-2789(87)90020-0. |
[2] |
C. R. Doering, J. D. Gibbon, D. Holm and B. Nicolaenko, Low-dimensional behavior in the complex Ginzburg-Landau equation,, Nonlinearity, 1 (1988), 279.
doi: 10.1088/0951-7715/1/2/001. |
[3] |
K. Promislow, Induced trajectories and approximate inertial manifolds for the Ginzburg-Landau partial differential equation,, Physica D, 41 (1990), 232.
doi: 10.1016/0167-2789(90)90125-9. |
[4] |
C. Bu, On the Cauch problem for the 1+2 complex Ginzburg-Landau equation,, J. Austral Math. Soc. Ser. B, 36 (1994), 313.
doi: 10.1017/S0334270000010468. |
[5] |
S. J. Lü, The dynamical behavior of the Ginzburg-Landau equation and its Fourier spectral approximation,, Numerical Mathematics, 22 (2000), 1.
|
[6] |
H. J. Gao and J. Q. Duan, On the initial value problem for the generalized 2D Ginzburg-Landau equation,, J. Math. Anal. Appl., 216 (1997), 536.
doi: 10.1006/jmaa.1997.5682. |
[7] |
H. J. Gao and J. Q. Duan, Asymptotics for the generalized two-dimensional Ginzburg-Landau equation,, J. Math. Anal. Appl., 247 (2000), 198.
doi: 10.1006/jmaa.2000.6848. |
[8] |
S. J. Lü and Q. S. Lu, A linear discrete scheme for the Ginzburg-Landau equation,, International Journal of Computer Mathematics, 85 (2008), 745.
doi: 10.1080/00207160701253810. |
[9] |
B. L. Guo and B. X. Wang, Finite dimensional behavior for the derivative Ginzburg-Landau equation in two spatial dimensions,, Phys. D, 89 (1995), 83.
doi: 10.1016/0167-2789(95)00216-2. |
[10] |
J. D. Carter, "Stability and Existence of Traveling Wave Solutions of the Two-dimensional Nonlinear Schrödinger Equation and its Higher-order Generalizations,", Ph.D. Thesis, (2001). Google Scholar |
[11] |
J. D. Carter, C. C. Contreras, Stability of plane-wave solutions of a dissipative generalization of the nonlinear Schrödinger equation,, Physica D, 237 (2008), 3292.
doi: 10.1016/j.physd.2008.07.016. |
[12] |
R. S. Mackay and S. Aubry, Proof of existence of breathers for time-reversibler or Hamilotonian network of weakly coupled oscillators,, Nonlinearity, 7 (1994), 1623.
doi: 10.1088/0951-7715/7/6/006. |
[13] |
S. J. Lü and Q. S. Lu, Exponential attractor for the 3D Ginburg-Landau type equation,, Nonl. Anal., 67 (2007), 3116.
doi: 10.1016/j.na.2006.10.005. |
[14] |
S. J. Lü, Q. S. Lu, Q. G. Meng and Z. Feng, Regularity of attactor for 3D Ginzburg-Landau equation,, Dyn. of Partial Differ. Equ., 6 (2009), 185.
|
[15] |
S. J. Lü and Q. S. Lu, Fourier spectral approximation to long-time behavior of three dimensional Ginzburg-Landau type equation,, Adv. Comp. Math., 27 (2007), 293.
doi: 10.1007/s10444-005-9004-x. |
[16] |
Z. Feng and Q. G. Meng, Exact solution for a two-dimensional kdv-burgers-type equation with nonlinear terms of any order,, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 397. Google Scholar |
[17] |
Z. Feng and G. Chen, Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth,, Discrete Contin. Dyn. Syst., 24 (2009), 763.
doi: 10.3934/dcds.2009.24.763. |
[18] |
M. L. Wang, Exact solutions for a compound KdV-Burgers equation,, Phys. Lett. A, 213 (1996), 279.
doi: 10.1016/0375-9601(96)00103-X. |
[19] |
M. L. Wang, Solitary wave solutions for variant Boussinesq equations,, Phys. Lett. A, 199 (1995), 169.
doi: 10.1016/0375-9601(95)00092-H. |
show all references
References:
[1] |
J. M. Ghidaglia and B. Héron, Dimension of the attractor associated to the Ginzburg-Landau equation,, Phys. D, 28 (1987), 282.
doi: 10.1016/0167-2789(87)90020-0. |
[2] |
C. R. Doering, J. D. Gibbon, D. Holm and B. Nicolaenko, Low-dimensional behavior in the complex Ginzburg-Landau equation,, Nonlinearity, 1 (1988), 279.
doi: 10.1088/0951-7715/1/2/001. |
[3] |
K. Promislow, Induced trajectories and approximate inertial manifolds for the Ginzburg-Landau partial differential equation,, Physica D, 41 (1990), 232.
doi: 10.1016/0167-2789(90)90125-9. |
[4] |
C. Bu, On the Cauch problem for the 1+2 complex Ginzburg-Landau equation,, J. Austral Math. Soc. Ser. B, 36 (1994), 313.
doi: 10.1017/S0334270000010468. |
[5] |
S. J. Lü, The dynamical behavior of the Ginzburg-Landau equation and its Fourier spectral approximation,, Numerical Mathematics, 22 (2000), 1.
|
[6] |
H. J. Gao and J. Q. Duan, On the initial value problem for the generalized 2D Ginzburg-Landau equation,, J. Math. Anal. Appl., 216 (1997), 536.
doi: 10.1006/jmaa.1997.5682. |
[7] |
H. J. Gao and J. Q. Duan, Asymptotics for the generalized two-dimensional Ginzburg-Landau equation,, J. Math. Anal. Appl., 247 (2000), 198.
doi: 10.1006/jmaa.2000.6848. |
[8] |
S. J. Lü and Q. S. Lu, A linear discrete scheme for the Ginzburg-Landau equation,, International Journal of Computer Mathematics, 85 (2008), 745.
doi: 10.1080/00207160701253810. |
[9] |
B. L. Guo and B. X. Wang, Finite dimensional behavior for the derivative Ginzburg-Landau equation in two spatial dimensions,, Phys. D, 89 (1995), 83.
doi: 10.1016/0167-2789(95)00216-2. |
[10] |
J. D. Carter, "Stability and Existence of Traveling Wave Solutions of the Two-dimensional Nonlinear Schrödinger Equation and its Higher-order Generalizations,", Ph.D. Thesis, (2001). Google Scholar |
[11] |
J. D. Carter, C. C. Contreras, Stability of plane-wave solutions of a dissipative generalization of the nonlinear Schrödinger equation,, Physica D, 237 (2008), 3292.
doi: 10.1016/j.physd.2008.07.016. |
[12] |
R. S. Mackay and S. Aubry, Proof of existence of breathers for time-reversibler or Hamilotonian network of weakly coupled oscillators,, Nonlinearity, 7 (1994), 1623.
doi: 10.1088/0951-7715/7/6/006. |
[13] |
S. J. Lü and Q. S. Lu, Exponential attractor for the 3D Ginburg-Landau type equation,, Nonl. Anal., 67 (2007), 3116.
doi: 10.1016/j.na.2006.10.005. |
[14] |
S. J. Lü, Q. S. Lu, Q. G. Meng and Z. Feng, Regularity of attactor for 3D Ginzburg-Landau equation,, Dyn. of Partial Differ. Equ., 6 (2009), 185.
|
[15] |
S. J. Lü and Q. S. Lu, Fourier spectral approximation to long-time behavior of three dimensional Ginzburg-Landau type equation,, Adv. Comp. Math., 27 (2007), 293.
doi: 10.1007/s10444-005-9004-x. |
[16] |
Z. Feng and Q. G. Meng, Exact solution for a two-dimensional kdv-burgers-type equation with nonlinear terms of any order,, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 397. Google Scholar |
[17] |
Z. Feng and G. Chen, Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth,, Discrete Contin. Dyn. Syst., 24 (2009), 763.
doi: 10.3934/dcds.2009.24.763. |
[18] |
M. L. Wang, Exact solutions for a compound KdV-Burgers equation,, Phys. Lett. A, 213 (1996), 279.
doi: 10.1016/0375-9601(96)00103-X. |
[19] |
M. L. Wang, Solitary wave solutions for variant Boussinesq equations,, Phys. Lett. A, 199 (1995), 169.
doi: 10.1016/0375-9601(95)00092-H. |
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