Traveling wave solutions and its stability for 3D Ginzburg-Landau type equation

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September 2011, 16(2): 507-527. doi: 10.3934/dcdsb.2011.16.507

Traveling wave solutions and its stability for 3D Ginzburg-Landau type equation

Shujuan Lü ^1, , Chunbiao Gan ^2, , Baohua Wang ^3, , Linning Qian ^3, and Meisheng Li ^3,

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

Received February 2010 Revised December 2010 Published June 2011

Abstract
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In this paper, a three dimensional Ginburg-Landau type equation is considered. Firstly, two families of new traveling wave solutions in term of explicit functions are presented by using the homogeneous balance method, in which one consists of variable-amplitude solutions and the other constant-amplitude solutions (namely, plane wave solutions). Moreover, the stability of plane wave solutions is analyzed by using the regular phase plane techniques.

Keywords: stability., traveling solution, Complex Ginzburg-Landau equation, phase analysis, homogeneous balance method.

Mathematics Subject Classification: Primary: 34C05; Secondary: 35B4.

Citation: Shujuan Lü, Chunbiao Gan, Baohua Wang, Linning Qian, Meisheng Li. Traveling wave solutions and its stability for 3D Ginzburg-Landau type equation. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 507-527. doi: 10.3934/dcdsb.2011.16.507

References:

[1]	J. M. Ghidaglia and B. Héron, Dimension of the attractor associated to the Ginzburg-Landau equation, Phys. D, 28 (1987), 282-304. 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), 279-309. 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), 232-252. 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), 313-324. 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), 1-9.
[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), 536-548. 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), 198-216. 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), 745-758. 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), 83-99. 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, University of Colorado at Boulder, 2001.
[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), 3292-3296. 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), 1623-1643. 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), 3116-3135. 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), 185-201.
[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), 293-318. 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), 397-413.
[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), 763-780. doi: 10.3934/dcds.2009.24.763.
[18]	M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A, 213 (1996), 279-287. doi: 10.1016/0375-9601(96)00103-X.
[19]	M. L. Wang, Solitary wave solutions for variant Boussinesq equations, Phys. Lett. A, 199 (1995), 169-172. 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), 282-304. 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), 279-309. 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), 232-252. 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), 313-324. 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), 1-9.
[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), 536-548. 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), 198-216. 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), 745-758. 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), 83-99. 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, University of Colorado at Boulder, 2001.
[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), 3292-3296. 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), 1623-1643. 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), 3116-3135. 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), 185-201.
[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), 293-318. 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), 397-413.
[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), 763-780. doi: 10.3934/dcds.2009.24.763.
[18]	M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A, 213 (1996), 279-287. doi: 10.1016/0375-9601(96)00103-X.
[19]	M. L. Wang, Solitary wave solutions for variant Boussinesq equations, Phys. Lett. A, 199 (1995), 169-172. doi: 10.1016/0375-9601(95)00092-H.

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