American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

S. J. Lü, The dynamical behavior of the Ginzburg-Landau equation and its Fourier spectral approximation, Numerical Mathematics, 22 (2000), 1-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. 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), 3292-3296. doi: 10.1016/j.physd.2008.07.016.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. 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), 763-780. doi: 10.3934/dcds.2009.24.763.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

S. J. Lü, The dynamical behavior of the Ginzburg-Landau equation and its Fourier spectral approximation, Numerical Mathematics, 22 (2000), 1-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. 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), 3292-3296. doi: 10.1016/j.physd.2008.07.016.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. 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), 763-780. doi: 10.3934/dcds.2009.24.763.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure & Applied Analysis, 2021, 20 (5) : 2021-2038. doi: 10.3934/cpaa.2021056
[2]

Yueling Jia, Zhaohui Huo. Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation. Kinetic & Related Models, 2014, 7 (1) : 57-77. doi: 10.3934/krm.2014.7.57
[3]

Hans G. Kaper, Peter Takáč. Bifurcating vortex solutions of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 871-880. doi: 10.3934/dcds.1999.5.871
[4]

Noboru Okazawa, Tomomi Yokota. Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 311-341. doi: 10.3934/dcds.2010.28.311
[5]

Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825
[6]

Jungho Park. Bifurcation and stability of the generalized complex Ginzburg--Landau equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1237-1253. doi: 10.3934/cpaa.2008.7.1237
[7]

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
[8]

Michael Stich, Carsten Beta. Standing waves in a complex Ginzburg-Landau equation with time-delay feedback. Conference Publications, 2011, 2011 (Special) : 1329-1334. doi: 10.3934/proc.2011.2011.1329
[9]

N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711
[10]

Hong Lu, Shujuan Lü, Mingji Zhang. Fourier spectral approximations to the dynamics of 3D fractional complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2539-2564. doi: 10.3934/dcds.2017109
[11]

Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129
[12]

Feng Zhou, Chunyou Sun. Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3767-3792. doi: 10.3934/dcdsb.2016120
[13]

Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801
[14]

Noboru Okazawa, Tomomi Yokota. Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains. Conference Publications, 2001, 2001 (Special) : 280-288. doi: 10.3934/proc.2001.2001.280
[15]

N. I. Karachalios, H. E. Nistazakis, A. N. Yannacopoulos. Remarks on the asymptotic behavior of solutions of complex discrete Ginzburg-Landau equations. Conference Publications, 2005, 2005 (Special) : 476-486. doi: 10.3934/proc.2005.2005.476
[16]

Yuta Kugo, Motohiro Sobajima, Toshiyuki Suzuki, Tomomi Yokota, Kentarou Yoshii. Solvability of a class of complex Ginzburg-Landau equations in periodic Sobolev spaces. Conference Publications, 2015, 2015 (special) : 754-763. doi: 10.3934/proc.2015.0754
[17]

Satoshi Kosugi, Yoshihisa Morita. Phase pattern in a Ginzburg-Landau model with a discontinuous coefficient in a ring. Discrete & Continuous Dynamical Systems, 2006, 14 (1) : 149-168. doi: 10.3934/dcds.2006.14.149
[18]

N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647
[19]

Jingna Li, Li Xia. The Fractional Ginzburg-Landau equation with distributional initial data. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2173-2187. doi: 10.3934/cpaa.2013.12.2173
[20]

Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences