American Institute of Mathematical Sciences

Advanced
2011, 2011(Special): 1329-1334. doi: 10.3934/proc.2011.2011.1329

Standing waves in a complex Ginzburg-Landau equation with time-delay feedback

Michael Stich 1, and  Carsten Beta 2,

1. 

Centro de Astrobiología (CSIC-INTA), Ctra. de Ajalvir, km. 4, 28850 Torrejón de Ardoz (Madrid), Spain
2. 

Institut für Physik und Astronomie, Universität Potsdam, Karl-Liebknecht-Str. 24/25, Haus 28, 14476 Potsdam, Germany

Received  August 2010 Revised  April 2011 Published  October 2011

Standing waves are studied as solutions of a complex Ginzburg- Landau equation subjected to local and global time-delay feedback terms. The onset of standing waves is studied at the instability of the homogeneous periodic solution with respect to spatially periodic perturbations. The solution of this spatiotemporal wave pattern is given and is compared to the homogeneous periodic solution.
Keywords: reaction-diffusion system, control, pattern formation.
Mathematics Subject Classification: Primary: 35K57; 35B10; Secondary: 35Q92.
Citation: 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
[1]

Joseph G. Yan, Dong-Ming Hwang. Pattern formation in reaction-diffusion systems with $D_2$-symmetric kinetics. Discrete & Continuous Dynamical Systems, 1996, 2 (2) : 255-270. doi: 10.3934/dcds.1996.2.255
[2]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589
[3]

Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170
[4]

Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031
[5]

Hongyan Zhang, Siyu Liu, Yue Zhang. Dynamics and spatiotemporal pattern formations of a homogeneous reaction-diffusion Thomas model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1149-1164. doi: 10.3934/dcdss.2017062
[6]

Wilhelm Stannat, Lukas Wessels. Deterministic control of stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2021, 10 (4) : 701-722. doi: 10.3934/eect.2020087
[7]

Wanli Yang, Jie Sun, Su Zhang. Analysis of optimal boundary control for a three-dimensional reaction-diffusion system. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 325-344. doi: 10.3934/naco.2017021
[8]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631
[9]

Maxime Breden, Christian Kuehn, Cinzia Soresina. On the influence of cross-diffusion in pattern formation. Journal of Computational Dynamics, 2021, 8 (2) : 213-240. doi: 10.3934/jcd.2021010
[10]

Weihua Jiang, Xun Cao, Chuncheng Wang. Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain. Discrete & Continuous Dynamical Systems - B, 2022, 27 (2) : 1163-1178. doi: 10.3934/dcdsb.2021085
[11]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245
[12]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493
[13]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3785-3801. doi: 10.3934/dcdss.2020433
[14]

Heather Finotti, Suzanne Lenhart, Tuoc Van Phan. Optimal control of advective direction in reaction-diffusion population models. Evolution Equations & Control Theory, 2012, 1 (1) : 81-107. doi: 10.3934/eect.2012.1.81
[15]

Mohamed Ouzahra. Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control. Discrete & Continuous Dynamical Systems - B, 2022, 27 (2) : 1075-1090. doi: 10.3934/dcdsb.2021081
[16]

Thomas I. Seidman. Optimal control of a diffusion/reaction/switching system. Evolution Equations & Control Theory, 2013, 2 (4) : 723-731. doi: 10.3934/eect.2013.2.723
[17]

Sze-Bi Hsu, Junping Shi, Feng-Bin Wang. Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3169-3189. doi: 10.3934/dcdsb.2014.19.3169
[18]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. Determination of initial data for a reaction-diffusion system with variable coefficients. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 771-801. doi: 10.3934/dcds.2019032
[19]

Nicolas Bacaër, Cheikh Sokhna. A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Mathematical Biosciences & Engineering, 2005, 2 (2) : 227-238. doi: 10.3934/mbe.2005.2.227
[20]

José-Francisco Rodrigues, Lisa Santos. On a constrained reaction-diffusion system related to multiphase problems. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 299-319. doi: 10.3934/dcds.2009.25.299

 Impact Factor: 

Tools

Metrics

  • PDF downloads (92)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy