April  2021, 26(4): 1827-1842. doi: 10.3934/dcdsb.2020364

Traveling waves in quadratic autocatalytic systems with complexing agent

1. 

Department of Applied Mathematics, National Chiao Tung University, No. 1001, Ta Hsueh Road, Hsinchu 300093, Taiwan

2. 

Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawinskiego 5B, 02-106 Warsaw, Poland

* Corresponding author: Bogdan Kazmierczak

Dedicated to Prof. Sze-Bi Hsu in appreciation of his inspiring ideas

Received  October 06, 2020 Revised  October 25, 2020 Published  April 2021 Early access  December 2020

The quadratic autocatalytic reaction forms a key step in a number of chemical reaction systems, and traveling waves are observed in such systems. In this study, we investigate the effect of complexation reactions on traveling waves in the quadratic autocatalytic reaction system. More precisely, under the assumption that the complexation reaction is fast relative to the autocatalytic reaction, we show that the governing system is reduced to a two-component reaction-diffusion system with density-dependent diffusivity. Further, the numerical evidence suggests that for some parameter values, a traveling wave solution of this reduced two-component system is nonlinearly selected. This is contrast to that associated with the quadratic autocatalytic reaction (without complexation reactions).

Citation: Wei-Chieh Chen, Bogdan Kazmierczak. Traveling waves in quadratic autocatalytic systems with complexing agent. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1827-1842. doi: 10.3934/dcdsb.2020364
References:
[1]

J. Billingham and D. J. Needham, The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion coefficient. I. Permanent form travelling waves, Phil. Trans. R. Soc. A, 334 (1991), 1-24. doi: 10.1098/rsta.1991.0001.  Google Scholar

[2]

J. Billingham and D. J. Needham, The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. â…¢. Large time development in quadratic autocatalysis, Quart. Appl. Math, 50 (1992), 343-372. doi: 10.1090/qam/1162280.  Google Scholar

[3]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[4]

B. Kaźmierczak and Z. Peradzyński, Calcium waves with fast buffers and mechanical effects, J. Math. Biol., 62 (2011), 1-38. doi: 10.1007/s00285-009-0323-2.  Google Scholar

[5]

B. Kazmierczak and V. Volpert, Travelling calcium waves in systems with non-diffusing buffers, Mathematical Models and Methods in Applied Sciences, 18 (2008), 883-912.  doi: 10.1142/S0218202508002899.  Google Scholar

[6]

J. H. Merkin and H. Ševčíková, Reaction fronts in an ionic autocatalytic system with an applied electric field, J.Math. Chem., 25 (1999), 111-132. doi: 10.1023/A: 1019124231138.  Google Scholar

[7]

J. H. Merkin and H. Ševčíková, The effects of a complexing agent on travelling waves in autocatalytic systems with applied electric fields, IMA J. Appl. Math., 70 (2005), 527-549. doi: 10.1093/imamat/hxh045.  Google Scholar

[8]

J. H. Merkin and H. Ševčíková, D. Snita, The effect of an electric field on the local stoichiometry of front waves in an ionic chemical system, IMA J. Appl. Math., 64 (2000), 157-188. Google Scholar

[9]

K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2.  Google Scholar

[10]

J. H. Merkin and D. J. Needham, Propagating reaction-diffusion waves in a simple isothermal quadratic autocatalytic chemical system, J. Engng. Math., 23 (1989), 343-356. doi: 10.1007/BF00128907.  Google Scholar

show all references

References:
[1]

J. Billingham and D. J. Needham, The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion coefficient. I. Permanent form travelling waves, Phil. Trans. R. Soc. A, 334 (1991), 1-24. doi: 10.1098/rsta.1991.0001.  Google Scholar

[2]

J. Billingham and D. J. Needham, The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. â…¢. Large time development in quadratic autocatalysis, Quart. Appl. Math, 50 (1992), 343-372. doi: 10.1090/qam/1162280.  Google Scholar

[3]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[4]

B. Kaźmierczak and Z. Peradzyński, Calcium waves with fast buffers and mechanical effects, J. Math. Biol., 62 (2011), 1-38. doi: 10.1007/s00285-009-0323-2.  Google Scholar

[5]

B. Kazmierczak and V. Volpert, Travelling calcium waves in systems with non-diffusing buffers, Mathematical Models and Methods in Applied Sciences, 18 (2008), 883-912.  doi: 10.1142/S0218202508002899.  Google Scholar

[6]

J. H. Merkin and H. Ševčíková, Reaction fronts in an ionic autocatalytic system with an applied electric field, J.Math. Chem., 25 (1999), 111-132. doi: 10.1023/A: 1019124231138.  Google Scholar

[7]

J. H. Merkin and H. Ševčíková, The effects of a complexing agent on travelling waves in autocatalytic systems with applied electric fields, IMA J. Appl. Math., 70 (2005), 527-549. doi: 10.1093/imamat/hxh045.  Google Scholar

[8]

J. H. Merkin and H. Ševčíková, D. Snita, The effect of an electric field on the local stoichiometry of front waves in an ionic chemical system, IMA J. Appl. Math., 64 (2000), 157-188. Google Scholar

[9]

K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2.  Google Scholar

[10]

J. H. Merkin and D. J. Needham, Propagating reaction-diffusion waves in a simple isothermal quadratic autocatalytic chemical system, J. Engng. Math., 23 (1989), 343-356. doi: 10.1007/BF00128907.  Google Scholar

Figure 1.  Time-evolution of the solution $ (A, B) $ of system (1.9) with $ L = 1600 $. The initial data is that $ A_0(x) = 1\; (0 \leq x \leq L) $, and $ B_0(x) = 0\; (20 \leq x \leq L) $ and $ 1\; (0 \leq x < 20) $. Here the parameters are $ d = 2, K = 2 $, and $ \sigma = 4 $
Figure 2.  The dependence of wave speed $ v_m $ on $ \sigma $. The parameter $ K = 2 $ and the diffusivity parameter $ d $ is $ 0.5 $, $ 1 $, $ 2 $ and $ 4 $ for panels (a), (b), (c) and (d), respectively
[1]

Joseph Thirouin. Classification of traveling waves for a quadratic Szegő equation. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3099-3122. doi: 10.3934/dcds.2019128

[2]

Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure & Applied Analysis, 2011, 10 (1) : 141-160. doi: 10.3934/cpaa.2011.10.141

[3]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382

[4]

Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21

[5]

Yaping Wu, Niannian Yan. Stability of traveling waves for autocatalytic reaction systems with strong decay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1601-1633. doi: 10.3934/dcdsb.2017033

[6]

Grigori Chapiro, Lucas Furtado, Dan Marchesin, Stephen Schecter. Stability of interacting traveling waves in reaction-convection-diffusion systems. Conference Publications, 2015, 2015 (special) : 258-266. doi: 10.3934/proc.2015.0258

[7]

Tianran Zhang. Traveling waves for a reaction-diffusion model with a cyclic structure. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1859-1870. doi: 10.3934/dcdsb.2020006

[8]

Xiao-Biao Lin, Stephen Schecter. Traveling waves and shock waves. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : i-ii. doi: 10.3934/dcds.2004.10.4i

[9]

Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029

[10]

Yicheng Jiang, Kaijun Zhang. Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations. Kinetic & Related Models, 2018, 11 (5) : 1235-1253. doi: 10.3934/krm.2018048

[11]

Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126

[12]

Chiun-Chuan Chen, Li-Chang Hung. An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1503-1521. doi: 10.3934/dcdsb.2018054

[13]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405

[14]

Jonatan Lenells. Traveling waves in compressible elastic rods. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 151-167. doi: 10.3934/dcdsb.2006.6.151

[15]

Dieter Bothe, Michel Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 49-59. doi: 10.3934/dcdss.2012.5.49

[16]

Tyrone E. Duncan. Some partially observed multi-agent linear exponential quadratic stochastic differential games. Evolution Equations & Control Theory, 2018, 7 (4) : 587-597. doi: 10.3934/eect.2018028

[17]

Laurent Imbert, Michael J. Jacobson, Jr., Arthur Schmidt. Fast ideal cubing in imaginary quadratic number and function fields. Advances in Mathematics of Communications, 2010, 4 (2) : 237-260. doi: 10.3934/amc.2010.4.237

[18]

Tong Li, Jeungeun Park. Traveling waves in a chemotaxis model with logistic growth. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6465-6480. doi: 10.3934/dcdsb.2019147

[19]

Guangyu Zhao. Multidimensional periodic traveling waves in infinite cylinders. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 1025-1045. doi: 10.3934/dcds.2009.24.1025

[20]

Matthew S. Mizuhara, Peng Zhang. Uniqueness and traveling waves in a cell motility model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2811-2835. doi: 10.3934/dcdsb.2018315

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (122)
  • HTML views (123)
  • Cited by (0)

Other articles
by authors

[Back to Top]