# American Institute of Mathematical Sciences

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 and 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. [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. [3] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York-Berlin, 1982. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [3] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York-Berlin, 1982. [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. [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. [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. [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. [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. [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. [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.
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 $\sigma = 4$
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
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