| 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 |
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).
| [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. |
show all 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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