Volume 49, Issue 2
The Traveling Wave of Auto-Catalytic Systems-Monotone and Multi-Peak Solutions

J. Math. Study, 49 (2016), pp. 149-168.

Published online: 2016-07

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• Abstract

This article studies propagating wave fronts of a reaction-diffusion system modeling an isothermal chemical reaction $A+2B → 3B$ involving two chemical species, a reactant $A$ and an auto-catalyst $B$, whose diffusion coefficients, $D_A$ and $D_B$, are unequal due to different molecular weights and/or sizes. Explicit bounds $c_∗$ and $c^∗$ that depend on $D_B/D_A$ are derived such that there is a unique travelling wave of every speed $c ≥ c^∗$ and there does not exist any travelling wave of speed $c < c_∗$. Furthermore, the reaction-diffusion system of the Gray-Scott model of $A+2B → 3B$, and a linear decay $B → C$, where $C$ is an inert product is also studied. The existence of multiple traveling waves which have distinctive number of local maxima or peaks is shown. It shows a new and very distinctive feature of Gray-Scott type of models in generating rich and structurally different traveling pulses.

• Keywords

Qubic autocatalysis, travelling wave, minimum speed, Gray-Scott, multi-peak waves.

34C20, 34C25, 92E20

Yuanwei.Qi@ucf.edu (Yuanwei Qi)

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@Article{JMS-49-149, author = {Yuanwei and Qi and Yuanwei.Qi@ucf.edu and 7102 and Department of Mathematics, University of Central Florida, Orlando, Florida 32816, USA and Yuanwei Qi}, title = {The Traveling Wave of Auto-Catalytic Systems-Monotone and Multi-Peak Solutions}, journal = {Journal of Mathematical Study}, year = {2016}, volume = {49}, number = {2}, pages = {149--168}, abstract = {

This article studies propagating wave fronts of a reaction-diffusion system modeling an isothermal chemical reaction $A+2B → 3B$ involving two chemical species, a reactant $A$ and an auto-catalyst $B$, whose diffusion coefficients, $D_A$ and $D_B$, are unequal due to different molecular weights and/or sizes. Explicit bounds $c_∗$ and $c^∗$ that depend on $D_B/D_A$ are derived such that there is a unique travelling wave of every speed $c ≥ c^∗$ and there does not exist any travelling wave of speed $c < c_∗$. Furthermore, the reaction-diffusion system of the Gray-Scott model of $A+2B → 3B$, and a linear decay $B → C$, where $C$ is an inert product is also studied. The existence of multiple traveling waves which have distinctive number of local maxima or peaks is shown. It shows a new and very distinctive feature of Gray-Scott type of models in generating rich and structurally different traveling pulses.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v49n2.16.04}, url = {http://global-sci.org/intro/article_detail/jms/996.html} }
TY - JOUR T1 - The Traveling Wave of Auto-Catalytic Systems-Monotone and Multi-Peak Solutions AU - Qi , Yuanwei JO - Journal of Mathematical Study VL - 2 SP - 149 EP - 168 PY - 2016 DA - 2016/07 SN - 49 DO - http://doi.org/10.4208/jms.v49n2.16.04 UR - https://global-sci.org/intro/article_detail/jms/996.html KW - Qubic autocatalysis, travelling wave, minimum speed, Gray-Scott, multi-peak waves. AB -

This article studies propagating wave fronts of a reaction-diffusion system modeling an isothermal chemical reaction $A+2B → 3B$ involving two chemical species, a reactant $A$ and an auto-catalyst $B$, whose diffusion coefficients, $D_A$ and $D_B$, are unequal due to different molecular weights and/or sizes. Explicit bounds $c_∗$ and $c^∗$ that depend on $D_B/D_A$ are derived such that there is a unique travelling wave of every speed $c ≥ c^∗$ and there does not exist any travelling wave of speed $c < c_∗$. Furthermore, the reaction-diffusion system of the Gray-Scott model of $A+2B → 3B$, and a linear decay $B → C$, where $C$ is an inert product is also studied. The existence of multiple traveling waves which have distinctive number of local maxima or peaks is shown. It shows a new and very distinctive feature of Gray-Scott type of models in generating rich and structurally different traveling pulses.

Yuanwei Qi. (2020). The Traveling Wave of Auto-Catalytic Systems-Monotone and Multi-Peak Solutions. Journal of Mathematical Study. 49 (2). 149-168. doi:10.4208/jms.v49n2.16.04
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