Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay

Abstract

Abstract. The reaction-diffusion systems which are based on an isothermal autocatalytic chemical reaction involving both an autocatalytic step of the (m + 1)th order (A + mB → (m + 1)B) and a decay step of the same order (B → C) have very rich and interesting dynamics. Previous studies in the literature indicate that traveling waves play a key role in understanding these interesting dynamical phenomena. However, there is a lack of rigorous proof of the existence of traveling waves to this system. Here we generalize this isothermal autocatalytic chemical reaction model and provide a rigorous proof of the existence of traveling waves for the resulting reaction-diffusion system which also includes the systems arising from epidemiology and the microbial growth in a flow reactor. Introduction.Since the pioneering works of Fisher [9] and of Kolmogorov, Petrovsky and Piskunov [23], traveling waves in a wide array of biological and chemical systems, based on the interaction between reaction and diffusion processes, have been extensively investigated. In many realistic systems, traveling waves can be initiated by a highly localized disturbance of a stable rest state (e.g., a highly localized increase in ion concentrations). A typical example is the Hodgkin-Huxley model [13] which describes the electrical activity across membranes of never cells. It is known that if a short, but sufficiently large, burst of simulating current is injected into a nerve axon, a so-called action potential wave can then be generated. In this paper, we will discuss a system which enjoys a similar phenomenon. Specifically, this system is governed by the following equations:

Computational Study of Traveling Wave Solutions of Isothermal Chemical Systems

“…Other related results appeared in [11,12]. Whereas m > 1 case has been studied in [10,25]. The results again are in the classical mono-stable category for existence of traveling waves.…”

Traveling waves solutions to general isothermal diffusion systems

“…A type of center manifold argument, see [14][15][16] shows there is a unique positive solution coming out of (0, 1, 0), whose asymptotic behavior we describe in more detail in what follows.…”

The Traveling Wave of Auto-Catalytic Systems-Monotone and Multi-Peak Solutions

“…Moreover, v losses its monotonicity. For n = m, it was demonstrated in [15,16,26,27] that any traveling wave has the property that v is bellshaped. But, when n > m, our results show that v can oscillate a large number of times, which increases the difficulty of rigorous analysis.…”