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Instability of solutions in a degenerate reaction diffusion equation

R. Marangell, J. J. Wylie, B. H. Bradshaw-Hajek

TL;DR

The paper analyzes spectral stability of travelling and stationary fronts and pulses in a class of degenerate reaction-diffusion systems with two interacting components. By combining analytical Evans-function results in special cases (notably $D=1$ and $D=0$) with robust numerical Riccati-Evans computations for the general case, it characterizes the essential spectrum and identifies when the left-half-plane condition holds, linking stability primarily to the point spectrum. The main findings are that stable travelling fronts can occur (in contrast to travelling pulses, which are typically unstable), while many nonstationary cases (especially pulses) are unstable; the degeneracy and special limits ($D=0$, $D=1$) create distinct stability behaviors with important implications for modeling, such as ionic transport models. The work provides both explicit Evans-function expressions in select scenarios and a practical numerical framework for computing spectra in degenerate RD systems, offering insight into when degenerate reaction terms can yield stable wave propagation in applications.

Abstract

We study the spectral stability of travelling and stationary front and pulse solutions in a class of degenerate reaction-diffusion systems. We characterise the essential spectrum of the linearised operator in full generality and identify conditions under which it lies entirely in the left-half plane. For a number of special cases we obtain analytical results, including explicit Evans functions, and complete spectral descriptions for certain stationary waves. In regimes where analytical methods are not available, we compute the point spectrum numerically using a Riccati-Evans function approach. Our results show that stable travelling fronts can occur, while travelling pulses are typically unstable.

Instability of solutions in a degenerate reaction diffusion equation

TL;DR

The paper analyzes spectral stability of travelling and stationary fronts and pulses in a class of degenerate reaction-diffusion systems with two interacting components. By combining analytical Evans-function results in special cases (notably and ) with robust numerical Riccati-Evans computations for the general case, it characterizes the essential spectrum and identifies when the left-half-plane condition holds, linking stability primarily to the point spectrum. The main findings are that stable travelling fronts can occur (in contrast to travelling pulses, which are typically unstable), while many nonstationary cases (especially pulses) are unstable; the degeneracy and special limits (, ) create distinct stability behaviors with important implications for modeling, such as ionic transport models. The work provides both explicit Evans-function expressions in select scenarios and a practical numerical framework for computing spectra in degenerate RD systems, offering insight into when degenerate reaction terms can yield stable wave propagation in applications.

Abstract

We study the spectral stability of travelling and stationary front and pulse solutions in a class of degenerate reaction-diffusion systems. We characterise the essential spectrum of the linearised operator in full generality and identify conditions under which it lies entirely in the left-half plane. For a number of special cases we obtain analytical results, including explicit Evans functions, and complete spectral descriptions for certain stationary waves. In regimes where analytical methods are not available, we compute the point spectrum numerically using a Riccati-Evans function approach. Our results show that stable travelling fronts can occur, while travelling pulses are typically unstable.
Paper Structure (11 sections, 76 equations, 11 figures, 2 tables)

This paper contains 11 sections, 76 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Essential spectra
  4. Analytical results
  5. Equal diffusivity, $D=1$
  6. Stationary waves, $v$ does not diffuse, $D=c=0$
  7. Numerical results
  8. Nonstationary waves, $v$ does not diffuse, $D=0,\,c>0$
  9. Stationary waves, $v$ diffuses, $D>0$, $c=0$
  10. The general case: nonstationary waves, $v$ diffuses, $D>0$, $c\ne0$
  11. Summary of results and Discussion

Figures (11)

  • Figure 1: The six types of typical Fredholm borders and the number of roots with positive real part for $\nu$ are shown for various values of $\hat{g}_u^\pm$ and $\hat{g}_v^\pm$. The results are shown for $c>0$ and $0<D<1$. Results for $c<0$ and $D>1$ can be obtained using symmetry considerations.
  • Figure 2: The behaviour as $c\to0$ of the Fredholm boundaries and the number of roots with positive real part for $\nu$ are shown for a typical case in which $\hat{g}_u^\pm\le0$ or $\hat{g}_v^\pm\le0$. The value of $c$ decreases from the upper subfigure to the lower subfigure. As $c\to0$ the Fredholm borders collapse onto a portion of the real axis.
  • Figure 3: The behaviour as $c\to0$ of the Fredholm boundaries and the number of roots with positive real part for $\nu$ are shown for a typical case in which $\hat{g}_u^\pm>0$ and $\hat{g}_v^\pm\>0$. The value of $c$ decreases from the upper subfigure to the lower subfigure. As $c\to0$ the Fredholm borders collapse onto a portion of the real axis and an ellipse.
  • Figure 4: The essential spectrum for a travelling front solution whose asymptotic states far behind and far in front of the wave correspond to Figures \ref{['fig:FredholmBoundaries']}a and c .
  • Figure 5: Example 5: Nonstationary pulse, $D=0$. Real (blue online) and imaginary (gold) parts of the Riccati-Evans function, \ref{['eq:ricevans3d']} for the waves given in \ref{['eq:d0trav1sols']} for various values of $c$. In all cases for $c>0$ we have a positive eigenvalue, and hence point spectrum in the right-half plane.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Remark
  • Remark