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Exact solutions describing very slow layer oscillations in a shadow reaction-diffusion system

Shin-Ichiro Ei, Yasuhito Miyamoto, Tatsuki Mori

TL;DR

This work analyzes a shadow activator-inhibitor reaction-diffusion system and constructs exact single- and multi-layer stationary solutions. It proves Hopf bifurcations as the time constant $\tau$ crosses a critical value, yielding oscillations with period $2\pi/\lambda_{I}$ that are exponentially slow in $\varepsilon$, i.e., $O\left(e^{C/\varepsilon}\right)$. The authors derive exact critical values and periods, establish metastability for multi-layer states, and predict anti-phase horizontal oscillations of layers, all supported by numerical simulations that corroborate the theory. The results illuminate the mechanism by which slow layer oscillations arise from single-layer dynamics and provide a precise, verifiable framework for pattern evolution in shadow RD systems, with potential extensions to related activator-inhibitor models and reduced low-dimensional descriptions.

Abstract

We show in a rigorous way that a stable internal single-layer stationary solution is destabilized by the Hopf bifurcation as the time constant exceeds a certain critical value. Moreover, the exact critical value and the exact period of oscillatory solutions can be obtained. The exact period indicates that the oscillation is very slow, i.e., the period is of order $O(e^{C/\varepsilon})$. We also rigorously prove that Hopf bifurcations from multi-layer stationary solutions occur. In this case anti-phase horizontal oscillations of layers are shown by formal calculations. Numerical experiments show that the exact period agrees with the numerical period of a nearly periodic solution near the Hopf bifurcation point. Anti-phase (out of phase) horizontal oscillations of layers are numerically observed.

Exact solutions describing very slow layer oscillations in a shadow reaction-diffusion system

TL;DR

This work analyzes a shadow activator-inhibitor reaction-diffusion system and constructs exact single- and multi-layer stationary solutions. It proves Hopf bifurcations as the time constant crosses a critical value, yielding oscillations with period that are exponentially slow in , i.e., . The authors derive exact critical values and periods, establish metastability for multi-layer states, and predict anti-phase horizontal oscillations of layers, all supported by numerical simulations that corroborate the theory. The results illuminate the mechanism by which slow layer oscillations arise from single-layer dynamics and provide a precise, verifiable framework for pattern evolution in shadow RD systems, with potential extensions to related activator-inhibitor models and reduced low-dimensional descriptions.

Abstract

We show in a rigorous way that a stable internal single-layer stationary solution is destabilized by the Hopf bifurcation as the time constant exceeds a certain critical value. Moreover, the exact critical value and the exact period of oscillatory solutions can be obtained. The exact period indicates that the oscillation is very slow, i.e., the period is of order . We also rigorously prove that Hopf bifurcations from multi-layer stationary solutions occur. In this case anti-phase horizontal oscillations of layers are shown by formal calculations. Numerical experiments show that the exact period agrees with the numerical period of a nearly periodic solution near the Hopf bifurcation point. Anti-phase (out of phase) horizontal oscillations of layers are numerically observed.

Paper Structure

This paper contains 12 sections, 18 theorems, 164 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Critical eigenvalues and Hopf bifurcation
  4. Asymptotic formulas
  5. Proofs of Theorems \ref{['T1']} and \ref{['T2']}
  6. Antiphase horizontal oscillation
  7. Numerical simulations
  8. Conluding remarks
  9. Appendix: Elliptic integrals and functions
  10. Complete elliptic integrals
  11. Elliptic functions
  12. Formulas

Key Result

Theorem 1.2

Suppose that $0<\varepsilon<1/\pi$. Let us consider single-layer stationary solutions $(u^{\pm}_1,0)$ and let $k_1$ be defined by (e) with $n=1$. Let $\delta:=\alpha\beta/\gamma$ and where $K(k)$ and $E(k)$ are Jacobi's complete elliptic integrals of the first and second kinds, respectively. They are defined by (K) and (E) in Appendix. Let $\chi_n$ be defined by Here, $\chi_n>0$ because of Lemma

Figures (5)

  • Figure 1: Two initial data. The left function is used as an initial function $u_0$ in (A1) and (B1) in Figure \ref{['fig2']}. The right function is used as an initial function $u_0$ in (C1) and (D1) in Figure \ref{['fig2']}. In both initial data, $\xi_0=0.03$.
  • Figure 2: Bird views of $u(x,t)$. $(u_1^-,0)$ is stable in (A1), but it is unstable in (B1). In both (C1) and (D1) $(u_2^-,0)$ is unstable. Anti-phase horizontal oscillations of two layers are observed in (C1) and (D1).
  • Figure 3: A graph of $\int_0^1u(x,\,\cdot\,)dx$ in four cases.
  • Figure 4: Enlarged view of (A2) for $0<t<100$. The solution is nearly periodic and the first period is approximately $38.12$.
  • Figure 5: The relative error percentage of the numerical period of (A2) i.e.,$1-\textrm{Numerical period}/\textrm{Exact period}$. The relative error starts from $2.1\%$ near $t=0$ and it is increasing in $t$ up to $2.9\%$.

Theorems & Definitions (32)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 22 more