<p>We propose a shadow reaction-diffusion system with a symmetric bistable nonlinearity and consider the system in the unit interval with the Neumann boundary condition. This problem has internal <i>n</i>-layer stationary solutions. The spectral set for the linearization around these solutions can be rigorously determined. Using exact complex conjugate eigenvalues, we show that the 1-layer stationary solution is destabilized by Hopf bifurcation as a time constant exceeds a certain value. Then, time periodic solutions appear and their period depends on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$k\in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> which represents the sharpness of the layer. We show that the minimum of the period is attained at an intermediate point <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$k\in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> when a coupling constant <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\gamma >3/4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>></mo> <mn>3</mn> <mo stretchy="false">/</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>. Proofs are based on explicit calculations using Jacobi’s elliptic functions and integrals.</p>

Exact eigenvalues and Hopf bifurcations for a 1D shadow reaction-diffusion system with a symmetric bistable nonlinearity

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2026年7月22日

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Yasuhito Miyamoto,
Hayato Nakamura

摘要

We propose a shadow reaction-diffusion system with a symmetric bistable nonlinearity and consider the system in the unit interval with the Neumann boundary condition. This problem has internal n-layer stationary solutions. The spectral set for the linearization around these solutions can be rigorously determined. Using exact complex conjugate eigenvalues, we show that the 1-layer stationary solution is destabilized by Hopf bifurcation as a time constant exceeds a certain value. Then, time periodic solutions appear and their period depends on $k\in (0,1)$ $k \in (0, 1)$ which represents the sharpness of the layer. We show that the minimum of the period is attained at an intermediate point $k\in (0,1)$ $k \in (0, 1)$ when a coupling constant $\gamma $ $γ$ satisfies $\gamma >3/4$ $γ > 3 / 4$ . Proofs are based on explicit calculations using Jacobi’s elliptic functions and integrals.