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

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 $$k\in (0,1)$$ which represents the sharpness of the layer. We show that the minimum of the period is attained at an intermediate point $$k\in (0,1)$$ when a coupling constant $$\gamma $$ satisfies $$\gamma >3/4$$ . Proofs are based on explicit calculations using Jacobi’s elliptic functions and integrals.