Code for the investigation of quenching of stable pulses in FitzHugh-Nagumo using the Dedalus Project.

Instructions:

1. install dedalus (version 2.2207) in an environment (conda recommended -- see environment.lock.yml). Environment may be instantiated using

conda env create -f environment.lock.yml

2. activate the conda environment with conda activate dedalus2

3. Either:

   • run waves.sh to generate the input data for the stable and unstable pulses; this involves the solution¹² of a nonlinear boundary value problem (NLBVP) for $(u,c)$: $D u'' + c u' + f(u) = 0, B_l(u, u') = 0, B_r(u, u') = 0.$ This code then solves the left (AVP) and right (EVP) eigenproblems: $v \sigma = L v, w^\dagger \sigma^* = L^\dagger w^\dagger,$ for the leading eigenmodes (e.g., $Re(\sigma) > -Re(\sigma_1) $), whose boundary conditions are inherited from the NLBVP formulation of $B_l(u,u')$ and $B_r(u,u')$.³
   • Or use the HDF files in the repository (recommended).

4. run quench.sh which solves the quenching problem for a selected stable pulse and (presently hard-coded) family of perturbations. This uses a bisection root-finding procedure to solve for the root of the function, $f(U_q) = \lim_{t\to\infty}\psi(u(t)) - \hat{\psi}$, where $u(0) = \check{u} + U_q \check{X}(x-\theta; x_s)$, for $0 > U_q \geq U_q^{max}$, where $|U_q^{max}| \gg 1$ is 'large', yielding $(x_s, \theta, U_q)$ tuples.

The corresponding linear theory calculation code is available here.