Numerical computation of an Evans function for travelling waves

• A geometrically inspired technique for computing Evans functions.
• Evans functions are computed for scalar and systems of PDEs.
• Numerically examine stability of travelling waves in a chemotaxis model.
• Includes a new proof of stability of travelling waves in the F-KPP equation.

We demonstrate a geometrically inspired technique for computing Evans functions for the linearised operators about travelling waves. Using the examples of the F-KPP equation and a Keller–Segel model of bacterial chemotaxis, we produce an Evans function which is computable through several orders of magnitude in the spectral parameter and show how such a function can naturally be extended into the continuous spectrum. In both examples, we use this function to numerically verify the absence of eigenvalues in a large region of the right half of the spectral plane. We also include a new proof of spectral stability in the appropriate weighted space of travelling waves of speed c≥2δ in the F-KPP equation.