Numerical computation of an Evans function for travelling waves

Area of Science:

• Mathematical analysis
• Applied mathematics
• Numerical analysis

Background:

• Traveling waves are crucial solutions in reaction-diffusion systems, like the Fisher-KPP equation and Keller-Segel models.
• Understanding the spectral stability of these waves is essential for predicting system behavior and pattern formation.

Purpose of the Study:

• To develop and demonstrate a geometrically inspired technique for computing Evans functions.
• To apply this method to analyze the spectral stability of traveling waves in the F-KPP and Keller-Segel models.
• To provide a new proof of spectral stability for specific F-KPP traveling waves.

Main Methods:

• A novel geometric approach for calculating Evans functions associated with linearized operators around traveling waves.
• Numerical computation of the Evans function across a wide range of spectral parameters.
• Extension of the Evans function into the continuous spectrum.
• Analysis of eigenvalues in the right half of the spectral plane.

Main Results:

• The developed Evans function is computable over many orders of magnitude of the spectral parameter.
• The Evans function was successfully extended into the continuous spectrum for both models.
• Numerical verification confirmed the absence of eigenvalues in a significant region of the right spectral plane for both models.
• A new proof of spectral stability was established for F-KPP traveling waves with speed c≥2√δ.

Conclusions:

• The geometric technique provides a robust method for computing Evans functions and assessing spectral stability.
• The findings support the spectral stability of traveling waves in the studied F-KPP and Keller-Segel models.
• This work contributes to a deeper understanding of wave stability in important biological and physical models.