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Evans function and Fredholm determinants.

Issa Karambal1, Simon J A Malham1

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Summary
This summary is machine-generated.

This study clarifies the equivalence between the Evans function, transmission coefficient, and Fredholm determinant for linear differential operators. These findings are crucial for analyzing stability in nonlinear partial differential equations like reaction-diffusion and solitary wave equations.

Keywords:
Evans functionFredholm determinanttravelling waves

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Area of Science:

  • Mathematical analysis
  • Applied mathematics
  • Numerical analysis

Background:

  • Linear stability analysis is crucial for understanding traveling wave solutions in nonlinear PDEs.
  • The Evans function, transmission coefficient, and Fredholm determinant are key tools for eigenvalue problems.
  • Recent attention has focused on the interrelationships between these determinant-based methods.

Purpose of the Study:

  • To explore the relationship between the Evans function, transmission coefficient, and Fredholm determinant.
  • To clarify the equivalence between these mathematical tools.
  • To investigate their application in linear stability problems.

Main Methods:

  • Analysis of first-order linear differential operators on the real line.
  • Focus on systems where the Fredholm operator is a trace class perturbation of the identity.
  • Inversion of eigenvalue problems using the free-state operator.

Main Results:

  • Clarification of the equivalence between the Evans function and the transmission coefficient.
  • Proof of the equivalence between the transmission coefficient and the Fredholm determinant, especially for distinct far fields.
  • Demonstration of their utility in eigenvalue determination.

Conclusions:

  • The Evans function and transmission coefficient are shown to be equivalent under specific conditions.
  • The transmission coefficient and Fredholm determinant are proven to be equivalent, offering a unified approach.
  • These results enhance the analytical and numerical tools for studying stability in nonlinear PDEs.