Instability of pulses in gradient reaction-diffusion systems: a symplectic approach

Area of Science:

• Nonlinear dynamics
• Mathematical physics
• Chemical kinetics

Background:

• The Maslov index (MI) determines steady-state stability in scalar reaction-diffusion equations.
• Pulse solutions in these systems are known to be unstable.
• Gradient nonlinearity in reaction-diffusion systems presents unique stability challenges.

Purpose of the Study:

• To extend the concept of the Maslov index to reaction-diffusion systems with gradient nonlinearity.
• To establish a link between the Maslov index and the stability of pulse solutions.
• To generalize the Maslov index for asymptotically constant states and homoclinic orbits.

Main Methods:

• Generalizing the definition of the Maslov index for asymptotically constant states.
• Analyzing linearized evolution equations to count unstable eigenvalues.
• Employing symmetry arguments to prove properties of pulse solutions.

Main Results:

• A generalized Maslov index is associated with asymptotically constant states in reaction-diffusion systems.
• The Maslov index is shown to equal the number of unstable eigenvalues.
• Pulse solutions are demonstrated to possess a non-zero Maslov index.

Conclusions:

• The Maslov index provides a robust tool for assessing the stability of pulse solutions.
• All pulse solutions in these systems are inherently unstable due to a non-zero Maslov index.
• This work extends topological methods for stability analysis in nonlinear wave phenomena.