[Paper Review] A Stability Index for Traveling Waves in Activator-Inhibitor Systems

Paul Cornwell, Christopher K. R. T. Jones|arXiv (Cornell University)|Mar 22, 2017

Advanced Differential Equations and Dynamical Systems25 references3 citations

TL;DR

This paper introduces a stability index for traveling waves in activator-inhibitor reaction-diffusion systems using a symplectic formulation of the Evans function and the Maslov index. By showing that the parity of the Maslov index determines the sign of the Evans function's derivative at zero, the method provides a geometric criterion for spectral stability, with the index computed via conjugate points detected by a symplectic detection form.

ABSTRACT

We consider the stability of nonlinear traveling waves in a class of activator-inhibitor systems. The eigenvalue equation arising from linearizing about the wave is seen to preserve the manifold of Lagrangian planes for a nonstandard symplectic form. This allows us to define a Maslov index for the wave corresponding to the spatial evolution of the unstable bundle. We formulate the Evans function for the eigenvalue problem and show that the parity of the Maslov index determines the sign of the derivative of the Evans function at the origin. The connection between the Evans function and the Maslov index is established by a "detection form," which identifies conjugate points for the curve of Lagrangian planes.

Motivation & Objective

• To develop a geometric stability criterion for traveling waves in activator-inhibitor systems using topological invariants.
• To establish a connection between the Evans function's derivative at zero and the Maslov index of the wave's unstable bundle.
• To formulate a symplectic version of the Evans function using a nonstandard symplectic structure preserving Lagrangian planes.
• To introduce a detection form that identifies conjugate points along curves of Lagrangian planes in the spatial evolution of the unstable bundle.
• To apply the theory to singular perturbation regimes, such as fast-slow dynamics, to compute the Maslov index for traveling wave solutions.

Proposed method

• The eigenvalue problem for linearized traveling waves is recast in a symplectic framework using a nonstandard symplectic form that preserves the manifold of Lagrangian planes.
• The Maslov index is defined as the topological invariant counting the signed crossings of the unstable bundle with a fixed Lagrangian subspace under the spatial evolution.
• A detection form is introduced to identify conjugate points along the curve of Lagrangian planes, where the unstable bundle intersects a fixed subspace nontrivially.
• The Evans function is reformulated in symplectic terms, and its derivative at λ=0 is linked to the parity of the Maslov index via intersection theory.
• The method is applied to a singularly perturbed activator-inhibitor system with fast and slow dynamics, computing the Maslov index piecewise along the wave profile.
• The contribution to the Maslov index is computed at each segment (fast jump and slow manifold) using the crossing form, with sign determined by the symplectic structure.

Experimental results

Research questions

• RQ1 How can the Maslov index be used to determine the sign of the derivative of the Evans function at zero for traveling waves in activator-inhibitor systems?
• RQ2 What is the role of the symplectic structure in preserving the Lagrangian property of the unstable bundle during spatial evolution?
• RQ3 How can conjugate points along the unstable bundle be detected using a differential form in the symplectic setting?
• RQ4 What is the contribution of fast and slow segments of a singular traveling wave to the total Maslov index?
• RQ5 Can the parity of the Maslov index predict spectral stability in activator-inhibitor systems with Turing-type nonlinearity?

Key findings

• The parity of the Maslov index determines the sign of the derivative of the Evans function at λ=0, with even index implying positive derivative.
• The Maslov index for the traveling wave in the studied activator-inhibitor system is computed as zero, indicating even parity and thus a positive Evans function derivative at zero.
• There is one conjugate point on each of the two fast jumps (contributing -1 each) and one on each slow manifold (contributing +1 each), yielding a total Maslov index of zero.
• The crossing form calculation confirms that the fast jump crossings are negative-definite and the slow manifold crossings are positive-definite, consistent with the index contributions.
• The result implies that D′(0) > 0, which, combined with D(λ) > 0 for large λ and no positive eigenvalues, supports the spectral stability of the fast traveling wave.
• The method provides a geometric, topological alternative to direct eigenvalue computation for stability analysis in reaction-diffusion systems.