[Paper Review] Competing Interactions and Traveling Wave Solutions in Lattice Differential Equations

Erik S. Van Vleck, Aijun Zhang|arXiv (Cornell University)|Oct 28, 2012

Differential Equations and Numerical Methods3 citations

TL;DR

This paper establishes the existence of traveling wave solutions in vector lattice differential equations with bistable nonlinearities and competing first- and second-neighbor interactions, using a perturbation approach based on Fredholm theory for mixed-type functional differential equations. The key contribution is a general persistence result for traveling waves under small perturbations, even in the absence of a comparison principle, extending prior scalar results to vector systems and infinite-range interactions.

ABSTRACT

The existence of traveling front solutions to bistable lattice differential equations in the absence of a comparison principle is studied. The results are in the spirit of those in Bates, Chen, and Chmaj in[1], but are applicable to vector equations and to more general limiting systems. An abstract result on the persistence of traveling wave solutions is obtained and is then applied to lattice differential equations with repelling first and/or second neighbor interactions and to some problems with infinite range interactions.

Motivation & Objective

• To establish the existence of traveling wave solutions in lattice differential equations with competing repelling first- and second-neighbor interactions.
• To extend the perturbation-based existence framework of Bates, Chen, and Chmaj from scalar to vector equations.
• To analyze systems without a comparison principle, where standard monotonicity techniques fail.
• To apply the theory to lattice equations with infinite-range interactions and periodic media.
• To prove persistence of monotonicity and exponential tail behavior under small perturbations.

Proposed method

• Uses a traveling wave ansatz $ u_j(t) = \varphi(j - ct) $ to reduce the PDE-like lattice system to a functional differential equation in $ \varphi $.
• Applies Fredholm theory for mixed-type functional differential equations to analyze the linearized operator around the wave solution.
• Employs the implicit function theorem in a Banach space framework to prove persistence of solutions under small perturbations of the interaction coefficients $ d_1, d_2 $.
• Analyzes the kernel structure of the linearized operator at the limiting system, particularly the principal eigenvalues and eigenvectors at equilibria $ \vec{0} $ and $ \vec{1} $.
• Establishes exponential decay of wave tails using principal eigenvalue analysis and asymptotic behavior of solutions.
• Imposes conditions on interaction coefficients (e.g., $ |d_2| \ll 1 $, $ |d_1| \ll 1 $) to ensure the Fredholm property and solution persistence.

Experimental results

Research questions

• RQ1 Under what conditions does a traveling wave solution persist in a vector lattice differential equation with competing first- and second-neighbor interactions?
• RQ2 How can the existence of traveling waves be established when no comparison principle is available?
• RQ3 What role does the kernel structure of the linearized operator play in the existence and stability of traveling waves?
• RQ4 Can the perturbation framework be extended to systems with infinite-range interactions?
• RQ5 Does monotonicity of the wave profile persist under small perturbations of the interaction coefficients?

Key findings

• There exists an $ \epsilon^* > 0 $ such that for all $ \epsilon \in (0, \epsilon^*] $, the perturbed system admits a traveling wave solution $ (c, \vec{w}) $ with $ \vec{w}(-\infty) = \vec{0} $, $ \vec{w}(\infty) = \vec{1} $, and $ \vec{0} < \vec{w}(\xi) < \vec{1} $ for all $ \xi \in \mathbb{R} $.
• For small $ \epsilon $, the wave speed $ c $ satisfies $ c \dot{\phi}_n(\xi) < 0 $ for all $ n \in \mathbb{Z} $ and $ \xi \in \mathbb{R} $, indicating monotonicity.
• Traveling waves exhibit exponential decay at both ends: $ \frac{u_i(t)}{e^{(i-ct)\lambda_0}\phi_i^0} \to h^- $ as $ i - ct \to -\infty $, and $ \frac{u_i(t)}{e^{(i-ct)\lambda_1}\phi_i^1} \to h^+ $ as $ i - ct \to \infty $.
• The principal eigenvalues $ \lambda_0 > 0 $ and $ \lambda_1 < 0 $ ensure the correct asymptotic behavior and stability of the wave profile.
• The persistence of monotonicity is preserved under small perturbations due to the sign stability of the principal eigenvalues.
• The results apply to systems with infinite-range interactions under conditions such as $ \sum_{|k| > k_0} a_{n,k} < \Pi(k_0) $ and $ K_2 < 1/C_0 $, ensuring the Fredholm property.