Quantitative Stability of Two Weakly Interacting Kinks in the Stationary phi^6 Model

TL;DR

This paper establishes precise stability estimates for configurations close to two weakly interacting kinks in the stationary phi^6 model, demonstrating how solutions remain stable under small perturbations and quantifying their interactions.

Contribution

The paper provides the first sharp quantitative stability estimates for two weakly interacting kinks in the phi^6 model, including explicit bounds and decay rates.

Findings

Stability estimates depend exponentially on the separation between kinks.

Existence of translation parameters aligning the solution with kink configurations.

Quantitative bounds relate solution deviations to residuals in the differential equation.

Abstract

We study the stationary phi^6 model given by the equation -phi''(x) + 2 phi(x) - 8 phi(x)^3 + 6 phi(x)^5 = 0 for x in R, and establish sharp quantitative stability estimates for configurations close to two weakly interacting kinks. More precisely, there exist constants a > 0 and epsilon > 0 such that, for any function u in L-infinity satisfying || u - H_{0,1}(x + x1) - H_{-1,0}(x + x2) ||_{H1} < epsilon with x2 - x1 > a, there exist constants y1, y2 such that || u - H_{0,1}(x + y1) - H_{-1,0}(x + y2) ||_{H1} + exp(-sqrt(2) (y2 - y1)) <= C * || u'' - 2 u + 8 u^3 - 6 u^5 ||_{L2}.