Quantitative Stability of Two Weakly Interacting Kinks in the Stationary phi^6 Model
Xin Liao
TL;DR
This work analyzes the stationary phi^6 equation $-\phi''+2\phi-8\phi^3+6\phi^5=0$ and establishes a sharp quantitative stability result for configurations near two weakly interacting kinks $H_{0,1}$ and $H_{-1,0}$ with large separation. The authors develop a two-kink modulation framework, derive an explicit residual equation for the linearized dynamics, and prove that the $H^1$-distance to the two-kink manifold plus the exponential separation between the kinks is controlled by the $L^2$-norm of the residual $\mathscr{F}(u)=u''-2u+8u^3-6u^5$. This yields a corollary for the time-dependent equation, showing that initial data near the two-kink configuration remain close to a modulated two-kink profile with a small remainder, and it provides explicit exponential interaction bounds that quantify how the kinks influence each other as their separation grows. The results rely on modulation analysis, spectral properties of the linearized operator, and precise estimates of inter-kink interactions, contributing to a detailed understanding of multi-soliton stability in nonintegrable scalar field theories.
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}.