Kozyreff G, Chapman S J
Optique Nonlinéaire Théorique, Université Libre de Bruxelles, CP 231, Campus Plaine, B-1050 Bruxelles, Belgium.
Phys Rev Lett. 2006 Jul 28;97(4):044502. doi: 10.1103/PhysRevLett.97.044502. Epub 2006 Jul 25.
We analyze stationary fronts connecting uniform and periodic states emerging from a pattern-forming instability. The size of the resulting periodic domains cannot be predicted with weakly nonlinear methods. We show that what determine this size are exponentially small (but exponentially growing in space) terms. These can only be computed by going beyond all orders of the usual multiple-scale expansion. We apply the method to the Swift-Hohenberg equation and derive analytically a snaking bifurcation curve. At each fold of this bifurcation curve, a new pair of peaks is added to the periodic domain, which can thus be seen as a bound state of localized structures. Such scenarios have been reported with optical localized structures in nonlinear cavities and localized buckling.
我们分析了由图案形成不稳定性产生的连接均匀态和周期态的静止前沿。所得周期域的大小无法用弱非线性方法预测。我们表明，决定这个大小的是指数小（但在空间中指数增长）的项。这些项只能通过超越通常多尺度展开的所有阶数来计算。我们将该方法应用于Swift-Hohenberg方程，并解析地导出了一条蛇行分岔曲线。在这条分岔曲线的每个折点处，一对新的峰值被添加到周期域中，因此可以将其视为局部结构的束缚态。这种情况在非线性腔中的光学局部结构和局部屈曲中已有报道。
