A wide variety of stationary or moving spatially localized structures is present in evolution problems on unbounded domains, governed by higher-than-second-order reversible spatial interactions. This work provides a generic unfolding in one spatial dimension of a certain codimension-three singularity that explains the organization of bifurcation diagrams of such localized states in a variety of contexts, ranging from nonlinear optics to fluid mechanics, mathematical biology and beyond. The singularity occurs when a cusp bifurcation associated with the onset of bistability between homogeneous steady states encounters a pattern-forming, or Turing, bifurcation. The latter corresponds to a Hamiltonian-Hopf point of the corresponding spatial dynamics problem. Such codimension-three points are sometimes called Lifshitz points in the physics literature. In the simplest case where the spatial system conserves a first integral, the system is described by a canonical fourth order scalar system. The problem contains three small parameters, two that unfold the cusp bifurcation and one that unfolds the Turing bifurcation. Several cases are revealed, depending on open conditions on the signs of the lowest-order nonlinear terms. Taking the case in which the Turing bifurcation is subcritical, various parameter regimes are considered and the bifurcation diagrams of localized structures are elucidated. A rich bifurcation structure is revealed, which involves transitions between regions of localized periodic patterns generated by homoclinic snaking, and mesa-like patterns with uniform cores. The theory is shown to unify previous numerical results obtained in models arising in nonlinear optics, fluid mechanics, and excitable media more generally.

各种各样的静止或移动的空间局部结构是 出现在无界区域上的演化问题中，由 高于二阶的可逆空间相互作用。这项工作提供了一个 在某个余维-3的一个空间维度中的一般展开 奇点，它解释了这种情况的分叉图的组织 各种情况下的局域态，从非线性光学到 流体力学、数学生物学和其他学科。奇点发生在一个 尖点分叉与齐次系统之间双稳态的开始 稳态会遇到图案形成或图灵分叉。后者 对应于相应空间动力学的哈密顿-霍普夫点 有问题。这种余维三点有时被称为Lifshitz点 物理学文献。在最简单的情况下，空间系统守恒 作为第一积分，系统用正则四阶标量来描述 系统。这个问题包含三个小参数，其中两个是展开尖端的 分叉和展开图灵分叉的分支。有几个案例是 透露，取决于开盘条件的标志最低顺序 非线性项。以图灵分叉为例 考虑了亚临界、不同参数的区域和分叉 阐明了局域结构的图解。丰富的分叉结构 ，这涉及到局域周期区域之间的跃迁 同宿蛇形花纹和均匀台面花纹 核心。该理论统一了文献[1]中得到的数值结果。 非线性光学、流体力学和可激发介质中的模型更多 一般说来。