Defects are weak and self-dual solutions of the Cross-Newell phase diffusion equation for natural patterns

Twist grain boundaries in three-dimensional lamellar Turing structures

“…When the aspect ratio of the system (its size measured in terms of the wavelength of the pattern) is large, or the boundaries not congruent, defects such as dislocations, disclinations, foci, grain boundaries, etc. (2,4) are formed as a result of the underlying rotational symmetry and local biases present during the nucleation of the pattern.…”

Global description of patterns far from onset: a case study

“…In the weak bending limit we proved [11] that this equipartition was a fact in general; i.e., solutions of the associated self-dual equations were also solutions of the fourth-order variational equations. Although this is not the case here, because of the Hessian obstruction, it turns out that the self-dual solutions are either a good approximation (in the small ε limit, the Hessian has its support at point defects [11] and along curved line defects) or, at worst, serve as good test functions for bounding the free energy in the ε → 0 limit. This is the fundamental idea underlying our analysis.…”

“…In earlier papers [1,11,12], we had proved that self-dual (or anti-self-dual) solutions, namely phase functions Θ(X, Y) which satisfy…”

“…One way to do this concretely would be to consider single-valued functions on domains which are Riemann surfaces rather than simply connected planar domains. In [11], we carried out such numerical and analytical studies on a double cover of a disc Ω with vectorial boundary data whose winding number is half integral. In this way, we were able to obtain stationary solutions of the RCN equation corresponding to concave (V) and convex (X) disclinations, which are the canonical point defects of two-dimensional patterns (see Fig.…”

Global Description of Patterns Far from Onset: A Case Study

“…7, 8 and 11. For example, a string of dislocations located along the x-axis, at positions ±x n is θ = y − sign(y) log 1 + This is an exact solution of the phase equation with a "weak bending" assumption [11]. In fact, if a multi-dislocation with spacing seen in Fig.…”