The phase structure of grain boundaries

Abstract

This article discusses numerical and analytical results on grain boundaries, which are line defects that separate roll patterns oriented in different directions. The work is set in the context of a canonical pattern-forming system, the Swift–Hohenberg (SH) equation, and of its phase diffusion equation, the regularized Cross–Newell equation. It is well known that, as the angle made by the rolls on each side of a grain boundary is decreased, dislocations appear at the core of the defect. Our goal is to shed some light on this transition, which provides an example of defect formation in a system that is variational. Numerical results of the SH equation that aim to analyse the phase structure of far-from-threshold grain boundaries are presented. These observations are then connected to properties of the associated phase diffusion equation. Outcomes of this work regarding the role played by phase derivatives in the creation of defects in pattern-forming systems, about the role of harmonic analysis in understanding the phase structure in such systems, and future research directions are also discussed.

This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

1. Introduction

Pattern-forming systems, their solutions, defects and instabilities, have driven the field of nonlinear phenomena for more than 100 years (see [1–3] as well as for instance [4–6] and references therein). Particularly fascinating is the role played by nonlinearities and symmetries in defining the properties of emerging patterns and of their defects.

This article is concerned with two-dimensional stationary stripe patterns and their line defects, which are grain boundaries. The existence of the latter is due to the almost invariably present rotational symmetry of the system before rolls emerge, resulting in a degeneracy associated with the orientation of the pattern.

As a consequence, in a spatially extended system, regions of rolls of different orientations are likely to appear, thereby giving rise to grain boundaries, which are curves separating these regions. In what follows, we focus on symmetric grain boundaries, which are invariant under reflection across their core, the latter then taking the form of a straight line. We are particularly interested in a transition numerically observed in these defects [7], whereby dislocations appear at the core of grain boundaries when the angle made by the rolls on each side is decreased. Our goal is to explore the causes of this transition and its intrinsic mechanism. To this end, we restrict our study to patterns described by the Swift–Hohenberg (SH) equation [8], which is the simplest known stripe pattern-forming differential equation one may construct.

(a). The Swift–Hohenberg and regularized Cross–Newell equations

The SH equation [8],

1.1

is a canonical pattern-forming model that was initially derived to model thermal fluctuations near the appearance of Rayleigh–Bénard convection rolls. The parameter R is a measure of distance from the convection threshold and the variable ψ, which is a real function of time t and of space coordinates, denoted hereafter by x and y, represents a measurable physical quantity, such as temperature or vertical fluid velocity. The SH equation can be rewritten in variational form as

For R>0 small and , it possesses [9] a stable [10–13] family of stationary roll solutions of the form

1.2

where , is small, and ψ₀ is a periodic function of the phase

which is itself a linear function of space. In the (k,R) plane, the region of roll stability is known as the Busse balloon [14], and is bounded by the zig-zag [15] and Eckhaus [16] instability curves. The former occurs when rolls are ‘stretched’ too much (or when their wavenumber k is smaller than k_Z≃1); the latter takes place when rolls are too compressed (and their wavenumber k exceeds the Eckhaus instability threshold k_E).

Far from threshold, that is for values of , roll patterns are well described by the regularized Cross–Newell (RCN) equation [17,18],

1.3

which can be formally derived from the SH equation by means of a multiple scales expansion, assuming ψ(x,y)≃ψ₀(Θ/ϵ), letting

rewriting the solvability condition obtained at order ϵ³ in terms of the original variables θ, x, y and t (see eqn (18) of Ercolani et al. [18] for details), and rescaling the fourth-order regularizing term (as in [19]). When the wavenumber k of the roll solution is such that k² is close to 1, two simplifications occur, τ(k²)≃1 and B(k²)≃2(1−k²), and together they transform the RCN equation into

1.4

which may also be written in the following variational form:

It was shown in [18] that self-dual solutions, that is solutions satisfying Δθ=±(1−|∇θ|²), are solutions of the fourth-order RCN equation (1.4), provided their Hessian, or equivalently the Gaussian curvature of the graph of their solution θ, vanishes. Moreover, the work of [18,20] suggests that, even though the RCN equation is obtained in a regime where phase gradients are assumed to vary slowly, it is able to capture the structure of many defects observed in pattern-forming systems, including disclinations and grain boundaries.

(b). Defects in roll patterns

Roll patterns admit a variety of point and line defects, similar to defects found in nematic liquid crystals (see e.g. [21]), and which can be classified in a similar fashion [22]. In two spatial dimensions, point defects consist of disclinations and dislocations characterized by a rotation of the direction of the rolls about the core of the defect equal to ±π and ±2π. Line defects are grain boundaries that separate patches of rolls oriented in different directions. Figure 1a shows a snapshot of a time-evolving solution of the SH equation, where disclinations, dislocations and grain boundaries may be seen. Our numerical simulations indicate that, when R=1, the quantity

is a good diagnostic to locate point defects of the SH equation, because the latter are characterized by relatively large values of σ. Indeed, figure 1a–c identify regions where σ>σ_th, for different values of the threshold σ_th. Setting σ_th=0.35 (figure 1a) conveniently isolates the cores of dislocations and disclinations.

Figure 1.

Snapshots of a time-evolving solution ψ(t,x,y) of equation (1.1) with superimposed regions where σ>0.35 (a), σ>0.25 (b) and σ>0.15 (c). (Online version in colour.)

For lower values of the threshold σ_th, the cores of grain boundaries start to become visible (figure 1c). A complete numerical study of grain boundaries of the SH equation was recently provided in [23]. As mentioned before, we focus on symmetric grain boundaries, whose cores are straight lines. They connect roll patterns with asymptotic phases of the form θ₁(x,y) and θ₂(x,y) such that

and are parametrized by the angle α. The existence of such defects for the SH equation near threshold (that is with R small and bounded above by a quantity that vanishes as α→π/2) was recently proved in [24,25] for all values of the angle α. It has, however, been known since the early 2000s that in far-from-threshold numerical simulations with boundary conditions that enforce the reflection symmetry about the core of grain boundaries, the nature of these solutions changes when the angle α exceeds a critical value (or when k_y is larger than some threshold value) [7]. This transition manifests itself by the appearance of dislocations (or possibly pairs of disclinations) along the core of the grain boundary. The goal of this article is to shed light on the nature of this instability and to discuss its implications regarding connections between defect creation and large phase derivatives in pattern-forming systems.

(c). This article

The rest of this article is organized as follows. Section 2 presents a numerical investigation of the phase structure of stable grain boundary solutions of the SH equation. It reveals that, as the angle α increases, the transition occurring at the core of grain boundaries is associated with the presence of large phase gradients, which translates into a concentration of the RCN energy at points along the core of the grain boundary.

Section 3 discusses self-dual grain boundary solutions of the RCN equation that correspond to dislocation-free line defects for all values of the angle α. It also provides a proof of their stability in a solution space where gradients are vector fields. Specifically, we have the following result.

Theorem 1.1. —

Knee solutions of the RCN equation, given by

are the unique (up to a constant phase shift) minimizers of the RCN energy ℰ_RCN in the space ℱ of functions θ(x,y) such that

• — θ∈H²(Ω), Ω=[0,P]×[−L,L], ;

• — θ(x+P,y)=θ(x,y)+π, ∀ y∈[−L,L];

• —  at y=±L.

Remark 1.2. —

This theorem extends a result proven in [18] in that it pins down an appropriate function space for our analysis and, within that function space, establishes uniqueness of a minimizer. That uniqueness, in turn, yields a weak stability result, as it shows that it would be inconsistent for any perturbation preserving the function space to have a growth rate that is non-negative. Preliminary calculations indicate that in fact a strong stability result should hold here; i.e. that for any value of |μ|<1, there exists c>0, depending on μ, such that the infimum of the decay rate magnitudes of perturbations preserving the space is bounded below by c. However, this remains to be rigorously established by a complete analysis of the Hessian for the RCN variational problem.

The second part of §3 discusses how to reconcile this stability result with the presence of an instability in grain boundaries of the SH equation. Specifically, it reviews recent results on minimizers of the RCN equation when phase gradients are not vector fields.

Section 4 revisits the numerical simulations of §2 in the light of the results of §3 and proposes a new class of boundary conditions for potential minimizers of the RCN equation. It also places the latter equation in the perspective of a hierarchy of variational problems whose solutions increase in complexity, and it elaborates on the role of harmonic analysis in the results that we present. Finally, future research directions suggested by this work are discussed. The article ends with a brief summary of its content, in §5.

2. The phase structure of grain boundaries

In this section, we show numerical simulations of the SH equation using a pseudo-spectral code with periodic boundary conditions in both spatial directions. The numerical box has size L_x=2π/k₁, where , 0<μ<1, and L_y≃1000. Depending on L_x, we use up to 2⁸ Fourier modes in the x-direction and 2¹² in the y-direction. The initial conditions consist of two regions of rolls of wavevectors k₊=(k₁,μ) and k₋=(k₁,−μ), separated by two straight lines parallel to the x-axis. The time evolution is obtained by combining an exact integration of the linear part of the dynamics in Fourier space with an approximation of the nonlinear contributions, which are assumed to be constant over each (small) time step. For each value of μ and associated initial condition, the SH energy, approximated on the time-dependent numerical solution, decreases as a function of time towards a constant value. Grain boundary solutions shown in this article are configurations saved at time , equal to 10 000 or 20 000 units of time. At that point, these solutions are (for all practical purposes) stationary.

The use of a spectral code makes it convenient to extract the amplitude of each asymptotic pattern by shifting the part of the Fourier spectrum centred on one of the two asymptotic modes towards the origin, removing the rest of the spectrum, and applying an inverse Fourier transform. Figure 2 shows the pattern consisting of two grain boundaries (figure 2a) and the envelopes of the asymptotic rolls (figure 2b). In this case, these envelopes are translationally invariant in the x-direction along the core of the grain boundary. As the parameter μ increases, this translational invariance is broken and two dislocations appear per period at the core of the grain boundary, as illustrated in figure 3. These patterns are visually identical to those recently obtained in Lloyd & Scheel in [23] using a different numerical approach. In particular, the solution we observe for μ=0.85 corresponds to the upper branch of the bifurcation diagram shown in fig. 16 of [23]. All other solutions appear to be on the primary branch, which means the parity-shift bifurcation described in [23] was not observed on these solutions within the time frame of the simulations, and for boxes of length L_x corresponding to one period p of the pattern. For μ=0.9, changing the length of the box to 2p led to an instability corresponding to different concentrations of the SH energy at the core of each defect, which is consistent with the parity-shift bifurcation described in [23]. We did not see instabilities of this type at other values of μ, in boxes of length up to 8p (respectively, 2p) and for times up to 50 000 (respectively, 20 000) units for μ≤0.8 (respectively, 0.85<μ<0.9). In this article, we are only interested in the properties of symmetric grain boundaries on the primary branch of Lloyd & Scheel [23].

Figure 2.

Numerical solution of the SH equation corresponding to two grain boundaries with k₂=±μ=±0.2, showing both the pattern (a) and the envelopes of the asymptotic modes (b). (Online version in colour.)

Figure 3.

Solutions of the SH equation at t=20 000 (a,b) and t= 10 000 (c,d). The vertical extent of each picture corresponds to 60 units of length (recall L_y≃1000). (Online version in colour.)

The simulations reported here document changes in the phase structure of these grain boundaries as the angle of inclination of the rolls, or equivalently the parameter μ, is varied. Taking the Hilbert transform of a grain boundary solution ψ(x,y) of the SH equation in a direction parallel to its core (x in our case) for a fixed value of y, extends this solution to a complex field z(x,y) such that Re(z)=ψ(x,y). By carrying this out simultaneously for all values of y and introducing a phase shift of ±2π at the initiation/termination of each period, one can numerically evaluate a continuous function, which we define as the phase ϑ of the solution, such that and ψ(x,y)=Re(z(x,y)) everywhere in the numerical box.

The application of the Hilbert transform to construct phase-field representations in partial differential equation models of fluid dynamical phenomena is not unprecedented (see for instance [23,26–29]). We pause here to provide some analytical background for this numerical procedure. We recall from elementary complex function theory [30] the theorem of Schwarz, which states that any continuous periodic function, U(ϕ), of period 2π (which may be viewed as a function on the unit circle, |w|=1, in the complex w plane), has a unique extension to a harmonic function, u(w), on the disc |w|<1 such that the radial limit . This extension is explicitly given by Poisson’s formula

2.1

A complex-valued extension of this formula is very simply given by

2.2

which is analytic for |w|<1; v(w) is the conjugate harmonic function of u(w) [30]. It is then natural to ask about the properties of the boundary value function , which additionally is unique if one requires that v(0)=0. A theorem of M. Riesz [31] implies that if U(ϕ)∈L²(S¹), then so is V (ϕ) and so is representable by a Fourier series. In this setting, V (ϕ) is referred to as the Hilbert transform of U(ϕ). By elliptic regularity, for our smooth boundary conditions, the assumptions implicit in the above constructions are all satisfied so that we may take for fixed and then . It is straightforward to adjust all of this to the case when the period in x is different from 2π. In this setting, it is also straightforward to see that one may also directly define the Hilbert transform in terms of Fourier series: if U(ϕ) is a trigonometric polynomial, then the transform may be expressed as

This mapping extends to a bounded operator on L²(S¹), which, in our setting, is equal to the Hilbert transform. Matlab provides a code for implementing this representation in terms of the discrete Fourier transform which is used in our simulations. The ‘four-quadrant inverse tangent function’ provides ϑ mod 2π and a continuous version of ϑ is obtained by first eliminating discontinuities along the x-direction and then along the y-direction.

Figure 4 presents the outcome of this procedure for μ=0.5. Figure 4a shows ϑ as a function of x and y, and figure 4b shows ∇ϑ superimposed on the grain boundary. As illustrated in this figure, for small values of μ, the component k_y of k=∇ϑ smoothly vanishes and changes sign as it crosses the core of the grain boundary. This structure changes however as μ increases: instead of growing linearly towards the core of the defect, the phase of the solution increases in a stepwise fashion, leading to large y-derivatives on each side of the grain boundary. Specifically, k_y varies quickly from very large (corresponding to a step in ϑ along the y-direction) to very small (on the ‘flat’ part of the step). Moreover, because of the symmetry across the core of the defect, k_y, in those regions where it is not small, develops a ‘jump’ across the grain boundary. This is illustrated in figure 5.

Figure 4.

(a) Numerically estimated phase ϑ(x,y) for a grain boundary solution with μ=0.5. (b) Phase gradient superimposed on the grain boundary solution. The vectors do not appear perpendicular to the pattern due to the use of different scales in the x- and y-directions for the background picture. (Online version in colour.)

Figure 5.

Same as figure 4 but for μ=0.99. Note the large values of k_y near the core of the defect. (Online version in colour.)

The above results hint at a very interesting behaviour of the slowly varying phase function Θ described by the RCN equation for grain boundary solutions. Indeed, assuming that Θ_y should match the envelope of ϑ_y, figure 6 suggests that, as μ increases, Θ_y develops a jump across the core of the defect at y=0 (figure 6a), while concentrating at specific points in x, in a direction y=y_m parallel to the core of the grain boundary (figure 6b). Here, y_m is defined as the value of y near the core of the grain boundary where ϑ_y reaches its minimum. Moreover, as shown in figure 6c, ϑ alternates between regions where it is constant and regions where its y-derivative ϑ_y is constant along the line y=y_m.

Figure 6.

(a) Profile of k_y=ϑ_y at x=L_x/2 as a function of y, for different values of μ. The grain boundary is located at y=0. (b) Profile of ϑ_y at y=y_m, on one side of the core of the defect, plotted as a function of x for different values of μ. In this case, y_m is the value of y where ϑ_y reaches its minimum (see (a)). In these panels, distances along the x- and y-directions are measured in number of simulation points, to allow for comparison between different box sizes. (c) Profile of ϑ and ϑ_y as a function of x along y=y_m, superimposed on the grain boundary solution. (Online version in colour.)

3. Grain boundaries of the regularized Cross–Newell equation

One can easily show that the RCN equation (1.4) admits exact self-dual solutions given by

3.1

which correspond to SH grain boundaries that connect asymptotic roll patterns with and k₂=±μ. Figure 7 shows the pattern defined by , where θ_K is given in equation (3.1). We call these solutions knee solutions of the RCN equation. By construction, the envelopes of the asymptotic patterns are translationally invariant along the defect (at y=0) and, like the SH grain boundaries discussed in [24,25], these solutions do not have dislocations at their cores, regardless of the value of μ. Interestingly, they are the sole minimizers of the RCN energy, as stated in theorem 1.1. We now prove this theorem.

Figure 7.

Knee solutions of the RCN equation for different values of μ=k_y. (Online version in colour.)

(a). Proof of theorem 1.1.

Consider the following vector function of k=(f,g):

3.2

which has partial derivatives satisfying

3.3

Now assume k=∇θ, i.e. f=θ_x, g=θ_y, θ_xy=θ_yx, and evaluate the divergence of S(k). We have the following bound:

In other words,

3.4

for any function (as the limit of sequences of C² functions). The left-hand side of (3.4) may be evaluated using the boundary conditions satisfied by functions in ℱ. Specifically,

Therefore, all functions θ∈ℱ (such that ∇θ is globally defined as a vector field on Ω) satisfy

3.5

The last part of the argument consists in evaluating the right-hand side of (3.5) on the knee solution to the RCN equation given by equation (3.1). We have

so that

Solutions (3.1) have an energy equal to ℳ, and are therefore minimizers of ℰ_RCN on ℱ. In the limit as , we note that .

We can moreover show that θ_K is the unique (up to an additive constant) minimizer of ℰ_RCN in the space ℱ. The argument above indicates that ℰ_RCN(θ)≥ℰ_RCN(θ_K) for any with equality when

(a.e. = almost everywhere). That will be the case if

whence it is a standard exercise to show that

for some functions C₁∈H²([0,P]) and C₂∈H²([−L,L]). From the boundary conditions, we conclude that , so that the first equation gives, for g=θ_y=dC₂/dy,

Given in the trace sense, the equation has the unique solution

(b). Grain boundaries of the regularized Cross–Newell equation whose gradients are not vector fields

The numerical results of §2 suggest that, if grain boundaries with dislocations at their cores can be described by a solution Θ of the RCN equation, then Θ_y jumps at the core of the defect. As a consequence, such solutions cannot belong to the space ℱ defined in theorem 1.1. Because they are symmetric under reflection about the core of the grain boundary, they may however be regarded as elements of a function space of director fields. Such spaces have certainly been envisioned in other physical contexts such as liquid crystals, but their systematic global theory has yet to be fully developed. As an intermediate step, however, one may consider functions on with appropriate boundary conditions along the line y=0. We thus define the space of functions θ(x,y) such that

• — , , ;

• — θ(x+P,y)=θ(x,y)+π, ∀ y∈[0,L];

• —  at y=L;

• — there are appropriate boundary conditions along the line y=0.

This formulation is consistent with the picture of a director field on the whole plane which is symmetric about the x-axis, and which therefore has no jump singularity. Such a space was considered in [19] where the boundary conditions were taken of mixed Dirichlet–Neumann type:

There, it was shown that there exists μ_th∈(0,1) such that for each μ>μ_th, there exists a value of a∈[0,1] such that solutions have lower energy than the knee solution (3.1) with the same asymptotic behaviour. The threshold value μ_th was numerically estimated to lie between 0.89 and 0.92. This is very promising because it suggests that a transition similar to what is observed for SH grain boundaries also occurs in the RCN equation, provided the space of admissible solutions is taken to be . We however note that the boundary conditions for used in [19] are slightly different from those suggested by the phase structure of SH grain boundaries described in §2, where Dirichlet boundary conditions (ϑ(x,0)=0 up to an arbitrary constant) alternate with regions of constant ϑ_y.

4. Discussion

We believe that the combination of numerical explorations and analytical considerations presented in this article defines a roadmap for understanding the instability undergone by symmetric grain boundaries of the SH equation, as the angle made by the two roll patterns on each side of the core of the defect is decreased. The path is paved with analytical challenges, but the journey would be quite rewarding because it would provide the first rigorous bridge between the SH and RCN equations, and between large phase derivatives and the emergence of defects. The latter connection is also supported by numerical simulations of the Ginzburg–Landau equation [32], wherein the phase instability of oscillatory or plane wave solutions gives rise to the spontaneous formation of zeros of the complex order parameter. In two spatial dimensions, the resulting dynamic state was named defect-mediated turbulence [33–35].

(a). A search for additional grain boundary solutions of the regularized Cross–Newell equation

As mentioned in §3b, the lower-energy RCN grain boundary solutions found in [19] have different boundary conditions from those inferred by the numerical study of §2. It therefore seems reasonable to extend the approach of Ercolani & Venkataramani [19] to try to identify solutions of the RCN equation in a space that requires these alternate boundary conditions. The methodology developed in [19] relies on the use of self-dual solutions as test functions, and on seeking minimizers that only locally deviate from self-duality.

We estimated the self-duality of the phase of the numerically obtained grain boundary solutions of the SH equation discussed in §2, as a function of the parameter μ. Figure 8a shows that, for μ near 0.1, the phase of these solutions is self-dual, up to numerical errors. As μ gets larger, the quantity grows linearly with μ until μ≃0.85, after which the growth rate increases and solutions are clearly not self-dual. Further inspection, however, reveals that deviations from self-duality are localized near the core of the grain boundary, even for large μ, as illustrated in figure 8b. We therefore believe that the methodology of Ercolani & Venkataramani [19] could apply and hence that it is reasonable to search for almost-self-dual solutions of the RCN equation that satisfy the mixed boundary conditions identified in this article. Such a study is beyond the scope of the present work and will be the topic of further investigations.

Figure 8.

Deviation from self-duality of the phase of SH grain boundaries. (a) As a function of μ. The quantity plotted is , where , measured on the entire numerical domain. (b) As a function of x and y for μ=0.96. Here, the quantity plotted is local and defined as SD=|(Δϑ)²−(1−|∇ϑ|²)²|. (Online version in colour.)

(b). Knee solutions of the Swift–Hohenberg and regularized Cross–Newell equations

The results discussed in this article suggest a strong connection between the knee solutions of the RCN equation given in equation (1.4) and the ‘knee-like’ grain boundaries whose existence was proved in [24,25] for all values of μ. Their analysis was done close to threshold in the SH equation, and it seems that a natural first step would be to extend it to far-from-threshold (i.e. in equation (1.1)) situations. Second, the question of the stability of these solutions is still open. The spatial-dynamics approach developed in [24,25] might bring some insight into stability close to threshold (when R≪1), but a new methodology is required for values of R of order 1. Third, as knee solutions of the RCN equation exist for all values of μ, the existence of these two sets of grain boundary solutions provides a paradigm to build a rigorous connection between the SH equation and its phase diffusion equation. These questions are currently under investigations by some of the authors and postdoctoral collaborators at the University of Arizona.

(c). Methods of harmonic analysis

In [18, section 4], it was noted that the RCN variational problem fits into a natural hierarchy of variational equations whose members include the harmonic map problem and the Ginzburg–Landau problem. Asymptotic minimizers of the latter are harmonic vector fields away from point defects, which arise from the limiting locations of zeros of vector fields in the associated minimizing sequence. It should therefore not be surprising that techniques of harmonic analysis might enter into the study of SH minimizers, as we saw in §2. Motivated by the Ginzburg–Landau analogue, it is natural to think that the appearance of zeros of the complex field z(x,y) constructed in §2 should relate to the formation of dislocations in the real part of the field. Going further, one could explore the emergence of zeros in the analytical extension u(w)+iv(w) that was defined in that section. Such ‘internal’ zeros could be signatures of emerging disclination pairs if they nucleate near the real boundary of the harmonic functions u and v. Happily, there is an elegant analytical description of such zeros due to Beurling in terms of factorization into inner and outer Hardy functions [31] which extend the Poisson–Jensen representation formula [30]. We intend to investigate this further in future work.

(d). Large phase derivatives and defects

A careful inspection of figure 1a reveals that dislocations are associated with two local maxima of σ, whereas disclinations show only one, providing support to the theory that a dislocation may be viewed as a pair of disclinations (see e.g. [21]). The profile of ϑ_y plotted in figure 6 shows two locations where |ϑ_y| varies fast in the vicinity of each dislocation, joined by a segment where ϑ_y is constant. This may be a strong indication that the appearance of dislocations at the core of grain boundaries is driven by a local concentration of phase gradients. It may also provide guidance for an analytical approach to this transition, which could consist of following how phase gradients of initially self-dual solutions of the RCN equation concentrate along the core of the defect as μ is increased, eventually leaving the self-dual approximation as these derivatives get large. Work in this direction is in progress.

5. Conclusion

The purpose of this article was to present numerical and analytical results on symmetric grain boundaries, specifically regarding the transition that occurs at the core of these defects as the angle of the rolls on each side of the core is decreased. We discussed the coherent picture that numerically emerged regarding the phase structure of grain boundaries, as they went through this transition. We emphasized the parallel between grain boundary solutions of the SH and RCN equations, as well as the somewhat simpler nature of self-dual solutions to the RCN equation, and considered how they may pave a way to analyse the transition observed in SH grain boundaries. We also addressed the question of the role played by large phase derivatives in this transition, as suggested by our numerical investigations. The results of theorem 1.1 are added to the knowledge about solutions of the RCN equation, which itself fits nicely into a hierarchy of variational equations of harmonic character, and presents analytical challenges of its own because it is of fourth degree. Finally, §4 reviewed various avenues for further inquiries that build on the results of this work.

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Authors' contributions

J.L. designed and performed the numerical explorations, and wrote a first draft of the manuscript. N.M.E. provided expertise in the analysis of the RCN equation, including theorem 1.1 and its proof, as well as background on the Hilbert transform. N.K. proved uniqueness of the minimizer of theorem 1.1. All authors discussed the simulations and the results of this study and contributed to the writing of the final article, which they also read and approved.

Competing interests

The authors declare that they have no competing interests.

Funding

This material is based upon the work supported by the National Science Foundation under Grant No. DMS-1615921.