Pattern-forming fronts in a Swift-Hohenberg equation with directional quenching — parallel and oblique stripes
We study the effect of domain growth on the orientation of striped phases in a Swift-Hohenberg equation. Domain growth is encoded in a step-like parameter dependence that allows stripe formation in a half plane, and suppresses patterns in the complement, while the boundary of the pattern-forming region is propagating with fixed normal velocity. We construct front solutions that leave behind stripes in the pattern-forming region that are parallel to or at a small oblique angle to the boundary.
Technically, the construction of stripe formation parallel to the boundary relies on ill-posed, infinite-dimensional spatial dynamics. Stripes forming at a small oblique angle are constructed using a functional-analytic, perturbative approach. Here, the main difficulties are the presence of continuous spectrum and the fact that small oblique angles appear as a singular perturbation in a traveling-wave problem. We resolve the former difficulty using a farfield-core decomposition and Fredholm theory in weighted spaces. The singular perturbation problem is resolved using preconditioners and boot-strapping.
We are interested in the growth of crystalline phases in macro- or mesoscopic systems, subject to directional quenching. More precisely, we are interested in systems that exhibit stable or metastable ordered states, such as stripes, or spots arranged in hexagonal lattices. Examples of such systems arise for example in di-block copolymers , phase-field models [30, 1], and other phase separative systems [9, 37, 19], as well as in phyllotaxis , and reaction diffusion systems [24, 2]. Throughout, we will focus on a paradigmatic model, the Swift-Hohenberg equation
where , , , subscripts denote partial derivatives, and . It is well known that for , (1.1) possesses stable striped patterns , with and wave vector , for . The wave vector can be thought of as the lattice parameter of the crystalline phase, encoding its strain and orientation.
The particular scenario of interest here is when such a system is quenched into a pattern-forming state in a growing half plane , choosing for instance . The main question is then if the growth process will select an orientation or strain in the crystalline phase. Roughly speaking, our analysis establishes such a selection mechanism, for the strain, as a function of an arbitrary orientation, at least for small oblique angles.
In a moving coordinate frame , (1.1) then reads
The growth process and the selection of stripes is encoded in the existence and stability of coherent structures, that is, traveling waves or time-periodic solutions to (1.2).
Stripes parallel to the boundary in a comoving frame are of the form , hence time-periodic. The simplest solutions enabling the creation of such stripes are therefore of the form , , , , solving the boundary-value problem on ,
for some .
Obliques stripes are stationary in a vertically comoving frame . The simplest solutions enabling the creation of oblique stripes are therefore of the form , with , , , and solve
for some . Note that, formally letting , the problem (1.4) limits on (1.3). The difficulty in this limiting process is two-fold. First, the perturbation is singular in that the highest derivatives in vanish at . Second, the linearization of (1.3) at a solution ,
is not Fredholm as a closed and densely defined operator on , say, where . This can be readily seen noticing that belongs to the kernel but does not converge to zero at infinity, such that a simple Weyl sequence construction shows that the range is not closed.
It turns out that Fredholm properties can be recovered by choosing exponentially weighted function spaces. We therefore introduce the space
and consider as a closed operator on this exponentially weighted space with small weights .
Definition 1.1 (Non-degenerate parallel stripe formation).
We say that a solution of (1.3) is non-degenerate if the linearization is Fredholm of index 0 in the weighted space for all , sufficiently small, and is algebraically simple as an eigenvalue in these spaces. Moreover, the asymptotic periodic patterns are stable with respect to coperiodic perturbations, with a simple zero eigenvalue of the coperiodic linearization induced by translations.
We refer the reader to  for background and a spatial dynamics illustration motivating such non-degeneracy conditions. Note also that this choice of exponential weights allows for exponential growth of functions and is hence generally ill-suited for nonlinear analysis.
We are now in a position to state our results. Our first result is concerned with a singular perturbation in the presence of essential spectrum.
Theorem 1 (Parallel oblique stripe formation).
Suppose there exists a solution of (1.3) for some fixed , , forming parallel stripes. Suppose furthermore that the solution is non-degenerate as stated in Definition 1.1. Then there exists a family of solutions to (1.4), depending on , sufficiently small, forming oblique stripes. At , this family coincides with . The dependence of and of the solutions on , measured in , is of class . At leading order, the wavenumber satisfies the expansion with defined in (2.29), below.
Our second result shows that the assumptions of Theorem 1 hold for , sufficiently small.
Theorem 2 (Existence of parallel stripe formation).
Together, these two results establish the existence of crystallization fronts forming stripes with small oblique angle to the interface . In particular, crystallization fronts select wavenumbers transverse to the interface, depending on prescribed wavenumbers parallel to the interface. Using that , one can equivalently parameterize strain in the crystalline phase as a function of grain orientation, that is, growth selects strain but not orientation in this case of near-parallel orientation; see Remark 2.8.
We expect non-degeneracy as in Definition 1.1 to hold generically for parallel stripe formation. In particular, we expect Theorem 1 to apply to parallel stripe formation at finite amplitude, not necessarily small, or in other systems exhibiting striped phases, as described above. Theorem 2, on the other hand, is intrinsically focused on small amplitudes. It seems difficult to obtain existence results of this type at finite amplitude. We expect, however, that the methods from Theorem 2 could be adapted to find solutions to (1.4), at small amplitude. We chose the alternative approach from Theorem 1 in order to illustrate the robust continuation from parallel to oblique stripes, independent of small amplitude assumptions and, to some extent, specific model problems.
The proof of Theorem 2 pairs the somewhat classical techniques of center manifold reduction and normal forms (see ) with heteroclinic matching techniques such as invariant foliations and Melnikov theory in an infinite dimensional setting. In particular, we perform a center manifold reduction in the spatial dynamical systems for and , separately. The parallel striped front is then constructed as a heteroclinic orbit from the intersection of the unstable manifold of a periodic orbit in the -center manifold with the stable manifold of the origin in the -center manifold. Since locally the two center manifolds only intersect at the origin, we construct the heteroclinic by finding intersections of the center-unstable manifold of the periodic orbit in the dynamics with the stable manifold of the origin in the dynamics. We use invariant foliations to reduce the infinite-dimensional nature of the problem, allowing us to project the dynamics onto one of the center manifolds and obtain a leading order intersection. We then use Melnikov theory and transversality arguments to obtain the desired heteroclinic. Transversality of the intersection implies non-degeneracy as stated in Definition 1.1.
Traditionally, the two main difficulties in proving a result like Theorem 1 have been addressed using spatial dynamics. To address the neutral continuous spectrum induced by the asymptotic roll state, one typically uses spatial dynamics in the -direction. That is one formulates the equation as an (ill-posed) dynamical system with evolutionary variable and studies roll states as periodic orbits and fronts as heteroclinic orbits. Essential spectrum corresponds to the lack of hyperbolicity of periodic orbits, and is resolved by focusing on strong stable foliations.
To address the singular limit , one might use spatial dynamics in the direction along the growth interface, formulating the equation as a fast-slow dynamical system in the -direction [33, 29]. One would then try to study bifurcations using a center-manifold reduction or the geometric methods pioneered by Fenichel .
Combining these two difficulties seems beyond the scope of spatial dynamics techniques, and we therefore resort to a more direct functional-analytic approach. To overcome the singularly perturbed nature of the problem, we use an approach similar to , preconditioning the problem with a constant-coefficient linear operator before applying the Implicit Function Theorem. To resolve the difficulties caused by the continuous spectrum, we use an ansatz of the form
to decompose the far-field patterns at from the “core” patterns near the interface at . Here, is a smooth, monotone function with for and for . Having accounted in this way for changes in the farfield wavenumber, we may require the correction to be exponentially localized. We then solve in exponentially localized spaces, where the linearization turns out to be Fredholm of index -1, using the free parameter as a variable to account for the cokernel. As a result, we obtain and as functions of the remaining free parameter using the Implicit Function Theorem after careful preconditioning; see Section 2. Regularity in is obtained via a bootstrapping procedure; see Section 2.5.
Research partially supported by the National Science Foundation through the grants NSF-DMS-1603416 (RG), and NSF-DMS-1612441, NSF-DMS-1311740 (AS), as well as a UMN Doctoral Dissertation Fellowship (RG). RG would like to thank C. E. Wayne for useful discussions about this work, as well as the Institute for Mathematics and its Applications for its kind hospitality during a weeklong visit where some of this research was performed.
2 From parallel to oblique stripes — proof of Theorem 1
We prove Theorem 1. Section 2.1 collects and reinterprets information on the primary profile and the linearization. We then describe the functional-analytic setup, in particular the farfield-core decomposition, Section 2.2. Section 2.3 introduces the second key ingredient to the proof, a nonlinear preconditioning to set up the Implicit Function Theorem. Section 2.4 concludes the existence proof and Section 2.5 establishes differentiability in .
2.1 Properties of the parallel trigger and its linearization
We establish smoothness, exponential convergence, and some properties of the linearization. First, notice that solves a pseudo-elliptic equation, such that and belong to , with a jump at . For , can readily seen to be smooth. By assumption, converges to the periodic pattern as .
By translation invariance in , is bounded and belongs to the kernel of the linearized equation in , for all , but not for , since .
Lemma 2.1 (Fredholm crossing).
The operator from (1.5) is Fredholm of index -1 in for , sufficiently small, with trivial kernel.
Proof. We rely on the characterization of Fredholm indices using Fredholm borders; see [32, 34, 8]. Since the asymptotic state at is linearly stable, its Morse index at can be calculated through a homotopy as follows. We linearize at the asymptotic periodic pattern and find, including a spectral homotopy parameter ,
A Floquet-Bloch ansatz , with , yields the periodic boundary value problem
We are interested in spatial eigenvalues , crossing the imaginary axis, that is, . In this case, is Hermitian, such that for homotopies we necessarily find and . Restricting to the fundamental Floquet domain , we further conclude . This however is impossible for by the assumption of coperiodic stability, and it implies that up to scalar multiples for , . Inspecting multiplicity of this spatial Floquet multiplier , one is looking for a generalized eigenfunction solving for , where denotes the derivative with respect to . By the quadratic dependency on , this reduces to , which in turn is impossible since is self-adjoint. This proves that there is precisely one zero Floquet exponent for and no Floquet exponent crossings for , which implies that is Fredholm of index 0 for , small, and Fredholm of index -1 for , small. Since the eigenfunction at is not decaying as , the kernel is trivial for .
There are , such that
as , where is as in (1.7).
Proof. We can rely on spatial dynamics; see for isntance . The asymptotic periodic orbit is hyperbolic up to the neutral Floquet exponent generated by translations, as seen in Lemma 2.1. Its local stable manifold is therefore given by the union of strong stable fibers, thus implying exponential convergence. Since the spatial dynamics can be formulated in spaces of arbitrary regularity, convergence is exponential in spaces with higher derivatives, too.
2.2 Setup and farfield-core decomposition
We start setting up our fixed point argument by performing a functional analytic farfield-core decomposition. We start from (1.4),
with the appropriate boundary conditions stated there. Recall that is a smooth cut-off function with for all and support contained in . Our ansatz is of the form
where . We consider perturbations defined in (1.6), with sufficiently small. Note that we introduced parameter dependence into the trigger front by simply scaling appropriately with , such that solves for all .
we find, suppressing dependence,
From this last line, we can then define the mapping
with the -independent residual term
and the nonlinear term
Since , we have
Exponential convergence of the primary trigger from Lemma 2.2 implies that for , is a locally well-defined nonlinear mapping from the anisotropic Sobolev space to . Indeed, one obtains using the exponential convergence of to and the fact that Then, since can readily be seen to be a Banach algebra and is bounded, we have that as well.
The existence assumption of implies that Also, we note that the linearization of in satisfies
which is closed and densely defined on . We also record for later use that
It is readily observed that is not continuous in as a map from to as it is not well-defined for due to higher-order terms such as and . We will therefore regularize the equation by preconditioning with a Fourier multiplier in the next section. We conclude this setup by collecting some properties of the linearization.
For all small, let span with . Algebraic simplicity of the eigenvalue 0 in , , as in Definition 1.1 then implies that
Proof. We argue as in [22, Lemma 6.3]. Namely, if one assumes that the above inner product is zero, then there exists and non-zero such that
We then have
Since is exponentially localized, and , we have and is a generalized eigenvector of the kernel element when considered in , contradicting the algebraic simplicity.
We also define the -adjoint of for later use,
2.3 Regularization of the nonlinear mapping
To prove Theorem 1, we shall study zeros of the regularized nonlinear mapping
with the regularizing operator The existence and boundedness of such an inverse can be obtained in by studying the associated Fourier multiplier,
noticing that the real part of the denominator has real part less than -1 for all . One can then extend existence to the exponentially weighted space by the use of conjugating isomorphisms . More explicitly one obtains the inverse by first obtaining the inverse on the one-sided weighted spaces , where and acts as , and then use the fact that with equivalent norm,
This inverse can also be found to have the continuity properties listed in the following proposition. For ease of notation in the following we let , , and . Also, since they are in fact functions of , we re-define and .
For , and speed , the mappings and are well-defined, bounded, and norm-continuous on .
Proof. Since we are considering as a mapping from to itself, it suffices to prove the result on . Furthermore we prove the result for as the result for can be obtained as discussed above. Hence, we consider .
We start by considering continuity of . Therefore let . For , continuity is easily established using smoothness properties of the symbol. We therefore focus on the case . We need to show
To estimate , we first define
where in the last line we used the fact that . Then
can be bounded from below by up to a constant independent of . It then readily follows that the above quotient is bounded uniformly in so that
For we further decompose
For we scale and define . By studying the real and imaginary parts of the product , one finds for and sufficiently large
with constant , independent of . Note that here we have used the fact that and are both non-zero.
On the complement , one bounds the denominator from below by 1 so that
A similar argument is used to bound , this time with the scaling . Indeed, for we have
Combining the bounds for and we conclude that the term converges to zero as as desired.
We next turn to the derivative, . First exists in a neighborhood of and has the expected Fourier multiplier
where is defined above. This is obtained by observing
as for any near 0 and near . This convergence is obtained using similar estimates as in (2.10).
To prove continuity in , we proceed in the same way as above, aiming to show
and thus once again decompose
It is readily found that