Proving periodic solutions and branches in the 2D Swift Hohenberg PDE with hexagonal and triangular symmetry

Abstract: In this article, we enforce space group symmetries in Fourier series to rigorously prove the existence of smooth, periodic solutions in partial differential equations (PDEs) with hexagonal and triangular symmetries. In particular, we provide the necessary analytical and numerical tools to construct Fourier series of functions on the hexagonal lattice. This allows one to build approximate solutions that are periodic. Moreover, to generate the periodic tiling, we can use one symmetric hexagon for $D_6$ symmetry and two symmetric triangles for $D_3$ symmetry. We derive a Newton-Kantorovich approach based on the construction of an approximate inverse around an approximate solution, $\overline{u}$. More specifically, we verify a condition based on the computation of explicit bounds. The strategy for constructing $\overline{u}$, the approximate inverse, and the computation of these bounds will be presented. We demonstrate our approach on the 2D Swift-Hohenberg PDE by proving the existence of $D_3$ and $D_6$ periodic solutions. We then perform proofs of branches of solutions by using Chebyshev series. The algorithmic details to perform the proof can be found on Github.

• The paper develops a computer-assisted proof framework that combines hexagonal-lattice Fourier symmetry reduction, Newton–Kantorovich validation, and radii-polynomial bounds to prove periodic Swift–Hohenberg solutions.
• The authors rigorously validate eight isolated solutions and five solution branches with triangular ($D_3$) or hexagonal ($D_6$) symmetry, including branches that pass through saddle-node folds and solutions with negative $bc$.
• The results show that $D_3$ patterns tile using two distinct equilateral triangles, whereas $D_6$ patterns repeat on a single hexagonal domain, while the Julia-based implementation enables future validated studies of pattern-forming PDEs.

This paper develops a computer-assisted proof (CAP) methodology for establishing the existence and local uniqueness of periodic steady states of the two-dimensional Swift–Hohenberg PDE

$$0 = (I_d + \nabla^2)^2 u + \mu u - \gamma u^2 + u^3$$

that carry $D_3$ (triangular) or $D_6$ (hexagonal) dihedral symmetry. The work extends the symmetry-reduction framework of van den Berg and Williams for space-group-symmetric Fourier series to the hexagonal lattice, implements the required sequence structures in Julia as an add-on to RadiiPolynomial.jl, and applies a Newton–Kantorovich validation argument both at isolated parameter values and along entire solution branches parameterized by Chebyshev series in pseudo-arclength.

Symmetric Fourier series on the hexagonal lattice

The foundational device is the correspondence between function-space symmetry and coefficient-space constraints. Writing Fourier series with respect to the hexagonal change of basis $$\mathcal{L} = \begin{bmatrix} 1 & -1/2 \ 0 & \sqrt{3}/2 \end{bmatrix}$$, each group element $g \in D_j$ acts on indices via $$\beta_g(n) = \mathcal{L}^{-1}\mathcal{A}\mathcal{L}n$$ with phase factor $\alpha_g(n)$; the function satisfies $u(x) = u(gx)$ if and only if $\alpha_g(n)u_{\beta_g(n)} = u_n$. This yields three index sets: the fundamental domain $\mathcal{Z}_{\mathrm{dom}}$ (one representative per orbit), the trivial set $D_3$0 (indices forced to zero), and their intersection $D_3$1, which indexes all nonzero independent coefficients. Working only on $D_3$2 enforces the symmetry numerically by construction, so any validated solution inherits $D_3$3-symmetry from the contraction argument — no a posteriori symmetry verification is needed.

A notable structural contribution is the characterization of the periodic tiling generated by such solutions. For $D_3$4, the period parallelogram of side $D_3$5 splits into two equilateral triangles $D_3$6 meeting at $D_3$7 angles; the paper proves that $D_3$8 is $D_3$9-symmetric about each triangle's centroid, but crucially $D_6$0: the two triangles generate distinct sublattices because $D_6$1 lacks the element relating them. For $D_6$2, by contrast, a single hexagon $D_6$3 (built from one shifted parallelogram plus its rotations by $D_6$4 and $D_6$5) is itself a domain of periodicity on which all $D_6$6 operations hold, and one can show $D_6$7.

Newton–Kantorovich validation

The analysis takes place in the weighted Banach algebra $D_6$8 over $D_6$9, with weights $$\mathcal{L} = \begin{bmatrix} 1 & -1/2 \ 0 & \sqrt{3}/2 \end{bmatrix}$$0. Given an approximate solution $$\mathcal{L} = \begin{bmatrix} 1 & -1/2 \ 0 & \sqrt{3}/2 \end{bmatrix}$$1 obtained from random initial data refined by Newton iteration, the approximate inverse $$\mathcal{L} = \begin{bmatrix} 1 & -1/2 \ 0 & \sqrt{3}/2 \end{bmatrix}$$2 uses the finite-dimensional Jacobian $$\mathcal{L} = \begin{bmatrix} 1 & -1/2 \ 0 & \sqrt{3}/2 \end{bmatrix}$$3 on $$\mathcal{L} = \begin{bmatrix} 1 & -1/2 \ 0 & \sqrt{3}/2 \end{bmatrix}$$4 and the linear operator $$\mathcal{L} = \begin{bmatrix} 1 & -1/2 \ 0 & \sqrt{3}/2 \end{bmatrix}$$5 on the tail; $$\mathcal{L} = \begin{bmatrix} 1 & -1/2 \ 0 & \sqrt{3}/2 \end{bmatrix}$$6 matches this structure with the numerically inverted finite block. The radii-polynomial theorem then requires explicit bounds $$\mathcal{L} = \begin{bmatrix} 1 & -1/2 \ 0 & \sqrt{3}/2 \end{bmatrix}$$7 satisfying a quadratic exclusion condition. The tail bound reduces to $$\mathcal{L} = \begin{bmatrix} 1 & -1/2 \ 0 & \sqrt{3}/2 \end{bmatrix}$$8 where $$\mathcal{L} = \begin{bmatrix} 1 & -1/2 \ 0 & \sqrt{3}/2 \end{bmatrix}$$9; notably, since the method targets periodic solutions, invertibility of the linear part is not required, and solutions are proven for negative $g \in D_j$0 (e.g., $g \in D_j$1), which is impossible for localized-pattern approaches that rely on linear invertibility.

Two implementation issues deserve emphasis. First, the orbits of indices in $g \in D_j$2 for $g \in D_j$3 and $g \in D_j$4 only cover all lattice points up to order roughly $g \in D_j$5, so convolutions performed by unfolding behave as though the sequence has full size $g \in D_j$6 despite many zero entries — inflating runtime relative to square-lattice symmetries, though memory savings persist. Second, several bounds ($g \in D_j$7, and the uniform estimate for $g \in D_j$8 via the auxiliary sequence $g \in D_j$9) are deliberately non-sharp to reduce computational cost, a trade-off the authors acknowledge explicitly.

Proven solutions

Eight isolated solutions are validated, spanning both symmetries and both signs of $$\beta_g(n) = \mathcal{L}^{-1}\mathcal{A}\mathcal{L}n$$0. Representative parameters:

Solution	Symmetry	$$\beta_g(n) = \mathcal{L}^{-1}\mathcal{A}\mathcal{L}n$$1	$$\beta_g(n) = \mathcal{L}^{-1}\mathcal{A}\mathcal{L}n$$2	$$\beta_g(n) = \mathcal{L}^{-1}\mathcal{A}\mathcal{L}n$$3	$$\beta_g(n) = \mathcal{L}^{-1}\mathcal{A}\mathcal{L}n$$4
Triangular 1	$$\beta_g(n) = \mathcal{L}^{-1}\mathcal{A}\mathcal{L}n$$5	0.01	1.6	$$\beta_g(n) = \mathcal{L}^{-1}\mathcal{A}\mathcal{L}n$$6	$$\beta_g(n) = \mathcal{L}^{-1}\mathcal{A}\mathcal{L}n$$7
Triangular 4	$$\beta_g(n) = \mathcal{L}^{-1}\mathcal{A}\mathcal{L}n$$8	−0.2	2.0	$$\beta_g(n) = \mathcal{L}^{-1}\mathcal{A}\mathcal{L}n$$9	$\alpha_g(n)$0
Hexagonal 1	$\alpha_g(n)$1	−0.01	1.6	$\alpha_g(n)$2	$\alpha_g(n)$3
Hexagonal 4	$\alpha_g(n)$4	0.25	2.0	$\alpha_g(n)$5	$\alpha_g(n)$6

Each theorem asserts existence and uniqueness of a true solution within radius $\alpha_g(n)$7 of $\alpha_g(n)$8 in $\alpha_g(n)$9, together with the tiling statement (two triangles for $u(x) = u(gx)$0, one hexagon for $u(x) = u(gx)$1). The validated balls are small — between $u(x) = u(gx)$2 and $u(x) = u(gx)$3 — reflecting the large $u(x) = u(gx)$4 constants (up to $u(x) = u(gx)$5), which stem from the deliberately coarse estimates rather than intrinsic ill-conditioning.

Branches via Chebyshev continuation

The second half extends validation to entire branches using pseudo-arclength continuation, which permits passage through saddle-node bifurcations. Both $u(x) = u(gx)$6 and $u(x) = u(gx)$7 are expanded in Chebyshev series on $u(x) = u(gx)$8 in the arclength variable, living in the Banach space $u(x) = u(gx)$9; the augmented system couples the PDE residual to the orthogonality condition $\alpha_g(n)u_{\beta_g(n)} = u_n$0. A uniform-in-$\alpha_g(n)u_{\beta_g(n)} = u_n$1 contraction theorem yields a unique $\alpha_g(n)u_{\beta_g(n)} = u_n$2 branch $\alpha_g(n)u_{\beta_g(n)} = u_n$3 within a tube of radius $\alpha_g(n)u_{\beta_g(n)} = u_n$4 around the numerical branch. Bounds are computed via Chebyshev FFTs, exploiting polynomial degree bookkeeping: products of degree-$\alpha_g(n)u_{\beta_g(n)} = u_n$5 polynomials require transforms at $\alpha_g(n)u_{\beta_g(n)} = u_n$6, $\alpha_g(n)u_{\beta_g(n)} = u_n$7, or $\alpha_g(n)u_{\beta_g(n)} = u_n$8 coefficients. The tail norm admits the clean uniform bound $\alpha_g(n)u_{\beta_g(n)} = u_n$9.

Five branches are proven: two $\mathcal{Z}_{\mathrm{dom}}$0 branches (e.g., $\mathcal{Z}_{\mathrm{dom}}$1, $\mathcal{Z}_{\mathrm{dom}}$2, $\mathcal{Z}_{\mathrm{dom}}$3; and $\mathcal{Z}_{\mathrm{dom}}$4, $\mathcal{Z}_{\mathrm{dom}}$5, $\mathcal{Z}_{\mathrm{dom}}$6) and three $\mathcal{Z}_{\mathrm{dom}}$7 branches, including one traversing a fold ($\mathcal{Z}_{\mathrm{dom}}$8 ranging through approximately $\mathcal{Z}_{\mathrm{dom}}$9) and another covering $D_3$00 with $D_3$01. Because the contraction holds uniformly in $D_3$02, every solution on each validated segment carries the asserted symmetry and tiling structure.

Limitations and open questions

Several caveats bear directly on the strength of the results. The approximate solutions are found by repeated random restarts of Newton's method, a procedure with no convergence guarantee; success depends on locating basins of attraction empirically. The $D_3$03 and $D_3$04 bounds are intentionally conservative, and the authors note that sharper estimates would likely enlarge the validated tubes. The zero-padded structure of $D_3$05/$D_3$06 sequence structures makes convolutions significantly more expensive than for square-lattice symmetries, limiting applicability to computationally intensive problems. Finally, the method addresses periodic patterns only; extending it to localized hexagonal patterns would require combining hexagonal-lattice symmetry reduction with the center-manifold techniques used for radially symmetric and non-radial localized proofs, since the parallelogram domain is not itself $D_3$07- or $D_3$08-invariant when extended by zero. Whether a rigorous branch of hexagonally periodic solutions converging to a $D_3$09-localized pattern can be constructed remains open.

Conclusion

The paper supplies the analytical infrastructure — symmetric Fourier series on the hexagonal lattice, tiling theorems for triangles and hexagons, and rigorously evaluated radii-polynomial bounds — needed to validate $D_3$10- and $D_3$11-symmetric periodic solutions of the planar Swift–Hohenberg equation without restrictions on the sign of $D_3$12. Its practical contributions include the dihedral.jl package extending RadiiPolynomial.jl beyond $D_3$13/$D_3$14 symmetry, and validated existence proofs for eight isolated solutions and five entire solution branches, each carrying its symmetry by construction. The approach positions hexagonal-lattice symmetry reduction as a viable tool for future computer-assisted studies of pattern-forming PDEs beyond the square-lattice setting.