Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift-Hohenberg Equation

David J.B. Lloyd

REVIEW 3 major objections 6 minor 62 references

Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift-Hohenberg Equation

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Invading stripe fronts in the Swift-Hohenberg equation select a unique speed and wavenumber, while retreating fronts form one-parameter families.

desk verdict A serious, publishable numerical study of depinning fronts with genuinely new results; the numerics are careful and the theory is honestly conditional on an unverified transversality assumption. read the letter →

arxiv 1908.08274 v1 pith:42AIDV7X submitted 2019-08-22 nlin.PS math.DS

David J.B. Lloyd This is my paper · ORCID

classification nlin.PSmath.DS MSC 35B3635B3235C0765P30

keywords Swift-Hohenbergequationdepinningfrontshomoclinicsnakingpatternselectiontravellingwavesnumericalcontinuationfar-fieldcoredecompositionstripepatterns

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper addresses the motion of stripe patterns into a quiescent state in the two-dimensional Swift-Hohenberg equation, focusing on the parameter region just outside the homoclinic snaking window. Its central claim, Proposition 1.1, is that an invading parallel front selects a unique invasion speed $c$ and far-field wavenumber $k_x$, whereas a retreating parallel front comes in a one-parameter family with speed $c=c(k_x)$. To make this principle usable, the paper builds a boundary-value problem from a far-field core decomposition and traces the fronts by numerical continuation. The numerics find that almost-planar invasion fronts in the cubic-quintic equation bifurcate off parallel fronts near the snaking region and do not exist far away, and that parallel fronts may regain transverse stability at high speed. If the claims are right, front propagation outside the snaking window is a structured pattern-selection phenomenon that can be mapped systematically.

What carries the argument

The load-bearing object is the far-field core decomposition, which writes a front as the product of a far-field stripe $u_s(k_x(\rho\pm\tau+\psi);k_x)$ with a cut-off $\chi(\rho)$ plus a remainder $v(\rho,\tau;\omega)$ that is localised near the interface. Posed in exponentially weighted spaces $L^2_\alpha$, this makes the linearised operator Fredholm so that Newton and continuation methods apply, with two integral phase conditions fixing the translational mode and the far-field wavenumber; the unknowns solved for are the remainder $v$, the wavenumber $k_x$, and the transition frequency $\omega=c k_x$. The proof of Proposition 1.1 uses a Lyapunov-Schmidt reduction in this setup, and the stationary selection theory enters through the conserved quantities $H(u)$ and $S(u)$ that force a stationary front's far-field stripes to sit on the zero-Hamiltonian set.

What would settle it

Compute, at a numerically converged invasion front for representative parameters in the bistable region, the eigenvalues and eigenfunctions of the linearised operator $L$ in the exponentially weighted space $L^2_{-\alpha}$; an extra near-zero eigenvalue, or a zero eigenvalue of algebraic multiplicity greater than one, would refute Hypothesis 3 and with it the uniqueness part of Proposition 1.1. A second, independent test is to run two time simulations starting from fronts with different initial stripe wavenumbers or phases and check whether both converge to the same $c$ and $k_x$.

Watch

Extended reading notes

Core claim

The paper's main result is a pattern-selection dichotomy for parallel depinning fronts. Assuming the far-field stripe is temporally stable and the front is transverse, an invading front survives small perturbations and selects a unique invasion speed $c$ and far-field wavenumber $k_x$ for fixed $\mu$ and $\nu$; a retreating front instead forms a one-parameter family parameterised by $k_x$, with $c=c(k_x)$. The numerical companion result, obtained by solving the far-field core decomposition boundary-value problem, is that almost-planar invasion fronts in the cubic-quintic Swift-Hohenberg equation (nonlinearity $f(u)=\nu u^3-u^5$) bifurcate from parallel invasion fronts and coexist close to the homoclinic snaking region, while far away no almost-planar fronts exist and parallel fronts can regain transverse stability once they propagate above a critical speed.

Load-bearing premise

The result assumes a transversality condition (Hypothesis 3): the kernel of the linearisation about the front in $L^2_{-\alpha}$ is exactly two-dimensional, spanned by spatial and temporal translations, and the zero eigenvalue is algebraically simple; the paper states this without proving it for the Swift-Hohenberg equation.

Editorial extensions

If this is right

Invading parallel depinning fronts can be path-followed in $(c,k_x,\mu,\nu)$-space, so folds and stability changes are located by continuation rather than inferred from time simulations.
The selected wavenumber begins at the Hamiltonian-selected value at the snaking edge, dips, then increases; this non-monotonic dip is a quantitative target for exponential-asymptotic theories of pinning.
Almost-planar invasion fronts exist only close to the snaking region, where they coexist with parallel fronts and travel slower; far away, parallel fronts are the only invasion mode and can become transversely stable at high speed.
Patch growth on the plane is governed by the parallel and perpendicular front speeds: the parallel front fixes the stepping rate in one direction while the perpendicular front sets the other, matching the observed bulging of growing stripe patches.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the pattern-selection dichotomy holds beyond the Swift-Hohenberg equation, then bistable pattern-forming systems generally have a computable invasion speed and wavelength, so patch morphology in reaction-diffusion or convection models could be predicted from front data alone.
Hypothesis 3 could be certified numerically by solving the adjoint eigenvalue problem at the computed fronts; a numerical proof of a simple algebraically isolated zero eigenvalue would upgrade Proposition 1.1 from assumed to verified for the computed branches.
The wavenumber dip near the snaking edge is consistent with exponentially small splitting between stable and unstable manifolds, suggesting that matched-asymptotics methods could predict the location and depth of the dip as a function of $\mu$ and $\nu$.
The same boundary-value construction should apply to fronts whose far field is a different cellular pattern, such as hexagons, and to systems with nonlocal or discrete spatial structures, where the essential-spectrum obstruction is analogous.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a serious numerical study of depinning fronts in the two-dimensional Swift–Hohenberg equation, and the main results hold up. I would send it to referees without hesitation.

What is new: the far-field core decomposition is extended from stationary grain boundaries to moving invasion fronts, which yields a BVP that can be continued for the front speed and the far-field wavenumber. That solves a problem the paper correctly identifies as open — nobody had been able to do numerical bifurcation analysis of these depinning fronts. The two headline observations, the dip in the selected wavenumber away from the snaking region and the bifurcation from parallel to almost-planar invasion fronts, are genuinely new and documented carefully. The comparison with the semi-analytical |δ|^(1/2) scaling is honest, and the weakly nonlinear comparison acts as a legitimate external benchmark, not a fitted result. The lineage from the author's own grain-boundary work is clear and acknowledged; that is self-reference, not circularity.

The soft spot worth naming: the analytical framework rests on Hypothesis 3 (Appendix A.1), a transversality condition not proved for SH. On the stress-test note you passed along: the charge of internal inconsistency is overstated. The wording of Hypothesis 3 is sloppy, but Lemma 2 clarifies the standard picture — the kernel in the unweighted space is two-dimensional from the two translations, while in the weighted space L^2_α only the localized combination survives, so the point-spectrum eigenvalue is simple. The proof goes through the weighted space, so the apparent contradiction is verbal, not load-bearing. What remains genuinely unverified is the transversality itself; the convergence studies in §6.4 certify discretization error, not the absence of extra neutral modes. That is a real caveat, and an extra neutral mode would break the selection conclusion. But the proposition is labeled conditional and the numerics are presented as numerical results, which is the right framing.

Minor complaints: no code or data shipped, which matters more for a numerical-methods paper; and most bifurcation diagrams carry no error bars, which is common in this community and not a blocker. The author is also candid about the limits of the semi-analytical perturbation theory in §5, which I counted in the paper's favor.

Who this is for: people working on localized patterns, depinning, and numerical continuation of fronts. I would cite it in the next year if I were working on those problems.

Recommendation: accept after revision, conditional on tightening the presentation of Hypothesis 3 and making the numerical details more accessible.

Referee Report

3 major / 6 minor

Summary. The paper develops a numerical continuation framework for depinning (invasion) fronts in the planar Swift-Hohenberg equation in the bistable region outside homoclinic snaking. The main theoretical result is a conditional selection principle (Proposition 1.1): transversely invading parallel fronts are claimed to select a unique speed and far-field wavenumber, whereas retreating fronts are claimed to form one-parameter families. The numerical methodology adapts the far-field core decomposition of Lloyd & Scheel to moving fronts, adding phase conditions for the transition frequency and far-field wavenumber. Extensive computations are reported for parallel, oblique, perpendicular, and almost-planar fronts in the quadratic-cubic and cubic-quintic SH equations, including two-parameter sweeps, comparison to semi-analytical theory near the snaking edge, and application to the invasion of localized stripe patches.

Significance. If the claims are correct, the paper makes a valuable methodological contribution: it provides a concrete, systematic way to compute pattern-selecting invasion fronts in a class of pattern-forming PDEs, going beyond time-stepping and beyond the weakly nonlinear regime. The far-field core decomposition with phase conditions is a natural and useful extension of prior work, and the convergence studies for the 1D parallel-front solver (Figures 12-13) are careful and informative. The comparison with the semi-analytical |δ|^{1/2} scaling law (Figure 15) gives a meaningful benchmark, and the paper is honest about several open issues, including the lack of a proof of the needed transversality. However, the central uniqueness theorem rests on an unproved and, as stated, internally inconsistent transversality hypothesis, and the main numerical examples for the quadratic-cubic equation are computed in a regime that appears to violate the theorem's stability hypothesis. The almost-planar front coexistence, one of the headline observations, is supported only by computations at a single transverse resolution without a published convergence check.

major comments (3)

[Appendix A.1, Hypothesis 3] Hypothesis 3 as stated is internally inconsistent: it says the kernel of L in L^2_{-α} is two-dimensional, spanned by ∂_ρ u* and ∂_τ u*, and then says the eigenvalue λ=0 is algebraically simple. A two-dimensional kernel in that space precludes algebraic simplicity of the zero eigenvalue. The proof of Proposition A.1 actually needs a different statement: in the exponentially weighted space L^2_α the kernel should be one-dimensional (e.g., spanned by an exponentially localized combination of ∂_ρ u* and ∂_τ u*) and the parameter derivatives (∂_{kx}, ∂_ψ, ∂_ω) F_v should cover the remaining cokernel directions of D F_v. The hypothesis as written is therefore not merely unverified but ill-posed, and it is load-bearing for the uniqueness part of Proposition 1.1. The convergence tests in Section 6.4 (Figures 12-13) certify discretization error of a single computed solution; they do not certify the absence of additional neutral modes. If an extra mode, such as a resonant stripe sideband, exists, the Lyapunov-Schmidt reduction of Proposition A.1 fails and the branches computed in Section 7 could be folds rather than genuine selection curves.
[Proposition 1.1 and Section 7.1] The uniqueness/selection theorem assumes temporal stability of the far-field stripe solution, but the main numerical study of parallel fronts in Section 7.1 uses the quadratic-cubic SH equation at ν=1.6, for which the paper itself states (Section 2) that stripes are typically unstable in the bistable region to hexagonal perturbations. No argument is given to show that the hexagon instability is irrelevant for the 1D front problem, nor is the stability hypothesis verified for the computed branches. Consequently, the interpretation of Figure 14 as a confirmation of Proposition 1.1 is not supported by the stated hypotheses of that proposition. The paper needs to either restrict the theoretical claim to stable regimes, prove that the relevant stability notion is only the longitudinal one, or explicitly mark the quadratic-cubic results as being outside the theorem's assumptions.
[Section 7.3, Figure 19] The claim that almost planar invasion fronts bifurcate off parallel fronts and coexist near the snaking region is a headline observation, but it is based on computations with N_γ = N_τ = 16 and no convergence study in the transverse direction. The parallel branch itself is reported to be zig-zag unstable for the whole bistable region (Figure 19, left panel), which makes it unclear whether the observed bifurcation is a genuine solution bifurcation or a numerical artifact from an under-resolved transverse discretization. The paper should provide a mesh-refinement study in γ and τ for the almost-planar branches, and it should report the extrapolated values of ω and k_x with error estimates at the reported fold location, before this claim can be considered quantitatively supported.

minor comments (6)

[Lemma 1 and Proposition A.1] The text says 'invading fronts with ω<0' in Lemma 1 and in the statement of Proposition A.1, but throughout Section 6 and the introduction ω is defined as a positive quantity (ω = c k_x, c>0 for invading fronts). Please clarify the sign convention.
[Section 1, paragraph after Figure 2] The manuscript says there are 'three types of possible depinning stripe fronts: parallel, oblique, perpendicular and almost-planar stripe fronts' but then lists four types; please correct the count or the list.
[Section 6.4, equation (6.9)] The symbol L is used both for the linearized operator and for the domain length L_ρ; also the definition of L in (6.9) uses a minus sign before the biharmonic term while the operator in (6.7a) has a plus sign before it; please harmonize the notation.
[Figure 15] The fitted scaling law is reported as ω = 1.18 |μ-μ_1|^{0.5292} with no error bars or fit-quality measures; given that the exponent is expected to be 0.5, a confidence interval on the fitted exponent should be stated, along with the number of data points used and the residuals.
[Section 7.3, last paragraph] The text states that 'IVP suggests stability for smaller ky values' but no supporting IVP results or stability computation is shown; please provide evidence or rephrase as a conjecture.
[References] Several references are listed as 'Submitted 2018' or 'Submitted 2019'; if any have since appeared, please update them; otherwise state the status explicitly in the bibliography.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the selection theorem is explicitly conditional on a transversality hypothesis, and the numerics are self-contained rather than fitted to the claimed predictions.

full rationale

The paper's central claim, Proposition 1.1, is a conditional statement: it assumes temporal stability of the far-field stripes, existence of a transverse invasion front, and the explicit Hypothesis 3 in Appendix A.1 (kernel of the linearization spanned by the two translation modes, with the relevant zero eigenvalue algebraically simple). The Lyapunov-Schmidt reduction in Proposition A.1 is then a standard parameter-counting argument: L is Fredholm of index -1 for invading fronts (Lemma 1), the free parameters (kx, psi, omega) plus the two translation modes balance the cokernel, and the algebraic simplicity assumption is used to solve for omega. This is not circular because the hypothesis is a spectral nondegeneracy condition, not the selected values c and kx themselves. The numerical method of Section 6.4 solves for (v, omega, kx) as genuine unknowns with phase conditions; no fitted value is inserted as a predicted output. The least-squares fit in Figure 15 is a diagnostic comparison with the semi-analytic |delta|^{1/2} law, not a parameter used in the selection computation. The author's prior work [37] supplies the far-field core decomposition and continuation methodology, but that decomposition is a representation of any front (u = us chi + v) and the numerical results are new, with independent checks against weakly nonlinear travelling-front formulas from [26,50] and against initial-value simulations in Section 6.1. The unproved and potentially internally inconsistent statement of Hypothesis 3 is a genuine correctness/rigor risk for reading Proposition 1.1 as unconditional, but it is not circular reasoning: the paper states it as an assumption, not as a consequence of its derivation.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central theorem is conditional on three explicitly stated hypotheses, of which Hypothesis 3 is the most fragile. No physical free parameter is fitted to obtain the selection result. The only fitted numbers are a diagnostic power-law fit in Figure 15. Numerical-domain parameters are validated by convergence studies.

free parameters (2)

prefactor and exponent in the omega-scaling fit of Figure 15 = 1.18 and 0.5292
Least-squares fit used only to compare with the semi-analytic |delta|^{1/2} prediction; not a parameter of the pattern-selection theorem.
domain truncation L_rho, cut-off d, grid sizes N_rho, N_tau, N_gamma = L_rho=40 pi, d=20 to 40, N_rho=400, N_tau=20, N_gamma=16
Chosen by hand for the boundary value problem; convergence and insensitivity are reported in Figures 12-13, so they do not become fitted physical quantities.

assumptions (5)

domain assumption Hypothesis 1: for an open region of kx the periodic stripe orbits are stable and the critical Floquet eigenvalue has expansion lambda = d_parallel eta^2 + O(eta^4).
Invoked in Appendix A.1 to derive the Fredholm crossing for the linearization; only weakly nonlinear analysis and numerical evidence support it.
domain assumption Hypothesis 2: the trivial state is linearly stable, so mu > 0.
Restricts the proof to the bistable region; the numerical continuation later extends below mu = 0 using convective stability, which is outside the stated theorem.
domain assumption Hypothesis 3: for a front u*, the kernel of L in L^2_{-alpha} is two-dimensional, spanned by d_rho u* and d_tau u*, and the zero eigenvalue is algebraically simple.
This transversality condition is the main unproved premise of Proposition 1.1; no verification for the Swift-Hohenberg equation is supplied.
domain assumption Exponential convergence of the front to its asymptotic stripe and trivial states with a well-defined asymptotic phase.
Used in Lemma 2 of Appendix A.1 to identify the localized eigenfunction and to justify the far-field core decomposition on a finite domain.
standard math Fredholm bordering and Lyapunov-Schmidt reduction for Fredholm index -1 problems.
Basis for proving persistence and uniqueness in Proposition 1.1; used in Appendix A.1 following Kielhoefer.

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Cite this review

Pith. "Pith review of Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift-Hohenberg Equation." pith.science (2026). https://pith.science/paper/42AIDV7X

@misc{pith2026190808274,
  author       = {Pith},
  title        = {Pith review of: Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift-Hohenberg Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/42AIDV7X}},
  note         = {Machine review of arXiv:1908.08274}
}

read the original abstract

In this paper, we carry out numerical bifurcation analysis of depinning of fronts near the homoclinic snaking region, involving a spatial stripe cellular pattern embedded in a quiescent state, in the two-dimensional Swift-Hohenberg equation with either a quadratic-cubic or cubic-quintic nonlinearity. We focus on depinning fronts involving stripes that are orientated either parallel, oblique and perpendicular to the front interface, and almost planar depinning fronts. We show that invading parallel depinning fronts select both a far-field wavenumber and a propagation wavespeed whereas retreating parallel depinning fronts come in families where the wavespeed is a function of the far-field wavenumber. Employing a far-field core decomposition, we propose a boundary value problem for the invading depinning fronts which we numerically solve and use path-following routines to trace out bifurcation diagrams. We then carry out a thorough numerical investigation of the parallel, oblique, perpendicular stripe, and almost planar invasion fronts. We find that almost planar invasion fronts in the cubic-quintic Swift-Hohenberg equation bifurcate off parallel invasion fronts and co-exist close to the homoclinic snaking region. Sufficiently far from the 1D homoclinic snaking region, no almost planar invasion fronts exist and we find that parallel invasion stripe fronts may regain transverse stability if they propagate above a critical speed. Finally, we show that depinning fronts shed light on the time simulations of fully localised patches of stripes on the plane. The numerical algorithms detailed have wider application to general modulated fronts and reaction-diffusion systems.

Figures

Figures reproduced from arXiv: 1908.08274 by the authors.

**Figure 1.** (a) Bifurcation diagram for 1D stationary localised and stripe patterns bifurcating from the trivial state for the quadratic-cubic SH equation with ν = 1.6. The homoclinic snaking region occurs between 0.184 < µ < 0.211 where stationary localised pulses are found shown in (b). In panel (b) we also plot how the depinning of the localised patterns occurs either side of the snaking region with a fast-slow propagation s… view at source ↗

**Figure 2.** Four types of stripe fronts we focus on in this paper with stripes (a) parallel to the front interface (these fronts select a non-constant front speed and far-field wavenumber) (b) oblique to the front interface with constant front speed (these fronts select a constant propagation speed and far-field wavenumber selected but angle free) (c) perpendicular to the front interface (these fronts select a constant front sp… view at source ↗

**Figure 3.** (a) Snaking of almost planar fronts shown inset 1 for the cubic-quintic SH equation with ν = 1.25. The snaking occurs between 0.256 < µ < 0.268 with two other folds at µ = 0.2656 and µ = 0.2665. (b) Stationary stable localised patches worm µ = 0.66, ν = 2 in the cubic-quintic SH equation. . affects the depinning fronts of almost planar localised structures has not been explored. Two types of bifurcations of depinnin… view at source ↗

Figures from the paper (19 more)

**Figure 4.** Figure 4: Time simulations of the cubic-quintic SH equation with ν = 2 starting from a localised stripe pattern with a small random perturbation. Panel (a) µ = 0.45 shows de-stabilisation to an almost planar invasion front while panel (b) µ = 0.35 shows an initial de-stabilisati…

**Figure 5.** Figure 5: (a) max(λ)-eigenvalue of the periodic orbit as the Floquet multiplier σ is varied for (µ, ν) = (0, 1.25). All other eigenvalues are strictly negative for all σ. For kx = 1.2 the stripe is Eckhaus stable and unstable for kx > 1.2365. (b) Stripe existence and 2D stabilit…

**Figure 6.** Figure 6: Hamiltonian selected wavenumber for (a) 1D stripes in the quadratic-cubic SH equation ν = 1.6, (b) 1D stripes in the cubic-quintic SH equation ν = 1.25. The co-periodic stable branches are depicted as solid lines while co-periodic unstable branches are shown as dashed …

**Figure 7.** Figure 7: (a) Bifurcation diagram for a stationary pulse of (4.2) with f2 = f5 = 0, b = 1. The bifurcation diagram asymptotes at the Maxwell point (b) plot of Re(A) near the Maxwell point. The modulated fronts that we seek correspond to front solutions A(X, T) = A(X − cT˜ ) =: A…

**Figure 8.** Figure 8: Bifurcation diagrams for traveling wave solution CGL (4.4) for the quadratic-cubic and cubicquintic SH equation with b = 1. (a) shows the square of the amplitude of the travelling wave (b) the selected wavespeed and (c) the selected far-field wavenumber. We see that f…

**Figure 9.** Figure 9: Bifurcation diagram for localised patterns in the perpendicular amplitude equation (4.8) with ν˜ = 1 (a) stationary pulse that bifurcates from the trivial state (b) travelling fronts in the frame Z = X − cT˜ . In panel (b) we also plot in dashed gold the linear spreadi…

**Figure 10.** Figure 10: Semi-analytical prediction of the transition time pre-factor (a) quadratic-cubic (b) cubic-quintic SH equation. Gold line denotes trivial state invading while blue line denotes patterned state invading. The integrals αi vary strongly for localised pulses taken at the …

**Figure 11.** Figure 11: A plot of the remainder function v and the computational domain for the parallel invasion fronts in (ρ, τ ) − space. For almost planar fronts, we solve the boundary value problem ω(uρ − uτ ) − (1 + k 2 x∂ 2 ρ + ky∂ 2 γ ) 2u − µu + f(u) =0, ρ ∈ [−Lρ, Lρ], τ ∈ (0, 2π], …

**Figure 12.** Figure 12: Absolute error from a high resolution solution at (µ, ν) = (0, 1.6), (a) & (b) Nτ = 20, Lρ = 40π, d = 40, (c) & (d) Nρ = 400, Lρ = 40π, d = 40, (e) & (f ) Nτ = 20, Nρ = 400, Lρ = 40π 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 -12 -24 -10 -22 d d 100 120 140 160 180…

**Figure 13.** Figure 13: Absolute error (a) & (b) Nτ = 20, Nρ = 400, Lρ = 40π, (c) & (d) Nτ = 20, Nρ = 800, d = 20. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]

**Figure 14.** Figure 14: (a) Bifurcation diagram showing the pattern selection wave number, kx, for the parallel invasion fronts in the quadratic-cubic SH equation with ν = 1.6. The edge of the snaking region occurs around µ = 0.181 with a Hamiltonian selected wavenumber kx ∼ 0.9905. The gold…

**Figure 15.** Figure 15: A plot of the log of the transition frequency for invasion fronts to the right of the snaking region ν = 1.6. Yellow line are least square fit. The semi-analytical prediction is ω = 1.29|µ − µ| 0.5 . the data compares reasonably well with the semi-analytical predictio…

**Figure 16.** Figure 16: Two-parameter bifurcation diagrams for parallel invasion fronts in the Cubic-Quintic SH equation (a) kx and (b) ω. The snaking region is shown as a shaded grey area and the location of the minimum selected kx wavenumber is plotted as a red curve in (a). We conclude ou…

**Figure 17.** Figure 17: Perpendicular stripe fronts in the cubic-quintic SH equation with ν = 1.25, (a) existence for various ky’s (b) two parameter bifurcation diagram depicting the region of existence of invading perpendicular fronts. The bifurcation point where the saddle node and c = 0 c…

**Figure 18.** Figure 18: Oblique stripe invasion fronts in the cubic-quintic SH equation with ν = 1.25 with (ky > 0) and parallel (ky = 0) stripe fronts (a) the selected wavenumber, k, (b) wavespeed, cx, selection in the x-direction, and (c) a zoom-in near the Maxwell point. ky = 0.5 fixed. H…

**Figure 19.** Figure 19: Bifurcation diagrams of the almost planar invasion fronts ν = 1.25, ky = 0.5, (Nγ = Nt = 16, Nρ = 400, Lρ = 40π) and the parallel invasion fronts. Stability is respect to perturbations with ky = 0.5 but IVP suggests stability for smaller ky values. Panel (a) shows the…

**Figure 20.** Figure 20: Two parameter bifurcation diagram for cubic-quintic SH equation. The red curve denotes the bifurcation from the parallel invasion front while the gold curve denotes the fold of the almost planar fronts. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_20.png]

**Figure 21.** Figure 21: Worm patch invasion in the cubic-quintic SH equation with (µ, ν) = (0.01, 1.25). (a) at t = 0, (b) t = 25, (c) t = 50 (d) the interface locations in the x- and y-directions at the midpoints given by dx and dy, respectively. The upper dashed black line has a fitted lin…

**Figure 22.** Figure 22: Worm patch invasion for the cubic-quintic SH equation with (µ, ν) = (0.24, 1.25). (a) t = 500, (b) t = 1000, (c) the interface locations in the x- and y-directions at the mid-points given by dx and dy, respectively. The upper dashed black line has a fitted line of 0.5…

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