Integral Asymptotics, Coalescing Saddles, and Multiple-scales Analysis of a Generalised Swift-Hohenberg Equation

TL;DR

This work analyzes spatially localized instabilities in a heterogeneous generalised Swift–Hohenberg equation using integral asymptotics. The outer problem is solved by WKBJ methods, while the inner region at turning points is resolved via coalescing saddles, with a Chester–Friedman–Ursell transformation to map to Airy functions; a detailed contour-integral framework underpins the analysis. An alternative multiple-scales approach is developed and shown to reproduce the leading inner behavior, suggesting the coalescing-saddles method can sometimes be circumvented. Numerical contour-integral evaluations validate the asymptotic results and confirm robust matching across turning points. The study highlights the role of integral asymptotics in fourth-order pattern-forming problems and proposes a broader conjecture that multiple-scales analysis may replace coalescing saddles in certain classes of integral problems, inviting further investigation.

Abstract

Integral asymptotics play an important role in the analysis of differential equations and in a variety of other settings. In this work, we apply an integral asymptotics approach to study spatially localized solutions of a heterogeneous generalised Swift-Hohenberg equation. The outer solution is obtained via WKBJ asymptotics, while the inner solution requires the method of coalescing saddles. We modify the classic method of Chester et al. to account for additional technicalities, such as complex branch selection and local transformation to a cubic polynomial. By integrating our results, we construct an approximate global solution to the generalised Swift-Hohenberg problem and validate it against numerical contour integral solutions. We also demonstrate an alternative approach that circumvents the complexity of integral asymptotics by analyzing the original differential equation directly through a multiple-scales analysis and show that this generates the same leading-order inner solution obtained using the coalescing saddles method at least for one of the cases considered via integral asymptotics. Our findings reinforce the significance of integral asymptotics in approximating the fourth order differential equations found in the linear stability analysis for generalisations of the Swift Hohenberg equations. This study has also highlighted a conjecture that, in certain cases, the method of coalescing saddles can be systematically replaced by multiple-scales analysis using an intermediary differential equation, a hypothesis for future investigation.

Figures (7)

• Figure 1: In plot (a) the steepest descent contour passing through $s_{+}$ is the solid curve for $\varepsilon^* z=0.2$, $\alpha = 3$, $\beta=1$. The dashed lines indicate the revealed asymptotes and tangent at the saddle. In particular, we have $s_{+}= 1.395$, the tangent angle of the SDC at $s_{+}$ is $\pi/4$ and the asymptotes are at the angles $\pi/10$ and $17\pi/10$. In plot (b) the real part of $\psi$ on the steepest descent contour is given.
• Figure 2: In the plot (a) the steepest descent contour for $\varepsilon^* z=0.2$, $\alpha = 3$, $\beta=1$ passing through $s_{-}$. The dashed lines indicate (i) the asymptote for large $s_r$ and (ii) the tangent at the saddle. In particular, we have $s_{-}= 1.026$, the tangent angle of the SDC at $s_{-}$ is $-\pi/4$ and the asymptote for large $s_r$ has angle $17\pi/10$, with another asymptote at angle $\pi/2.$ In plot (b) the real part of $\psi$ along the steepest descent curve is given.
• Figure 3: In plot (a) the steepest descent contour passing through $s_{+}$ is given by the solid curve for $\varepsilon^* z=-0.2$, $\alpha = 3$, $\beta=1$ . The dashed lines indicate the revealed asymptote for large $s_r$. In particular, we have $s_{+}= 1.231+ \mathrm{i} 0.128$, the asymptote for large $s_r$ has angle $\pi/10$ with the other asymptote at angle $\pi/2$ and the tangent angle of the steepest descent contour at $s_{+}$ given by $-0.023 \pi$, which is indistinguishable from the horizontal axis at this resolution. In plot (b) the real value of $\psi$ is given for the steepest descent contours passing through $s_+$, with the two curves corresponding to two different vertical axes.
• Figure 4: In the plot (a) the steepest descent contour for $\varepsilon^* z=-0.2$, $\alpha = 3$, $\beta=1$ passing through $s_{-}$. Note that the vertical axis is through the saddle point rather than the origin and that the dashed lines indicate the revealed asymptotes and tangent at the saddle. In particular, we have $s_{-}= 1.231 - \mathrm{i} ~ 0.128 = \bar{s}_{+}$, the tangent angle of the SDC at $s_{-}$ is $(1/2-0.023) \pi$, which is indistinguishable from the vertical axis at this resolution and the asymptotes are at the angles $\pi/10$ and $17\pi/10$. Also, observe that this steepest descent contour proceeds to the saddle at $s_+=\bar{s}_-$, depicted by the dot in Fig. \ref{['fig.SDC+neg']}(a). In Fig. \ref{['fig.SDC-neg']}, there is a discontinuity in the contour tangent at this point, and the steepest descent contours from plots Fig. \ref{['fig.SDC+neg']}(a), Fig. \ref{['fig.SDC-neg']}(a) subsequently coincide, due to the need to select an appropriate asymptote as $|s|\rightarrow \infty.$ Finally note that the contribution of the contour integral local to the saddle $s_+$ is subordinate as $\Re(\psi(s_+))=-\Re(\psi(s_-))<0<\Re(\psi(s_-)).$ In plot (b) the real part of $\psi$ along the steepest descent contour is given.
• Figure 5: Contours generating Airy functions. Contour $\gamma_{0}$ corresponds to $\mathrm{Ai}$, while the remaining two generate a linear combination of $\mathrm{Ai}$, $\mathrm{Bi}$.