arXiv subjects

Ryan Goh

Publications and source records attributed to Ryan Goh.

At least 19 recordsLinked to original sources

arXiv2026

Patterned fronts in the wake of a parameter ramp in the complex Ginzburg Landau equation

We study of the formation of pattern-forming fronts in the presence of a rigidly-propagating parameter ramp which is slowly-varying in space. In the context of the prototypical supercritical complex Ginzburg-Landau equation, we show that not only the leading order front interface, but also the selected spatial wave number is governed by the transition of the ramp between absolute and convective instability. The slow ramp then induces a further delay of the front interface and perturbation of the selected wave number, controlled by the slow passage near a complex fold of strong- and weak-stable eigenspaces. To analyze the behavior near this fold, we perform a multiple scales analysis to predict the higher-order front interface delay in terms of zeros and poles of a complex Airy quotient inner solution. We confirm these predictions with numerical continuation of heteroclinics in the associated traveling wave equation. We also numerically characterize their spectral stability, finding accumulation of eigenvalues consistent with previous results on slow absolute spectrum. We then show the leading-order absolute/convective instability heuristic accurately describes selected wave numbers in an analogous slowly-ramped Swift-Hohenberg equation.

nlin.PS↗

arXiv2025

Exponential Dichotomies for Elliptic Equations on Multidimensional Domains

The existence of exponential dichotomies has been well-established as a powerful tool to study existence, stability, and bifurcations of coherent structures. Currently, the application of exponential dichotomies to elliptic problems posed on multi-dimensional domains is predominately limited to the context of cylindrical spatial domains. Recent work by Beck et. al. (2021) has shown how to extend the method of spatial dynamics, in which one views a spatial variable as a time-like evolutionary variable, to general multi-dimensional spatial domains. In this paper, we show that exponential dichotomies exist for a class of spatial dynamical systems arising in this more general setting, thus allowing for their use in future analyses of coherent structures.

math.AP↗

arXiv2025

Coherent dynamics in soft-threshold integrate-and-fire networks

We study bifurcations in networks of integrate-and-fire neurons with stochastic spike emission, focusing on the effects of the spatial and temporal structure of the synaptic interactions. Using a deterministic mean-field approximation of the population dynamics, we characterize spatial, temporal, and spatiotemporal patterns of macroscopic activity. In the mean-field theory, synaptic delays give rise to uniform oscillations across the population through a subcritical Hopf bifurcation of the stationary uniform equilibrium. With local excitation and long-range inhibition the network undergoes a Turing bifurcation, resulting in a localized area of sustained activity, or stationary bump. When the coupling has both delays, local inhibition, and long range excitation, the network undergoes a Turing-Hopf bifurcation leading to spatiotemporal dynamics, such as standing and traveling waves. When multiple instabilities are excited, we observe other complex spatiotemporal dynamics. We confirm all these predictions of the mean-field theory in simulations of the underlying stochastic model.

q-bio.NC↗

arXiv2025

Fronts and patterns with a dynamic parameter ramp

We examine the effect of a slowly-varying time-dependent parameter on invasion fronts for which an unstable homogeneous equilibrium is invaded by either another homogeneous state or a spatially periodic state. We first explain and motivate our approach by studying asymptotically constant invasion fronts in a scalar FKPP equation with time-dependent parameter which controls the stability of the trivial state. Following recent works in the area, we use a linearized analysis to derive formal predictions for front position and leading-edge spatial decay. We then use a comparison principle approach to establish a rigorous spreading result in the case of an unbounded temporal parameter. We then consider patterned-invasion in the complex Ginzburg-Landau equation with dynamic bifurcation parameter, a prototype for slow passage through a spatio-temporal Hopf instability. Linearized analysis once again gives front position and decay asymptotics, but also the selected spatial wavenumber at the leading edge. We then use a Burger's modulation analysis to predict the slowly-varying wavenumber in the wake of the front. Finally, in both equations, we used the recently developed concept of a space-time memory curve to characterize delayed invasion in the case where the parameter is initially stable before a subsequent slow passage through instability and invasion. We also provide preliminary results studying invasion in other prototypical pattern formation models modified with a dynamic parameter, as well as numerical results for delayed transition between pushed and pulled fronts in Nagumo's equation with dynamic parameter.

nlin.PS↗

arXiv2025

Numerical Continuation and Bifurcation in Nonlinear PDEs: Stability, invasion and wavetrains in the Swift-Hohenberg equation

We discuss some aspects of numerical continuation and bifurcation for partial differential equations, specifically pattern formation and coherent structures. For the sake of clarity we focus on wavetrains, stability and associated invasion processes in the paradigmatic cubic Swift-Hohenberg equation (SHE). We do not aim at a review of numerical continuation for PDE or pattern formation in SHE in any generality, rather our goal is to provide an entry point for interested students and colleagues to the application of continuation methods. We provide access to numerical implementations and hope that our presentation provides an introductory guideline that can also be used for teaching.

nlin.PS↗

arXiv2024

Efficient numerical computation of spiral spectra with exponentially-weighted preconditioners

The stability of nonlinear waves on spatially extended domains is commonly probed by computing the spectrum of the linearization of the underlying PDE about the wave profile. It is known that convective transport, whether driven by the nonlinear pattern itself or an underlying fluid flow, can cause exponential growth of the resolvent of the linearization as a function of the domain length. In particular, sparse eigenvalue algorithms may result in inaccurate and spurious spectra in the convective regime. In this work, we focus on spiral waves, which arise in many natural processes and which exhibit convective transport. We prove that exponential weights can serve as effective, inexpensive preconditioners that result in resolvents that are uniformly bounded in the domain size and that stabilize numerical spectral computations. We also show that the optimal exponential rates can be computed reliably from a simpler asymptotic problem posed in one space dimension.

math.NA↗

arXiv2024

Pitchfork bifurcation along a slow parameter ramp: coherent structures in the critical scaling

We investigate the slow passage through a pitchfork bifurcation in a spatially extended system, when the onset of instability is slowly varying in space. We focus here on the critical parameter scaling, when the instability locus propagates with speed $c\sim \varepsilon^{1/3}$, where $\varepsilon$ is a small parameter that measures the gradient of the parameter ramp. Our results establish how the instability is mediated by a front traveling with the speed of the parameter ramp, and demonstrate scalings for a delay or advance of the instability relative to the bifurcation locus depending on the sign of $c$, that is on the direction of propagation of the parameter ramp through the pitchfork bifurcation. The results also include a generalization of the classical Hastings-McLeod solution of the Painlev\'e-II equation to Painlev\'e-II equations with a drift term.

math.DS↗

arXiv2023

Transverse modulational dynamics of quenched patterns

We study the modulational dynamics of striped patterns formed in the wake of a planar directional quench. Such quenches, which move across a medium and nucleate pattern-forming instabilities in their wake, have been shown in numerous applications to control and select the wavenumber and orientation of striped phases. In the context of the prototypical complex Ginzburg-Landau and Swift-Hohenberg equations, we use a multiple-scale analysis to derive a one-dimensional viscous Burgers' equation which describes the long-wavelength modulational and defect dynamics in the direction transverse to the quenching motion, that is along the quenching line. We show that the wavenumber selecting properties of the quench determines the nonlinear flux parameter in the Burgers' modulation equation, while the viscosity parameter of the Burgers' equation is naturally determined by the transverse diffusivity of the pure stripe state. We use this approximation to accurately characterize the transverse dynamics of several types of defects formed in the wake, including grain boundaries and phase-slips.

nlin.PS↗

arXiv2023

Growing patterns

Pattern forming systems allow for a wealth of states, where wavelengths and orientation of patterns varies and defects disrupt patches of monocrystalline regions. Growth of patterns has long been recognized as a strong selection mechanism. We present here recent and new results on the selection of patterns in situations where the pattern-forming region expands in time. The wealth of phenomena is roughly organized in bifurcation diagrams that depict wavenumbers of selected crystalline states as functions of growth rates. We show how a broad set of mathematical and numerical tools can help shed light into the complexity of this selection process.

nlin.PS↗

arXiv2023

Oblique and checkerboard patterns in the quenched Cahn-Hilliard model

We consider transversely modulated fronts in a directionally quenched Cahn-Hilliard equation, posed on a two-dimensional infinite channel, with both parameter and source-term type heterogeneities. Such quenching heterogeneities travel through the domain, excite instabilities, and can select the pattern formed in their wake. We in particular study striped patterns which are oblique to the quenching direction and checkerboard type patterns. Under generic spectral assumptions, these patterns arise via an $O(2)$-Hopf bifurcation as the quenching speed is varied, with symmetries arising from translations and reflections in the transverse variable. We employ an abstract functional analytic approach to establish such patterns near the bifurcation point. Exponential weights are used to address neutral continuous spectrum, and a co-domain restriction is used to address neutral mass-flux. We also give a method to determine the direction of bifurcation of fronts. We then give an explicit example for which our hypotheses are satisfied and for which bifurcating fronts can be investigated numerically.

math.DS↗

arXiv2022

Fronts in the wake of a parameter ramp: slow passage through pitchfork and fold bifurcations

This work studies front formation in the Allen-Cahn equation with a parameter heterogeneity which slowly varies in space. In particular, we consider a heterogeneity which mediates the local stability of the zero state and subsequent pitchfork bifurcation to a non-trivial state. For slowly-varying ramps which are either rigidly propagating in time or stationary, we rigorously establish existence and stability of positive, monotone fronts and give leading order expansions for their interface location. For non-zero ramp speeds, and sufficiently small ramp slopes, the front location is determined by the local transition between convective and absolute instability of the base state and leads to an O(1) delay beyond the instantaneous pitchfork location before the system jumps to a nontrivial state. The slow ramp induces a further delay of the interface controlled by a slow-passage through a fold of strong- and weak-stable eigenspaces of the associated linearization. We introduce projective coordinates to de-singularize the dynamics near the trivial state and track relevant invariant manifolds all the way to the fold point. We then use geometric singular perturbation theory and blow-up techniques to locate the desired intersection of invariant manifolds. For stationary ramps, the front is governed by the slow passage through the instantaneous pitchfork bifurcation with inner expansion given by the unique Hastings-McLeod connecting solution of Painlev\'{e}'s second equation. We once again use geometric singular perturbation theory and blow-up to track invariant manifolds into a neighborhood of the non-hyperbolic point where the ramp passes through zero and to locate intersections.

math.DS↗

arXiv2021

Strain and defects in oblique stripe growth

We study stripe formation in two-dimensional systems under directional quenching in a phase-diffusion approximation including non-adiabatic boundary effects. We find stripe formation through simple traveling waves for all angles relative to the quenching line using an analytic continuation procedure. We also present comprehensive analytical asymptotic formulas in limiting cases of small and large angles as well as small and large quenching rates. Of particular interest is a regime of small angle and slow quenching rate which is well described by the glide motion of a boundary dislocation along the quenching line. A delocalization bifurcation of this dislocation leads to a sharp decrease of strain created in the growth process at small angles. We complement our results with numerical continuation reliant on a boundary-integral formulation. We also compare results in the phase-diffusion approximation numerically to quenched stripe formation in an anisotropic Swift Hohenberg equation.

math.AP↗

arXiv2020

Asymptotic approximation of a modified compressible Navier-Stokes system

We study the long time asymptotics of a modified compressible Navier-Stokes system (mcNS) inspired by the previous work of Hoff and Zumbrun. We introduce a new decomposition of the momentum field into its irrotational and incompressible parts, and a new method for approximating solutions of the heat equation in terms of Hermite functions in which $n^{th}$ order approximations can be computed for solutions with $n^{th}$ order moments. We then obtain existence of solutions to the mcNS system and show that the approximation in terms of Hermite functions gives the leading order terms in the long-time asymptotics, and under certain assumptions can be evaluated explicitly.

math.AP↗

arXiv2020

Delayed Hopf bifurcation and space-time buffer curves in the Complex Ginzburg-Landau equation

In this article, the phenomenon of delayed Hopf bifurcations (DHB) in reaction-diffusion PDEs is analyzed in the cubic Complex Ginzburg-Landau equation with a slowly-varying parameter. We use the classical asymptotic methods of stationary phase and steepest descents to show that solutions which approach the attracting quasi-steady state (QSS) before the Hopf bifurcation remain near that state for long times after the Hopf bifurcation and the QSS has become repelling. In the complex time plane, the phase function of the linear PDE has a saddle point, and the Stokes and anti-Stokes lines are central to the asymptotics. The nonlinear terms are treated by applying an iterative method to the mild form of the PDE given by perturbations about the linear particular solution. This tracks the closeness of solutions near the attracting and repelling QSS. Next, we show that beyond a key Stokes line through the saddle there is a space-time buffer curve along which the particular solution of the linear PDE ceases to be exponentially small, causing the solution of the nonlinear PDE to diverge from the repelling QSS and exhibit large-amplitude oscillations. The homogeneous solution also stops being exponentially small in a spatially dependent manner, as determined also by the initial time. We find four different cases of DHB, depending on the competition between the homogeneous and particular solutions, and we quantify how these depend on system parameters. Examples are presented for each case, with uni-modal, spatially-periodic, smooth step, and algebraically-growing source terms. Also, rich spatio-temporal dynamics are observed in the post-DHB oscillations. Finally, it is shown that large-amplitude source terms can be designed so that solutions spend substantially longer times near the repelling QSS, and hence region-specific control over the delayed onset of oscillations can be achieved.

math.DS↗

arXiv2020

Spectral stability of pattern-forming fronts in the complex Ginzburg-Landau equation with a quenching mechanism

We consider pattern-forming fronts in the complex Ginzburg-Landau equation with a traveling spatial heterogeneity which destabilizes, or quenches, the trivial ground state while progressing through the domain. We consider the regime where the heterogeneity propagates with speed $c$ just below the linear invasion speed of the pattern-forming front in the associated homogeneous system. In this situation, the front locks to the interface of the heterogeneity leaving a long intermediate state lying near the unstable ground state, possibly allowing for growth of perturbations. This manifests itself in the spectrum of the linearization about the front through the accumulation of eigenvalues onto the absolute spectrum associated with the unstable ground state. As the quench speed $c$ increases towards the linear invasion speed, the absolute spectrum stabilizes with the same rate at which eigenvalues accumulate onto it allowing us to rigorously establish spectrally stability of the front in $L^2(\mathbb{R})$. The presence of unstable absolute spectrum poses a technical challenge as spatial eigenvalues along the intermediate state no longer admit a hyperbolic splitting and standard tools such as exponential dichotomies are unavailable. Instead, we projectivize the linear flow, and use Riemann surface unfolding in combination with a superposition principle to study the evolution of subspaces as solutions to the associated matrix Riccati differential equation on the Grassmannian manifold. Eigenvalues can then be identified as the roots of the meromorphic Riccati-Evans function, and can be located using winding number and parity arguments.

nlin.PS↗

arXiv2018

Vortices in stably-stratified rapidly rotating Boussinesq convection

We study the Boussinesq approximation for rapidly rotating stably-stratified fluids in a three dimensional infinite layer with either stress-free or periodic boundary conditions in the vertical direction. For initial conditions satisfying a certain quasi-geostrophic smallness condition, we use dispersive estimates and the large rotation limit to prove global-in-time existence of solutions. We then use self-similar variable techniques to show that the barotropic vorticity converges to an Oseen vortex, while other components decay to zero. We finally use algebraically weighted spaces to determine leading order asymptotics. In particular we show that the barotropic vorticity approaches the Oseen vortex with algebraic rate while the barotropic vertical velocity and thermal fluctuations go to zero as Gaussians whose amplitudes oscillate in opposite phase of each other while decaying with an algebraic rate.

math.AP↗

arXiv2017

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.

nlin.PS↗

arXiv2016

Universal wavenumber selection laws in apical growth

We study pattern-forming dissipative systems in growing domains. We characterize classes of boundary conditions that allow for defect-free growth and derive universal scaling laws for the wavenumber in the bulk of the domain. Scalings are based on a description of striped patterns in semi-bounded domains via strain-displacement relations. We compare predictions with direct simulations in the Swift-Hohenberg, the Complex Ginzburg-Landau, the Cahn-Hilliard, and reaction-diffusion equations.

nlin.PS↗