arXiv subjects

Shuangquan Xie

Publications and source records attributed to Shuangquan Xie.

11 recordsLinked to original sources

arXiv2026

On the location of the maximal gradient of the torsion function over some non-symmetric planar domains

We investigate the location of the maximal gradient of the torsion function on certain non-symmetric planar domains. First, by establishing uniform estimates for convex narrow domains, we show that as a planar domain bounded by two graphs becomes increasingly narrow, the location of the maximal gradient of its torsion function converges to the endpoints of the longest vertical segment, with smaller curvature among them. This result confirms that Saint-Venant's conjecture on the location of fail points holds for asymptotically narrow domains. Second, for triangles, we prove that the maximal gradient of the torsion function always occurs on the longest side, lying between the foot of the altitude and the midpoint of that side. Moreover, via nodal line analysis, we show that, restricted to each side, the critical point of the gradient is unique and non-degenerate. Additionally, by perturbation and barrier arguments, we establish that for a class of nearly equilateral triangles, this critical point is closer to the midpoint than to the foot of the altitude, and the maximal gradient at the midpoint exceeds that at the foot of the altitude. Third, employing the reflection method, we demonstrate that for a non-concentric annulus, the maximal gradient of the torsion function is always attained at the point on the inner boundary closest to the center of the outer boundary.

math.AP↗

arXiv2026

A flow approach to the monotonicity of shape functionals

We develop a geometric flow framework to investigate two classical shape functionals: the torsional rigidity and the first Dirichlet eigenvalue of the Laplacian. First, by constructing novel deformation paths governed by height-stretching flows, leg-stretching flows, and angle-bisector flows, we prove new monotonicity properties for these functionals under deformations of triangles and rhombuses. These results also lead to new and simpler proofs of some known results, without using the Steiner symmetrization argument. Second, we introduce a mean curvature flow approach to the Saint-Venant inequality, providing a new geometric proof for smooth convex domains. We establish a weak monotonicity property along the flow and characterize the equality case, which leads to the discovery of an intriguing new functional whose extremal properties suggest a further conjecture. Third, by discovering a gradient norm inequality for the sides of rectangles, we prove monotonicity and rigidity results of the torsional rigidity on rectangles.

math.AP↗

arXiv2025

Stability and slow dynamics of an interior spiky pattern in a one-dimensional spatial Solow model with capital-induced labor migration

One of the most significant findings in the study of spatial Solow-Swan models is the emergence of economic agglomeration, in which economic activities concentrate in specific regions. Such agglomeration provides a fundamental mechanism driving the spatial patterns of urbanization, labor migration, productivity growth, and resource allocation. In this paper, we consider the one-dimensional spatial Solow-Swan model with capital-induced labor migration, which captures the dynamic interaction between labor and capital through migration and accumulation. Focusing on the regime of sufficiently small capital diffusivity, we first construct an interior spike (spiky economic agglomeration) quasi-equilibrium. Next, we perform the linear stability of the corresponding spike equilibrium by using a hybrid asymptotic and numerical method. We show that a single interior spike remains stable for small reaction-time constants but undergoes a Hopf bifurcation when the constant is sufficiently large, leading to oscillations in spike height (economic fluctuation). Finally, we derive a differential-algebraic system to capture the slow drift motion of quasi-equilibrium (core-periphery shift). Numerical simulations are carried out to support our theoretical studies and reveal some intriguing yet unexplained dynamics.

math.AP↗

arXiv2024

Oscillatory Instabilities of a One-Spot Pattern in the Schnakenberg Reaction-Diffusion System in $3$-D Domains

For an activator-inhibitor reaction-diffusion system in a bounded three-dimensional domain $Ω$ of $O(1)$ volume and small activator diffusivity of $O(\varepsilon^2)$, we employ a hybrid asymptotic-numerical method to investigate two instabilities of a localized one-spot equilibrium that result from Hopf bifurcations: an amplitude instability leading to growing oscillations in spot amplitude, and a translational instability leading to growing oscillations of the location of the spot's center $\mathbf{x}_0 \in Ω$. Here, a one-spot equilibrium is one in which the activator concentration is exponentially small everywhere in $Ω$ except in a localized region of $O(\varepsilon)$ about $\mathbf{x}_0 \in Ω$ where its concentration is $O(1)$. We find that the translation instability is governed by a $3\times 3$ nonlinear matrix eigenvalue problem. The entries of this matrix involve terms calculated from certain Green's functions, which encode information about the domain's geometry. In this nonlinear matrix eigenvalue system, the most unstable eigenvalue determines the oscillation frequency at onset, while the corresponding eigenvector determines the direction of oscillation. We demonstrate the impact of domain geometry and defects on this instability, providing analytic insights into how they select the preferred direction of oscillation. For the amplitude instability, we illustrate the intricate way in which the Hopf bifurcation threshold $τ_H$ varies with a feed-rate parameter $A$. In particular, we show that the $τ_H$ versus $A$ relationship possesses two saddle-nodes, with different branches scaling differently with the small parameter $\varepsilon$. All asymptotic results are confirmed by finite elements solutions of the full reaction-diffusion system.

nlin.PS↗

arXiv2024

Localized patterns in the Gierer Meinhardt model on a cycle graph

In this study, we provide a detailed analysis of the spike solutions and their stability for theGierer-Meinhardt model on discrete lattices. We explore several phenomena that have no analogues in the continuum limit. For example in the discrete case, the system retains spike patterns even when diffusion of the activator is set to zero. In this limit, we derive a simplified algebraic system to determine the presence of a K-spike solution. The stability of this solution is determined by a K by K matrix. We further delve into the scenarios where K = 2 and K = 3, revealing the existence of stable asymmetric spike patterns. Our stability analysis indicates that the symmetric two-spike solution is the most robust. Furthermore, we demonstrate that symmetric K-spike solutions are locally the most stable configurations. Additionally, we explore spike solutions under conditions where the inhibitor's diffusion rate is not significantly large. In doing so, we uncover zigzag and mesa patterns that do not occur in the continuous system. Our findings reveal that the discrete lattices support a greater variety of stable patterns for the Gierer-Meinhardt model.

math.DS↗

arXiv2022

Dynamics of N-spot rings with oscillatory tails in a three-component reaction-diffusion system

In two-dimensional space, we investigate the slow dynamics of multiple localized spots with oscillatory tails in a specific three-component reaction-diffusion system, whose key feature is that the spots attract or repel each other alternatively according to their mutual distances, leading to rather complex patterns. One fundamental pattern is the ring pattern, consisting of $N$ equally distributed spots on a circle with a certain radius. Depending on the parameters of the system, stationary or moving (i.e., traveling and rotating) $N$-spot rings can be observed. In order to understand the emergence of these patterns, we describe the dynamics of $N$ spots by a set of reduced ordinary differential equations (ODEs) encoding the information of each spot's location and velocity. On the basis of the reduced system, we analytically study the existence and stability of stationary and moving $N$-spot ring solutions, similar to the collective motion of self-propelled particles. Numerical simulations are provided to verify our results with fixed parameters.

math.DS↗

arXiv2022

Oscillatory translational instabilities of localized spot patterns in the Schnakenberg reaction-diffusion system on general 2-D domains

For a bounded 2-D planar domain $Ω$, we investigate the impact of domain geometry on oscillatory translational instabilities of $N$-spot equilibrium solutions for a singularly perturbed Schnakenberg reaction-diffusion system with $\mO(\eps^2) \ll \mO(1)$ activator-inhibitor diffusivity ratio. An $N$-spot equilibrium is characterized by an activator concentration that is exponentially small everywhere in $Ω$ except in $N$ well-separated localized regions of $\mO(\eps)$ extent. We use the method of matched asymptotic analysis to analyze Hopf bifurcation thresholds above which the equilibrium becomes unstable to translational perturbations, which result in $\mO(\eps^2)$-frequency oscillations in the locations of the spots. We find that stability to these perturbations is governed by a nonlinear matrix-eigenvalue problem, the eigenvector of which is a $2N$-vector that characterizes the possible modes (directions) of oscillation. The $2N\times 2N$ matrix contains terms associated with a certain Green's function on $Ω$, which encodes geometric effects. For the special case of a perturbed disk with radius in polar coordinates $r = 1 + σf(θ)$ with \red{$0< \varepsilon \ll σ\ll 1$}, $θ\in [0,2π)$, and $f(θ)$ $2π$-periodic, we show that only the mode-$2$ coefficients of the Fourier series of $f$ impact the bifurcation threshold at leading order in $σ$. We further show that when $f(θ) = \cos2θ$, the dominant mode of oscillation is in the direction parallel to the longer axis of the perturbed disk. Numerical investigations on the full Schnakenberg PDE are performed for various domains $Ω$ and $N$-spot equilibria to confirm asymptotic results and also to demonstrate how domain geometry impacts thresholds and dominant oscillation modes.

math.AP↗

arXiv2022

Complex oscillatory motion of multiple spikes in a three-component Schnakenberg system

In this paper, we introduce a three-component Schnakenberg model. Its key feature is that it has a solution consisting of N spikes that undergoes a Hopf bifurcation with respect to N distinct modes nearly simultaneously. This results in complex oscillatory dynamics of the spikes, not seen in typical two-component models. For parameter values above the Hopf bifurcations, we derive reduced equations of motion which consist of coupled ordinary differential equations (ODEs) of order 2N for spike positions and their velocities. These ODEs fully describe the slow-time evolution of the spikes near the Hopf bifurcations. We then apply the method of multiple scales to the resulting ODEs to derive long-time dynamics. For a single spike, we find that its long-time motion consists of oscillations near the steady-state whose amplitude can be computed explicitly. For two spikes, the long-time behaviour can be either in-phase or out-of-phase oscillations. Both in and out of phase orbits are stable, coexist for the same parameter values, and the fate of orbit depends solely on the initial conditions. Further away from the Hopf bifurcation point, we offer numerical experiments indicating the existence of highly complex and chaotic oscillations.

nlin.PS↗

arXiv2017

Vortex precession dynamics in general radially symmetric potential traps in two-dimensional atomic Bose-Einstein condensates

We consider the motion of individual two-dimensional vortices in general radially symmetric potentials in Bose-Einstein condensates. We find that although in the special case of the parabolic trap there is a logarithmic correction in the dependence of the precession frequency $ω$ on the chemical potential $μ$, this is no longer true for a general potential $V(r) \propto r^p$. Our calculations suggest that for $p>2$, the precession frequency scales with $μ$ as $ω\sim μ^{-2/p}$. This theoretical prediction is corroborated by numerical computations, both at the level of spectral (Bogolyubov-de Gennes) stability analysis by identifying the relevant precession mode dependence on $μ$, but also through direct numerical computations of the vortex evolution in the large $μ$, so-called Thomas-Fermi, limit. Additionally, the dependence of the precession frequency on the radius of an initially displaced from the center vortex is examined and the corresponding predictions are tested against numerical results.

cond-mat.quant-gas↗

arXiv2017

Multi-vortex crystal lattices in Bose-Einstein Condensates with a rotating trap

We consider vortex dynamics in the context of Bose-Einstein Condensates (BEC) with a rotating trap, with or without anisotropy. Starting with the Gross-Pitaevskii (GP) partial differential equation (PDE), we derive a novel reduced system of ordinary differential equations (ODEs) that describes stable configurations of multiple co-rotating vortices (vortex crystals). This description is found to be quite accurate quantitatively especially in the case of multiple vortices. In the limit of many vortices, BECs are known to form vortex crystal structures, whereby vortices tend to arrange themselves in a hexagonal-like spatial configuration. Using our asymptotic reduction, we derive the effective vortex crystal density and its radius. We also obtain an asymptotic estimate for the maximum number of vortices as a function of rotation rate. We extend considerations to the anisotropic trap case, confirming that a pair of vortices lying on the long (short) axis is linearly stable (unstable), corroborating the ODE reduction results with full PDE simulations. We then further investigate the many-vortex limit in the case of strong anisotropic potential. In this limit, the vortices tend to align themselves along the long axis, and we compute the effective one-dimensional vortex density, as well as the maximum admissible number of vortices. Detailed numerical simulations of the GP equation are used to confirm our analytical predictions.

cond-mat.quant-gas↗

arXiv2014

Drunken robber, tipsy cop: First passage times, mobile traps, and Hopf bifurcations

For a random walk on a confined one-dimensional domain, we consider mean first passage times (MFPT) in the presence of a mobile trap. The question we address is whether a mobile trap can improve capture times over a stationary trap. We consider two scenarios: a randomly moving trap and an oscillating trap. In both cases, we find that a stationary trap actually performs better (in terms of reducing expected capture time) than a very slowly moving trap; however, a trap moving sufficiently fast performs better than a stationary trap. We explicitly compute the thresholds that separate the two regimes. In addition, we find a surprising relation between the oscillating trap problem and a moving-sink problem that describes reduced dynamics of a single spike in a certain regime of the Gray-Scott model. Namely, the above-mentioned threshold corresponds precisely to a Hopf bifurcation that induces oscillatory motion in the location of the spike. We use this correspondence to prove the uniqueness of the Hopf bifurcation.

cond-mat.stat-mech↗