arXiv subjects

Gabriela Jaramillo

Publications and source records attributed to Gabriela Jaramillo.

17 recordsLinked to original sources

arXiv2026

A Mountain-Pass Algorithm for Nonlocal Problems with Super-quadratic Nonlinearities

In this paper we consider a nonlinear equation $-\mathcal{L} u(x) = f(x, u(x))$ with a super-quadratic nonlinearity, $f$, and a nonlocal operator, $\mathcal{L}$, generated by a special class of radially symmetric $L^1$ convolution kernels with finite second moments. The assumptions on this operator are mild and allow for a variety of kernels used in biological and physical applications, including kernels with algebraic decay and sign changing kernels. Using the strong nonlinearities present in the equation, we prove the existence of nontrivial solutions using the classical Mountain Pass Theorem, a central result in minimax theory that equates solutions of our equation to critical points of a corresponding energy functional. This existence result holds with both homogeneous nonlocal Dirichlet and nonlocal Neumann boundary conditions. We supplement these theoretical results with numerical simulations for various nonlinearities with odd maximal degree in the unknown $u$. The numerical scheme exploits the resulting energy landscape which allows one to adapt a gradient descent algorithm.

math.AP↗

arXiv2025

The role of boundary constraints in simulating a nonlocal Gray-Scott model

We present a second-order algorithm for approximating solutions to nonlocal diffusive processes in reaction-diffusion equations. The numerical scheme relies on a quadrature method for the spatial discretization and a second-order Adams-Bashford method for the time marching. This algorithm is then used to simulate a nonlocal Gray-Scott model, known for generating interesting structures including periodic patterns, traveling waves, pulse and multi-pulse solutions. Our main goal is to study the impact of boundary constraints on the formation of stationary pulse solutions. We consider nonlocal Dirichlet and Neumann boundary constraints, as well as what we refer to as `free' boundary conditions. In addition, we investigate the effects of using different convolution kernels, fat- or thin-tailed, on the formation of these localized solutions. Our numerical results show that when the spread of the kernel is large, i.e. when the model is nonlocal, both the type of kernel and the type of boundary constraint used have a strong impact on the solution's profile.

math.NA↗

arXiv2025

Fredholm properties of singular elliptic operators arising in the study of point defects

Motivated by the dynamics of defects in planar pattern-forming systems, we study Fredholm properties of elliptic operators with singular coefficients in weighted Sobolev spaces. In particular, we consider a family of doubly weighted spaces that encode algebraic decay/growth of functions at infinity, and near the origin. Our results give conditions on the weights under which the operators are either injective, surjective, or isomorphisms. We also give a precise description of the kernel and range of these operators.

math.AP↗

arXiv2024

Existence of Weak Solutions for a Nonlocal Klausmeier Model

We establish the existence of weak solutions for a nonlocal Klausmeier model within a small time interval $[0, T)$. The Klausmeier model is a coupled, nonlinear system of partial differential equations governing plant biomass and water dynamics in semiarid regions. The original model posits that plants disperse their seed according to classical diffusion. Instead, we opt for a nonlocal diffusive operator in alignment with ecological field data that validates long-range dispersive behaviors of plants and seeds. The equations, defined on a finite interval in $\mathbb{R}$, feature homogeneous Dirichlet boundary conditions for the water equation and nonlocal Dirichlet volume constraints for the plant biomass equation. The nonlocal operator involves convolution with a symmetric and spatially extended convolution kernel possessing mild integrability and regularity properties. We employ the Galerkin method to establish the existence of weak solutions. The key challenge comes from the nonlocal operator; we define it on a subspace of $L^{2}$ instead of $H^{1}$, precluding the use of Aubin's compactness theorem to prove the weak convergence of nonlinear terms. To overcome this, we modify the model and introduce two new equations for the spatial derivatives of plant biomass and water. This procedure allows us to recover enough regularity to establish compactness and complete the proof.

math.AP↗

arXiv2024

Analysis and Simulations of a Nonlocal Gray-Scott Model

The Gray-Scott model is a set of reaction-diffusion equations that describes chemical systems far from equilibrium. Interest in this model stems from its ability to generate spatio-temporal structures, including pulses, spots, stripes, and self-replicating patterns. We consider an extension of this model in which the spread of the different chemicals is assumed to be nonlocal, and can thus be represented by an integral operator. In particular, we focus on the case of strictly positive, symmetric, $L^1$ convolution kernels that have a finite second moment. Modeling the equations on a finite interval, we prove the existence of small-time weak solutions in the case of nonlocal Dirichlet and Neumann boundary constraints. We then use this result to develop a finite element numerical scheme that helps us explore the effects of nonlocal diffusion on the formation of pulse solutions.

math.NA↗

arXiv2024

Existence of spiral waves in oscillatory media with nonlocal coupling

We prove existence of spiral waves in oscillatory media with nonlocal coupling. Our starting point is a nonlocal complex Ginzburg-Landau (cGL) equation, rigorously derived as an amplitude equation for integro-differential equations undergoing a Hopf bifurcation. Because this reduced equation includes higher order terms that are usually ignored in a formal derivation of the cGL, the solutions we find also correspond to solutions of the original nonlocal system. To prove existence of these patterns we use perturbation methods together with the implicit function theorem. Within appropriate parameter regions, we find that spiral wave patterns have wavenumbers, $κ$, with expansion $κ\sim C e^{-a/\varepsilon}$, where $a$ is a positive constant, $\varepsilon$ is the small bifurcation parameter, and the positive constant $C$ depends on the strength and spread of the nonlocal coupling. The main difficulty we face comes from the linear operators appearing in our system of equations. Due to the symmetries present in the system, and because the equations are posed on the plane, these maps have a zero eigenvalue embedded in their essential spectrum. Therefore, they are not invertible when defined between standard Sobolev spaces and a straightforward application of the implicit function theorem is not possible. We surpass this difficulty by redefining the domain of these operators using doubly weighted Sobolev spaces. These spaces encode algebraic decay/growth properties of functions, near the origin and in the far field, and allow us to recover Fredholm properties for these maps.

math.AP↗

arXiv2023

Rotating spirals in oscillatory media with nonlocal interactions and their normal form

Biological and physical systems that can be classified as oscillatory media give rise to interesting phenomena like target patterns and spiral waves. The existence of these structures has been proven in the case of systems with local diffusive interactions. In this paper the more general case of oscillatory media with nonlocal coupling is considered. We model these systems using evolution equations where the nonlocal interactions are expressed via a diffusive convolution kernel, and prove the existence of rotating wave solutions for these systems. Since the nonlocal nature of the equations precludes the use of standard techniques from spatial dynamics, the method we use relies instead on a combination of a multiple-scales analysis and a construction similar to Lyapunov-Schmidt. This approach then allows us to derive a normal form, or reduced equation, that captures the leading order behavior of these solutions.

math.AP↗

arXiv2023

Can large inhomogeneities generate target patterns?

We study the existence of target patterns in oscillatory media with weak local coupling and in the presence of an impurity, or defect. We model these systems using a viscous eikonal equation posed on the plane, and represent the defect as a perturbation. In contrast to previous results we consider large defects, which we describe using a function with slow algebraic decay, i.e., $g \sim {\mathcal O}(1/|x|^m)$ for $m \in (1,2]$. We prove that these defects are able to generate target patterns and that, just as in the case of strongly localized impurities, their frequency is small beyond all orders of the small parameter describing their strength. Our analysis consists of finding two approximations to target pattern solutions, one which is valid at intermediate scales and a second one which is valid in the far field. This is done using weighted Sobolev spaces, which allow us to recover Fredholm properties of the relevant linear operators, as well as the implicit function theorem, which is then used to prove existence. By matching the intermediate and far field approximations we then determine the frequency of the pattern that is selected by the system.

math.AP↗

arXiv2023

Dynamics of a linearly-perturbed May-Leonard competition model

The May--Leonard model was introduced to examine the behavior of three competing populations where rich dynamics, such as limit cycles and nonperiodic cyclic solutions, arise. In this work, we perturb the system by adding the capability of global mutations, allowing one species to evolve to the other two in a linear manner. We find that for small mutation rates the perturbed system not only retains some of the dynamics seen in the classical model, such as the three-species equal-population equilibrium bifurcating to a limit cycle, but also exhibits new behavior. For instance, we capture curves of fold bifurcations where pairs of equilibria emerge and then coalesce. As a result, we uncover parameter regimes with new types of stable fixed points that are distinct from the single- and dual-population equilibria characteristic of the original model. On the contrary, the linearly-perturbed system fails to maintain heteroclinic connections that exist in the original system. In short, a linear perturbation proves to be significant enough to substantially influence the dynamics, even with small mutation rates.

math.DS↗

arXiv2020

Numerical Methods for a Diffusive Class Nonlocal Operators

In this paper we develop a numerical scheme based on quadratures to approximate solutions of integro-differential equations involving convolution kernels, $ν$, of diffusive type. In particular, we assume $ν$ is symmetric and exponentially decaying at infinity. We consider problems posed in bounded domains and in $\R$. In the case of bounded domains with nonlocal Dirichlet boundary conditions, we show the convergence of the scheme for kernels that have positive tails, but that can take on negative values. When the equations are posed on all of $\R$, we show that our scheme converges for nonnegative kernels. Since nonlocal Neumann boundary conditions lead to an equivalent formulation as in the unbounded case, we show that these last results also apply to the Neumann problem.

math.NA↗

arXiv2020

A Modified Split Bregman Algorithm for Computing Microstructures Through Young Measures

The goal of this paper is to describe the oscillatory microstructure that can emerge from minimizing sequences for nonconvex energies. We consider integral functionals that are defined on real valued (scalar) functions $u(x)$ which are nonconvex in the gradient $\nabla u$ and possibly also in $u$. To characterize the microstructures for these nonconvex energies, we minimize the associated relaxed energy using two novel approaches: i) a semi-analytical method based on control systems theory, ii) and a numerical scheme that combines convex splitting together with a modified version of the split Bregman algorithm. These solutions are then used to gain information about minimizing sequences of the original problem and the spatial distribution of microstructure.

math.OC↗

arXiv2018

Target Patterns in a 2-D Array of Oscillators with Nonlocal Coupling

We analyze the effect of adding a weak, localized, inhomogeneity to a two dimensional array of oscillators with nonlocal coupling. We propose and also justify a model for the phase dynamics in this system. Our model is a generalization of a viscous eikonal equation that is known to describe the phase modulation of traveling waves in reaction-diffusion systems. We show the existence of a branch of target pattern solutions that bifurcates from the spatially homogeneous state when $\varepsilon$, the strength of the inhomogeneity, is nonzero and we also show that these target patterns have an asymptotic wavenumber that is small beyond all orders in $\varepsilon$. The strategy of our proof is to pose a good ansatz for an approximate form of the solution and use the implicit function theorem to prove the existence of a solution in its vicinity. The analysis presents two challenges. First, the linearization about the homogeneous state is a convolution operator of diffusive type and hence not invertible on the usual Sobolev spaces. Second, a regular perturbation expansion in $\varepsilon$ does not provide a good ansatz for applying the implicit function theorem since the nonlinearities play a major role in determining the relevant approximation, which also needs to be "correct" to all orders in $\varepsilon$. We overcome these two points by proving Fredholm properties for the linearization in appropriate Kondratiev spaces and using a refined ansatz for the approximate solution, which obtained using matched asymptotics.

math.AP↗

arXiv2016

The Effect of Impurities on Striped Phases

We study the effect of algebraically localized impurities on striped phases in one space-dimension. We therefore develop a functional-analytic framework which allows us to cast the perturbation problem as a regular Fredholm problem despite the presence of essential spectrum, caused by the soft translational mode. Our results establish the selection of jumps in wavenumber and phase, depending on the location of the impurity and the average wavenumber in the system. We also show that, for select locations, the jump in the wavenumber vanishes.

math.AP↗

arXiv2014

Inhomogeneities in 3 dimensional oscillatory media

We consider localized perturbations to spatially homogeneous oscillations in dimension 3 using the complex Ginzburg-Landau equation as a prototype. In particular, we will focus on heterogeneities that locally change the phase of the oscillations. In the usual translation invariant spaces and at $ \varepsilon=0$ the linearization about these spatially homogeneous solutions result in an operator with zero eigenvalue embedded in the essential spectrum. In contrast, we show that when considered as an operator between Kondratiev spaces, the linearization is a Fredholm operator. These spaces consist of functions with algebraical localization that increases with each derivative. We use this result to construct solutions close to the equilibrium via the Implicit Function Theorem and derive asymptotics for wavenumbers in the far field.

math.AP↗

arXiv2014

Pacemakers in large arrays of oscillators with nonlocal coupling

We model pacemaker effects of an algebraically localized heterogeneity in a 1 dimensional array of oscillators with nonlocal coupling. We assume the oscillators obey simple phase dynamics and that the array is large enough so that it can be approximated by a continuous nonlocal evolution equation. We concentrate on the case of heterogeneities with positive average and show that steady solutions to the nonlocal problem exist. In particular, we show that these heterogeneities act as a wave source, sending out waves in the far field. This effect is not possible in 3 dimensional systems, such as the complex Ginzburg-Landau equation, where the wavenumber of weak sources decays at infinity. To obtain our results we use a series of isomorphisms to relate the nonlocal problem to the viscous eikonal equation. We then use Fredholm properties of the Laplace operator in Kondratiev spaces to obtain solutions to the eikonal equation, and by extension to the nonlocal problem.

math.CA↗

arXiv2013

Deformation of Striped Patterns by Inhomogeneities

We study the effects of adding a local perturbation in a pattern forming system, taking as an example the Ginzburg-Landau equation with a small localized inhomogeneity in two dimensions. Measuring the response through the linearization at a periodic pattern, one finds an unbounded linear operator that is not Fredholm due to continuous spectrum in typical translation invariant or weighted spaces. We show that Kondratiev spaces, which encode algebraic localization that increases with each derivative, provide an effective means to circumvent this difficulty. We establish Fredholm properties in such spaces and use the result to construct deformed periodic patterns using the Implicit Function Theorem. We find a logarithmic phase correction which vanishes for a particular spatial shift only, which we interpret as a phase-selection mechanism through the inhomogeneity.

math.AP↗

arXiv2010

Study of multi black hole and ring singularity apparent horizons

We study critical black hole separations for the formation of a common apparent horizon in systems of $N$ - black holes in a time symmetric configuration. We study in detail the aligned equal mass cases for $N=2,3,4,5$, and relate them to the unequal mass binary black hole case. We then study the apparent horizon of the time symmetric initial geometry of a ring singularity of different radii. The apparent horizon is used as indicative of the location of the event horizon in an effort to predict a critical ring radius that would generate an event horizon of toroidal topology. We found that a good estimate for this ring critical radius is $20/(3π) M$. We briefly discuss the connection of this two cases through a discrete black hole 'necklace' configuration.

gr-qc↗