N. Whitaker

Statistical Equilibrium Computations of Coherent Structures in Turbulent Shear Layers

A numerical method is developed to treat the statistical equilibrium model of coherent structures in two-dimensional turbulence. In this model the vorticity, which fluctuates on a microscopic scale, is described macroscopically by a local probability distribution. A coherent vortex is identified with a most probable macrostate, which maximizes entropy subject to the constraints dictated by the complete family of conserved quantities for incompressible, inviscid flow. Attention is focused on the special case corresponding to vortex patches, and a simple, robust, and efficient algorithm is proposed in this case. The form of the iterative algorithm and its convergence properties are derived from the variational structure of the statistical equilibrium problem. Solution branches are computed for the shear layer configuration, and the results are interpreted in terms of the dynamical phenomena of rollup and coalescence.

Linear and nonlinear parity-time-symmetric oligomers: a dynamical systems analysis

In the present work, we focus on the cases of two-site (dimer) and three-site (trimer) configurations, i.e. oligomers, respecting the parity-time () symmetry, i.e. with a spatially odd gain–loss profile. We examine different types of solutions of such configurations with linear and nonlinear gain/loss profiles. Solutions beyond the linear -symmetry critical point as well as solutions with asymmetric linearization eigenvalues are found in both the nonlinear dimer and trimer. The latter feature is absent in linear -symmetric trimers, while both of them are absent in linear -symmetric dimers. Furthermore, nonlinear gain/loss terms enable the existence of both symmetric and asymmetric solution profiles (and of bifurcations between them), while only symmetric solutions are present in the linear -symmetric dimers and trimers. The linear stability analysis around the obtained solutions is discussed and their dynamical evolution is explored by means of direct numerical simulations. Finally, a brief discussion is also given of recent progress in the context of -symmetric quadrimers.

MAXIMUM-ENTROPY STATES FOR ROTATING VORTEX PATCHES

The statistical equilibrium theory of coherent structures in two‐dimensional turbulence is used to study the coalescence and symmetrization of rotating vortex patches. In this theory, most probable states, which are local probability distributions on the fluctuating vorticity field, are characterized by maximizing entropy subject to constraints derived from the conserved quantities for the Euler equations. The merger of a symmetric pair of patches and the axisymmetrization of an elliptical patch are computed by solving the constrained maximum entropy problem. In this way, these generic phenomena, which are usually simulated by direct time integration, are shown to be equilibration processes for the statistical mechanics model. Moreover, macroscopic features of the filamentation generated in these processes are captured by the statistical equilibrium theory. Namely, the delicate balance between the coalescence into a vortex core and the migration of filamentary vorticity away from the core, which is dictated by the conservation of angular impulse, is exhibited by the equilibrium solutions.

A hybrid model for tumor-induced angiogenesis in the cornea in the presence of inhibitors

The present work formulates and analyzes, by means of numerical experiments, a model for tumor-induced angiogenesis in the presence of inhibitors in the cornea. Our model is a generalization of the earlier work of Tong and Yuan [S. Tong, F. Yuan, Numerical simulations of angiogenesis in the cornea, Microvascular Research 61 (2001) 14-27] to incorporate the role of inhibitors, relevant to experimental assays. The derived set of hybrid equations consists of partial differential equations for the tumor angiogenic factors and the inhibitors with a particle model for the motion of the endothelial cells. This is analyzed numerically in the two-dimensional setting. The relevant results are discussed and qualitative agreement with the experimental work is illustrated.

Two-dimensional paradigm for symmetry breaking : the nonlinear Schrödinger equation with a four-well potential

We study the existence and stability of localized modes in the two-dimensional (2D) nonlinear Schrödinger/Gross-Pitaevskii (NLS/GP) equation with a symmetric four-well potential. Using the corresponding four-mode approximation, we trace the parametric evolution of the trapped stationary modes, starting from the linear limit, and thus derive a complete bifurcation diagram for families of the stationary modes. This provides the picture of spontaneous symmetry breaking in the fundamental 2D setting. In a broad parameter region, the predictions based on the four-mode decomposition are found to be in good agreement with full numerical solutions of the NLS/GP equation. Stability properties of the stationary states coincide with those suggested by the corresponding discrete model in the large-amplitude limit. The dynamics of unstable modes is explored by means of direct simulations. Finally, in addition to the full analysis for the case of the self-attractive nonlinearity, the bifurcation diagram for the case of self-repulsion is briefly considered too.

Two-component nonlinear Schrodinger models with a double-well potential

Abstract We introduce a model motivated by studies of Bose–Einstein condensates (BECs) trapped in double-well potentials. We assume that a mixture of two hyperfine states of the same atomic species is loaded in such a trap. The analysis is focused on symmetry-breaking bifurcations in the system, starting at the linear limit and gradually increasing the nonlinearity. Depending on values of the chemical potentials of the two species, we find numerous states, as well as symmetry-breaking bifurcations, in addition to those known in the single-component setting. These branches, which include all relevant stationary solutions of the problem, are predicted analytically by means of a two-mode approximation, and confirmed numerically. For unstable branches, outcomes of the instability development are explored in direct simulations.

Modeling foundation species in food webs

Foundation species are basal species that play an important role in determining community composition by physically structuring ecosystems and modulating ecosystem processes. Foundation species largely operate via non-trophic interactions, presenting a challenge to incorporating them into food web models. Here, we used non-linear, bioenergetic predator-prey models to explore the role of foundation species and their non-trophic effects. We explored four types of models in which the foundation species reduced the metabolic rates of species in a specific trophic position. We examined the outcomes of each of these models for six metabolic rate “treatments” in which the foundation species altered the metabolic rates of associated species by one-tenth to ten times their allometric baseline metabolic rates. For each model simulation, we looked at how foundation species influenced food web structure during community assembly and the subsequent change in food web structure when the foundation species was removed. When a foundation species lowered the metabolic rate of only basal species, the resultant webs were complex, species-rich, and robust to foundation species removals. On the other hand, when a foundation species lowered the metabolic rate of only consumer species, all species, or no species, the resultant webs were species-poor and the subsequent removal of the foundation species resulted in the further loss of species and complexity. This suggests that in nature we should look for foundation species to predominantly facilitate basal species.

Numerical solutions of boundary value problems for K-surfaces in R3

A K-surface is a surface whose Gauss curvature K is a positive constant. In this article, we will consider K-surfaces that are defined by a nonlinear boundary value problem. In this setting, existence follows from some recent results on nonlinear second-order elliptic partial differential equations. The analytical techniques used to establish these results motivate effective numerical methods for computing K-surfaces. In theory, the solvability of the boundary value problem reduces to the existence of a subsolution. In an analogous way, we find that if an approximate numerical subsolution can be determined, then the corresponding K-surface can be computed. We will consider two boundary value problems. In the first problem, the K-surface is a graph over a plane. In the second, the K-surface is a radial graph over a sphere. From certain geometrical considerations, it follows that there is a maximum Gauss curvature Kmax for these problems. Using a continuation method, we estimate Kmax and determine numerically the unique one-parameter family of K-surfaces that exist for K E (0,Kmax). This is the first time that this numerical method has been applied to the nonlinear partial differential equations for a K -surface. Sharp estimates for Kmax are not available analytically, except in special situations such as a surface of revolution, where the parametrization can be obtained explicitly in terms of elliptic functions. We find that our numerical estimates for Kmax are in close agreement with the expected values in these cases.

Short pulse equations and localized structures in frequency band gaps of nonlinear metamaterials

Abstract We consider short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response. Two short-pulse equations (SPEs) are derived for the high- and low-frequency “band gaps” (where linear electromagnetic waves are evanescent) with linear effective permittivity ϵ 0 and permeability μ > 0 . The structure of the solutions of the SPEs is also briefly discussed, and connections with the soliton solutions of the nonlinear Schrodinger equation are made.

Collisionally inhomogeneous Bose-Einstein condensates in double-well potentials

Abstract In this work, we consider quasi-one-dimensional Bose–Einstein condensates (BECs), with spatially varying collisional interactions, trapped in double-well potentials. In particular, we study a setup in which such a “collisionally inhomogeneous” BEC has the same (attractive–attractive or repulsive–repulsive) or different (attractive–repulsive) types of interparticle interactions. Our analysis is based on the continuation of the symmetric ground state and anti-symmetric first excited state of the non-interacting (linear) limit into their nonlinear counterparts. The collisional inhomogeneity produces a saddle–node bifurcation scenario between two additional solution branches; as the inhomogeneity becomes stronger, the turning point of the saddle–node tends to infinity and eventually only the two original branches remain, which is completely different from the standard double-well phenomenology. Finally, one of these branches changes its monotonicity as a function of the chemical potential, a feature especially prominent, when the sign of the nonlinearity changes between the two wells. Our theoretical predictions, are in excellent agreement with the numerical results.