Linear Stability Of Stationary Solutions Research Articles

A finite element toolbox for the Bogoliubov-de Gennes stability analysis of Bose-Einstein condensates

We present a finite element toolbox for the computation of Bogoliubov-de Gennes modes used to assess the linear stability of stationary solutions of the Gross-Pitaevskii (GP) equation. Applications concern one (single GP equation) or two-component (a system of coupled GP equations) Bose-Einstein condensates in one, two and three dimensions of space. An implementation using the free software ▪is distributed with this paper. For the computation of the GP stationary (complex or real) solutions we use a Newton algorithm coupled with a continuation method exploring the parameter space (the chemical potential or the interaction constant). Bogoliubov-de Gennes equations are then solved using dedicated libraries for the associated eigenvalue problem. Mesh adaptivity is proved to considerably reduce the computational time for cases implying complex vortex states. Programs are validated through comparisons with known theoretical results for simple cases and numerical results reported in the literature. Program summaryProgram Title: FFEM_BdG_toolbox.zipCPC Library link to program files:https://doi.org/10.17632/dgypyc34gb.1Licensing provisions: Apache 2.0Programming language:▪(v 4.12) free software (www.freefem.org)Nature of problem: The software computes Bogoliubov-de Gennes (BdG) complex modes of Bose-Einstein condensates described by the Gross-Pitaevskii (GP) equation. BdG equations are obtained by linearizing the GP equation (or the system of coupled GP equations) around a stationary solution. Obtained BdG modes are used to assess on the stability of stationary states.Solution method: Stationary states of the GP equation are obtained by a Newton algorithm. Parameter space is explored using a continuation on the chemical potential. Once the stationary (complex or real) state is captured accurately, BdG modes are computed by solving the associated eigenvalue problem with the ARPACK library. Complex eigenvalues and eigenvectors are computed and stored. The wave function is discretized by P2 (piece-wise quadratic) Galerkin triangular (in 2D) or tetrahedral (in 3D) finite elements. Mesh adaptation is implemented to reduce the computational time. Examples are given for stationary states in one- and two-component Bose-Einstein condensates.

Dynamical study of two predators and one prey system with fractional Fourier transform method

In this work, we investigate both the analytical and numerical studies of the dynamical model comprising of three species systems. We analyze the linear stability of stationary solutions in the one‐dimensional multisystem modeling the interactions of two predators and one prey species. The stability analysis has a lot of implications for understanding the various spatiotemporal and chaotic behaviors of the species in the spatial domain. The analysis results presented have established the possibility of the three‐interacting species to coexist harmoniously, this feat is achieved by combining the local and global analyses to determine the global dynamics of the system. In the presence of a fractional diffusion term, we introduced a fractional Fourier transform for solving the system modeled by fractional partial differential equations. The main advantages of this method are that it yields a fully diagonal representation of the fractional operator with exponential accuracy and a completely straightforward extension to high spatial dimensions. The scheme is described in detail and justified by a number of computational experiments.

On a non-isothermal model of free fluid films

The drawing of thin fluid films and fibers from highly viscous fluid melts is a common engineering process in the chemical and textile industry. A standard procedure for the manufacture of free thin films is film casting. The widely used, one-dimensional model of film casting considered here is based on a slender body approximation of the Navier–Stokes equations with moving boundaries, paired with kinematic assumptions. The model equations describe the flow of the viscous fluid between the die exit and a take-up point and permit variations in film width and film thickness. A heat transfer equation accounts for the heat loss due to cooling. We will address the following two objectives here: First we put existence and uniqueness of stationary solutions for the equations of non-isothermal film casting on a rigorous analytical basis. This objective is tackled with the help of continuity arguments and a variant of the maximum principle. Our second objective is the study of the linearized equations. We will prove semigroup results for the linearization about steady state and shed light on the long-term regularity of solutions. Among other things we obtain the validity of the spectral mapping theorem for the semigroup and the spectrally determined growth property. These results form the basis for computational studies in the literature about the linear stability of stationary solutions and the potential onset of physical instabilities.

Stability and regions of motion in the restricted three-body problem when both the primaries are finite straight segments

In this paper we have studied the locations and stability of the Lagrangian equilibrium points in the restricted three-body problem under the assumption that both the primaries are finite straight segments. We have found that the triangular equilibrium points are conditional stable for 0<μ<μc, and unstable in the range μc<μ≤1/2, where μ is the mass ratio. The critical mass ratio μc depends on the lengths of the segments and it is observed that the range of μc increases when compared with the classical case. The collinear equilibrium points are unstable for all values of μ. We have also studied the regions of motion of the infinitesimal mass. It has been observed that the Jacobian constant decreases when compared with the classical restricted three-body problem for a fixed value of μ and lengths l1 and l2 of the segments. Beside this we have found the numerical values for the position of the collinear and triangular equilibrium points in the case of some asteroids systems: (i) 216 Kleopatra-951 Gaspara, (ii) 9 Metis-433 Eros, (iii) 22 Kalliope-243 Ida and checked the linear stability of stationary solutions of these asteroids systems.

Persistent localized activity in a two-population neural-field model with spatio-temporal external input

Persistent localized activity in neural networks has been suggested to serve as the neural substrate of short-term memory [1]. Neural-field models [2] provide a powerful tool to study the existence, uniqueness and stability of this type of activity. Bump solutions are often considered as an example of localized activity. In [3], it has been shown that a two-population neural-field model (Fig.​(Fig.1A)1A) exhibits up to two coexisting bump-pair solutions in the presence of homogeneous external input (or in the absence of external input). Here, we show that the same system can generate up to four coexisting bump pairs if the external input is spatially localized. Figure 1 A. Sketch of the two population neural field model with excitatory (he) and inhibitory (hi) external inputs. B. Persistent excitatory activity (Ue) switched on and off by a brief external input pulses. Inhibitory activity behaves in a similar way. In general, neural-field models are formulated in terms of Volterra-equation systems or systems of integro-differential equations. The dynamics of bump (pulse) solutions is often studied in a simplified framework by means of ordinary differential equations describing the time evolution of the pulse widths (Amari approach, see [4]). It has been proven that in the absence of external input this simplified pulse-width system gives the correct prediction with respect to the linear stability of stationary solutions, with one notable exception. It is often conjectured that the Amari approach might correctly predict also nonlinear aspects of the dynamics (bifurcation analysis). Here, we show that the Amari approach fails in the presence of spatially localized external input: the criteria for linear-stability of bumps in the pulse-width system do not coincide with those obtained for the full system. Therefore, for localized external input the analysis must rely on the full system. We determine fixed-point solutions and their stability analytically and illustrate the results by means of numerical simulations. Further, we numerically show that persistent localized activity in a two-population neural-field model can be switched on and off by means of brief external-input pulses localized in space in and time (see Fig ​Fig1B1B).

Asymptotic behavior of size-structured populations via juvenile-adult interaction

In this work a size structured juvenile-adult population model is considered. The linearized dynamical behavior of stationary solutions is analyzed using semigroup and spectral methods. The regularity of the governing linear semigroup allows to derive biologically meaningful conditions for the linear stability of stationary solutions. The main emphasis in this work is on juvenile-adult interaction and resulting consequences for the dynamics of the system. In addition, we investigate numerically the eect of a non-zero population inflow, due to an external source of newborns, on the dynamical behavior of the system in a special case of model ingredients.

On the Effects of Spinline Cooling and Surface Tension in Fiber Spinning

In contrast to isothermal fiber spinning, nonisothermal fiber spinning is a remarkably stable process. In this note, we shall study the effects of continuous spinline cooling and surface tension on the stability of stationary nonisothermal fiber spinning flow. We systematically derive an appropriate extension of the one-dimensional Matovich-Pearson thin filament equations of viscous liquids. This model will account for the physical chemistry of the fiber surface and the temperature dependence of the material parameters. We employ semigroup theory to discuss the linear stability of stationary solutions. To this end, we prove the spectral determinacy of the associated semigroup. This approach is made viable by a reformulation of the governing equations to avoid a moving flow domain. Finally, we shall give some computational results to study the onset of surface tension instabilities and their suppression by cooling.

Linear stability of stationary solutions of the Vlasov-Poisson system in three dimensions

Rigorous results on the stability of stationary solutions of the Vlasov-Poisson system are obtained in the contexts of both plasma physics and stellar dynamics. It is proved that stationary solutions in the plasma physics (stellar dynamics) case are linearly stable if they are decreasing (increasing) functions of the local, i.e., particle, energy. The main tool in the analysis is the free energy, a conserved quantity of the linearized system. In addition, an appropriate global existence result is proved for the linearized Vlasov-Poisson system and the existence of stationary solutions which satisfy the above stability condition is established.

Stability of stationary solutions of extended reaction-diffusion-convection equations on a finite segment

A simple geometric criterion on the linear stability of stationary solutions of nonlinear second order parabolic equations on a finite segment is stated and proved.