Family Of Stationary Solutions Research Articles

N-Symmetric Interaction of N Hetons, II: Analysis of the Case of Arbitrary N

This paper seeks and examines N-symmetric vortical solutions of the two-layer geostrophic model for the special case when the vortices (or eddies) have vanishing summed strength (circulation anomaly). This study is an extension [Sokolovskiy et al. Phys. Fluids 2020, 32, 09660], where the general formulation for arbitrary N was given, but the analysis was only carried out for N=2. Here, families of stationary solutions are obtained and their properties, including asymptotic ones, are investigated in detail. From the point of view of geophysical applications, the results may help interpret the propagation of thermal anomalies in the oceans.

On the Stability of Equilibrium of a Mechanical System with Tracking, Potential, and Small Dissipative Forces

We consider a holonomic non-conservative mechanical system with two degrees of freedom that simulates the motion of a vane on the elastic bushing of the helicopter’s main or rudder screw in the thrust plane. The system experiences the action of potential, non-conservative positional forces and linear dissipative forces, which describe the effect of internal friction during deformation of the vane and the effect of the external environment. A multiparameter family of stationary solutions of the model is described, and an investigation of Lyapunov’s stability of a trivial solution in the presence of small linear friction forces is carried out. Ziegler’s areas were plotted for different values of the problem parameters.

Dynamics and stability of stationary states for the spin-1 Bose–Einstein condensates in a standing light wave

The spin-1 Bose–Einstein condensates trapped in a standing light wave can be described by three coupled Gross–Pitaevskii equations with a periodic potential. In this paper, nine families of stationary solutions without phase structures in the form of Jacobi elliptic functions are proposed, and their stabilities are analyzed by both linear stability analysis and dynamical evolutions. Taking the ferromagnetic 87Rb atoms and antiferromagnetic (polar) 23Na atoms as examples, we investigate the stability regions of the nine stationary solutions, which are given in term of elliptic modulus k. It is shown that for the same stationary solution the stability regions of condensates with antiferromagnetic (polar) spin-dependent interactions are larger than that of the condensates with ferromagnetic ones. The dn-dn-dn stationary solution is the most stable solution among the nine families of stationary solutions. Moreover, in the same standing light wave, the spin-1 Bose–Einstein condensates are more stable than the scalar Bose–Einstein condensate.

Stable Transition Layers to Singularly Perturbed Spatially Inhomogeneous Allen-Cahn Equation

Abstract We use the variational concept of Γ-convergence to prove existence, stability and exhibit the geometric structure of four families of stationary solutions to the singularly perturbed parabolic equation ut = ε2Δu + u[a2(x) − u2], for (t, x) ∈ ℝ+ × Ω, where Ω ⊂ ℝ2 and a is positive, supplied with no-flux boundary condition. Let γ ⊂ Ω be a smooth simple closed curve and N(γ) a narrow tubular neighborhood of γ. Roughly speaking, the sufficient condition found for existence of such solutions relates the geometric profile of the function a in N(γ) to the signed curvature of γ .

The Centrally Symmetric V-States for Active Scalar Equations. Two-Dimensional Euler with Cut-Off

We consider the truncated equation for the centrally symmetric $V$-states in the 2d Euler dynamics of patches and prove the existence of solutions $y(x,\lambda),\, x\in [-1,1],\,\lambda\in (0,\lambda_0)$. These functions $y(x,\lambda)\in C^\infty(-1,1)$ and $\|y(x,\lambda)-|x|\|_{C[-1,1]}\to 0, \lambda\to 0$.

Multidimensional Degenerate Keller–Segel System with Critical Diffusion Exponent $2n/(n+2)$

This paper deals with a degenerate diffusion Patlak–Keller–Segel system in $n\geq 3$ dimension. The main difference between the current work and many other recent studies on the same model is that we study the diffusion exponent $m=2n/(n+2)$, which is smaller than the usual exponent $m^{*}=2-2/n$ used in other studies. With the exponent $m=2n/(n+2)$, the associated free energy is conformal invariant, and there is a family of stationary solutions $U_{\lambda,x_0}(x)=C(\frac{\lambda} {\lambda^2+|x-x_0|^2})^{\frac{n+2}{2}}$ $\forall \lambda>0$, $x_0\in {\mathbb R}^n$. For radially symmetric solutions, we prove that if the initial data are strictly below $U_{\lambda,0}(x)$ for some $\lambda$, then the solution vanishes in $L^1_{loc}$ as $t\to\infty$; if the initial data are strictly above $U_{\lambda,0}(x)$ for some $\lambda$, then the solution either blows up at a finite time or has a mass concentration at $r=0$ as time goes to infinity. For general initial data, we prove that there is a global weak solution...

The non-resonant, relativistic dynamics of circumbinary planets

We investigate the non-resonant, 3-D (spatial) model of the hierarchical system composed of point-mass stellar (or sub-stellar) binary and a low-mass companion (a circumbinary planet or a brown dwarf). We take into account the leading relativistic corrections to the Newtonian gravity. The secular model of the system relies on the expansion of the perturbing Hamiltonian in terms of the ratio of semi-major axes $\alpha$, averaged over the mean anomalies. We found that the low-mass object in a distant orbit may excite large eccentricity of the inner binary when the mutual inclination of the orbits is larger than about of 60 deg. This is related to strong instability caused by a phenomenon which acts similarly to the Lidov-Kozai resonance (LKR). The secular system may be strongly chaotic and its dynamics unpredictable over the long-term time scale. Our study shows that in the Jupiter-- or brown dwarf-- mass regime of the low-massive companion, the restricted model does not properly describe the long-term dynamics in the vicinity of the LKR. The relativistic correction is significant for the parametric structure of a few families of stationary solutions in this problem, in particular, for the direct orbits configurations (with the mutual inclination less than 90 degrees). We found that the dynamics of hierarchical systems with small $\alpha \sim 0.01$ may be qualitatively different in the realm of the Newtonian (classic) and relativistic models. This holds true even for relatively large masses of the secondaries.

A Family of Stationary Solutions to the Euler Equations and Generalized Solutions

En este trabajo, presentamos una familia interesante de soluciones estacionarias para las ecuaciones de Euler, que se comportan de la misma manera que las soluciones aproximadas presentadas en [6].

1D CAHN–HILLIARD EQUATION: OSTWALD RIPENING AND MODULATED PHASE SYSTEMS

We use a family of stationary solutions of the Cahn–Hilliard dynamics with the aim of describing the process of coarsening during a first order phase transition. With this analytical ansatz, we can compute the characteristic time for one step of period doubling in Langer's self-similar scenario for Ostwald ripening. As an application, the same ansatz is also used to compute the thermodynamically stable period of a 1D modulated phase pattern, described by a modified Cahn–Hilliard dynamics with a long range interaction term.

Orbital stability for polytropic galaxies

In this paper, motivated by the Galilean invariance of the solutions, we use the concept of orbital stability for the Vlasov–Poisson system. We prove that a family of stationary solutions, called polytropic gas spheres, are orbitally L1 stable in the gravitational case by considering direct variational arguments.

Onset of chaotic symbolic synchronization between population inversions in an array of weakly coupled Bose-Einstein condensates

We investigate the onset of chaotic dynamics of the one-dimensional discrete nonlinear Schrödinger equation with periodic boundary conditions in the presence of a single on-site defect. This model describes a ring of weakly coupled Bose-Einstein condensates with attractive interactions. We focus on the transition to global stochasticity in three different scenarios as the defect is changed. We make use of a suitable Poincaré section and study different families of stationary solutions, where certain bifurcations lead to global stochasticity. The global stochasticity is characterized by chaotic symbolic synchronization between the population inversions of certain pairs of condensates. We have seen that the Poincaré cycles are useful to gain insight in the dynamics of this Hamiltonian system. Indeed, the return maps of the Poincaré cycles have been used successfully to follow the orbit along the stochastic layers of different resonances in the chaotic self-trapping regime. Moreover, the time series of the Poincaré cycles suggests that in the global stochasticity regime the dynamics is, to some extent, Markovian, in spite of the fact that the condensates are phase locked with almost the same phase. This phase locking induces a peculiar local interference of the matter waves of the condensates.

GLOBAL SOLUTIONS TO THE BOLTZMANN EQUATION WITH EXTERNAL FORCES

For the Boltzmann equation with an external potential force depending only on the space variables, there is a family of stationary solutions, which are local Maxwellians with space dependent density, zero velocity and constant temperature. In this paper, we will study the nonlinear stability of these stationary solutions by using the energy method. The analysis combines the analytic techniques used for the conservation laws using the fluid-type system derived from the Boltzmann equation (cf. [14]) and the dissipative effects on the fluid and non-fluid components of the Boltzmann equation through the celebrated H-theorem. To our knowledge, this is the first result on the global classical solutions to the Boltzmann equation with external force and non-trivial large time behavior in the whole space.

Stability of attractive Bose-Einstein condensates in a periodic potential.

Using a standing light wave potential, a stable quasi-one-dimensional attractive dilute-gas Bose-Einstein condensate can be realized. In a mean-field approximation, this phenomenon is modeled by the cubic nonlinear Schrödinger equation with attractive nonlinearity and an elliptic function potential of which a standing light wave is a special case. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schrödinger equation nor in the integrable nonlinear Schrödinger equation. Their stability is examined using analytic and numerical methods. Trivial-phase solutions are experimentally stable provided they have nodes and their density is localized in the troughs of the potential. Stable time-periodic solutions are also examined.

Stability of repulsive Bose-Einstein condensates in a periodic potential.

The cubic nonlinear Schrödinger equation with repulsive nonlinearity and an elliptic function potential models a quasi-one-dimensional repulsive dilute gas Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schrödinger equation nor in the integrable nonlinear Schrödinger equation. Their stability is examined using analytical and numerical methods. All trivial-phase stable solutions are deformations of the ground state of the linear Schrödinger equation. Our results show that a large number of condensed atoms is sufficient to form a stable, periodic condensate. Physically, this implies stability of states near the Thomas-Fermi limit.

Relative equilibria of a satellite subjected to gravitational and aerodynamic torques

The problem of determining all equilibria of a satellite in a circular orbit is solved in the case where the satellite is subjected to gravitational and aerodynamic torques. The number of isolated equilibria is shown to be no less than eight and no more than 24. The existence proof of one-parameter families of stationary solutions is given. Using Lyapunov's method sufficient conditions for stability of isolated equilibria are obtained.

Nonlinear dynamics of a beam on elastic foundation

The nonlinear dynamics of a simply supported beam resting on a nonlinear spring bed with cubic stiffness is analyzed. The continuous differential operator describing the mathematical model of the system is discretized through the classical Galerkin procedure and its nonlinear dynamic behavior is investigated using the method of Normal Forms. This model can be regarded as a simple system describing the oscillations of flexural structures vibrating on nonlinear supports and then it can be considered as a simple investigation for the analysis of more complex systems of the same type. Indeed, the possibility of the model to exhibit actually interesting nonlinear phenomena (primary, superharmonic, subharmonic and internal resonances) has been shown in a range of feasibility of the physical parameters. The singular perturbation approach is used to study both the free and the forced oscillations; specifically two parameter families of stationary solutions are obtained for the forced oscillations.

ON THE METHOD OF STATIONARY STATES FOR QUASILINEAR PARABOLIC EQUATIONS

A method is presented for investigating the space-time structure of unbounded nonnegative solutions of quasilinear parabolic equations of the form , where is a nonlinear elliptic operator. Three examples are considered in detail: the Cauchy problem for the equation where 0$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img4.gif/> and 1$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img5.gif/> are constants; the boundary value problem in 0\,\}$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img6.gif/> 0,\quad x\in\Omega;$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img7.gif/> 0,\quad x_3=0;\qquad \alpha=\textrm{const}>0;$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img8.gif/> and the Cauchy problem for the system It is assumed that at the point the solution grows without bound as . The derivation of an estimate of the solution near , is based on an analysis of an appropriate family of stationary solutions : , , 0$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img17.gif/> a parameter. It is shown that the behavior of a solution as depends to large extent on the structure of the envelope 0}U_\lambda(x)$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img19.gif/>. In particular, if , then grows without bound as at points arbitrarily far from . If for , then determines a lower bound for in a neighborhood of , .Bibliography: 28 titles.