Existence Of Stable Solutions Research Articles

Particle-like solutions in the generalized SU(2) Proca theory

The generalized SU(2) Proca theory is a vector-tensor modified gravity theory where the action is invariant under both diffeomorphisms and global internal transformations of the SU(2) group. This work constitutes the first approach to investigate the physical properties of the theory at astrophysical scales.We have found solutions that naturally generalize the particle-like solutions of the Einstein-Yang-Mills equations, also known as gauge boson stars.Under the requirement that the solutions must be static, asymptotically flat, and globally regular, the t'Hooft-Polyakov magnetic monopole configuration for the vector field rises as one viable possibility. The solutions have been obtained analytically through asymptotic expansions and numerically by solving the boundary value problem. We have found new features in the solutions such as regions with negative effective energy density and imaginary effective charge. We have also obtained a new kind of globally charged solutions for some region in the parameter space of the theory. Furthermore, we have constructed equilibrium sequences and found turning points in some cases. These results hint towards the existence of stable solutions which are absent in the Einstein-Yang-Mills case.

Qualitative analysis of a stock-effort parabolic model incorporating marine reserve

A stock-effort dynamical model of a fishery with spatial structure is considered to assess the impacts of reserve size on fishing catch. We study the existence, uniqueness and global character of the solutions. It is shown that when the cost of fishing effort is less than the principal eigenvalue for an elliptic operator defined on a bounded domain corresponding to the fishing zone, then the fishery is viable, while in the opposite case, the fishery is not viable and fishing effort disappears. For both cases, the existence of stable solutions is studied. We perform some numerical simulations to illustrate our results in this paper and give a brief concluding remark.

Reconsidering the Existence of Stable Solutions in Three-sided Matching Problems with Mixed Preferences

Reconsidering the Existence of Stable Solutions in Three-sided Matching Problems with Mixed Preferences

Stable Model Matching by Non-Regular Static State Feedback

A solution of the longstanding problem of model matching by non-regular static state feedback is presented. The problem formulation avoids restrictive hypotheses concerning system and target model. Necessary and sufficient conditions for the problem to have a solution are obtained which yield a construction of a matching feedback law. Several cases of special interest are subsequently discussed. Finally, the existence of stable solutions is established and a construction of static stable solutions is addressed.

Mathematical model of marine protected areas

W e consider two regions with a fish population that is dispersing between the two areas, and fishing takes place only in region 2, with region 1 established as no-fishing zone. Marine protected areas (MPAs) have been promoted as conservation and fishery management tools, and at present, there are over 1300 MPAs in the world. A new mathematical model of an MPA that reflects the complexity of the natural setting is presented. The resulting model of an age-structured fish population belongs to a class of non-linear systems of differential equations with delay. New easily verifiable sufficient conditions for the existence, boundedness, permanence and stability of the positive internal steady-state solutions are obtained. From the point of view of fishery managers, the existence of stable solutions is necessary for planning harvesting strategies and sustaining the fishing grounds. Numerical simulations illustrate qualitative behaviour of the model, including stability switches.

Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation

We study solutions of the 2D Ginzburg-Landau equation 1uC 1 u.juj 2 1/D 0 subject to semi-stiff boundary conditions: Dirichlet conditions for the modulus,juj D 1, and homogeneous Neumann conditions for the phase. The principal result of this work shows that there are stable solutions of this problem with zeros (vortices), which are located near the boundary and have bounded energy in the limit of small. For the Dirichlet boundary condition (stiff problem), the existence of stable solutions with vortices, whose energy blows up as! 0, is well known. By contrast, stable solutions with vortices are not established in the case of the homogeneous Neumann (soft) boundary condition. In this work, we develop a variational method which allows one to construct local minimizers of the corresponding Ginzburg-Landau energy functional. We introduce an approximate bulk degree as the key ingredient of this method; unlike the standard degree over the curve, it is preserved in the weakH 1 -limit.

Existence of steady stable solutions for the Ginzburg-Landau equation in a domain with nontrivial topology

Let N > 2 and Ω C ℝ N be a bounded domain with boundary ∂Ω. Let Γ C ∂Ω be closed. Our purpose in this paper is to consider the existence of stable solutions u ∈ H 1 (Ω, C) of the Ginzburg-Landau equation {―Δu(x) = λ(w 2 0 (x) ― |u| 2 )u in Ω, u = g on ∂Ω\Γ, ∂u ∂v = 0 on Γ where A > 0, w 0 ∈ C 2 (Ω,ℝ + ) and g ∈ C 2 (∂Ω\Γ) such that |g(x)| = wo(x) on ∂Ω\Γ.

Numerical simulation of a stratigraphic model

A multi-lithology diffusive stratigraphic model is considered, which simulates at large scales in space and time the infill of sedimentary basins governed by the interaction between tectonics displacements, eustatic variations, sediment supply, and sediment transport laws. The model accounts for the mass conservation of each sediment lithology resulting in a mixed parabolic, hyperbolic system of partial differential equations (PDEs) for the lithology concentrations and the sediment thickness. It also takes into account a limit on the rock alteration velocity modeled as a unilaterality constraint. To obtain a robust, fast, and accurate simulation, fully and semi-implicit finite volume discre tization schemes are derived for which the existence of stable solutions is proved. Then, the set of nonlinear equations is solved using a Newton algorithm adapted to the unilaterality constraint, and preconditioning strategies are defined for the solution of the linear system at each Newton iteration. They are based on an algebraic approximate decoupling of the sediment thickness and the concentration variables as well as on a proper preconditioning of each variable. These algorithms are studied and compared in terms of robustness, scalability, and efficiency on two real basin test cases.

Dynamics of constrained differential delay equations

A class of forced first-order differential delay equations with piecewise-affine right-hand sides is introduced, as a prototype model for the speed of a motor under control. A simple pure delay form is mainly considered. When forcing is zero, an exact stable periodic solution is exhibited. For large amplitude periodic forcing, existence of stable solutions, whose period is equal to that of the forcing function, is discussed, and these solutions are constructed for square wave forcing. Traditional numerical methods are discussed briefly, and a new approach based on piecewise-polynomial structure is introduced. Simulations are then presented showing a wide range of dynamics for intermediate values of forcing amplitude, when the natural period of the homogeneous equation and the period of the forcing function compete.

Stability analysis of the D-dimensional nonlinear Schrödinger equation with trap and two- and three-body interactions

Considering the static solutions of the D-dimensional nonlinear Schrödinger equation with trap and attractive two-body interactions, the existence of stable solutions is limited to a maximum critical number of particles, when D≥2. In case D=2, we compare the variational approach with the exact numerical calculations. We show that, the addition of a positive three-body interaction allows stable solutions beyond the critical number. In this case, we also introduce a dynamical analysis of the conditions for the collapse.

Existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature

Let f ( x ) f(x) be a given positive function in R n + 1 R^{n+1} . In this paper we consider the existence of convex, closed hypersurfaces X X so that its Gauss-Kronecker curvature at x ∈ X x\in X is equal to f ( x ) f(x) . This problem has variational structure and the existence of stable solutions has been discussed by Tso (J. Diff. Geom. 34 (1991), 389–410). Using the Mountain Pass Lemma and the Gauss curvature flow we prove the existence of unstable solutions to the problem.

Dynamic consideration of compromise solutions in a many-sided conflict

A DYNAMIC scheme for making decisions about the pay-offs in a three-sided conflict is proposed; the scheme is described as a sequence of plays of a three-person zero-sum game, the players' interests being defined by the mean pay-off per play. The existence of stable solutions (of the equilibrium point type) is proved; they generate periodic motions of the players' coalitions and ensure a mean share equal to Shapley's vector. The stable solutions are possible both in the class of strategies that include information about the past plays, and in the class of strategies that only take account of the players' running pay-offs.