Family Of Critical Points Research Articles

Dynamical and variational properties of the NLS-[formula omitted] equation on the star graph

We study the nonlinear Schrödinger equation with δs′ coupling of intensity β∈R∖{0} on the star graph Γ consisting of N half-lines. The nonlinearity has the form g(u)=|u|p−1u,p>1. In the first part of the paper, under certain restriction on β, we prove the existence of the ground state solution as a minimizer of the action functional Sω on the Nehari manifold. It appears that the family of critical points which contains a ground state consists of N profiles (one symmetric and N−1 asymmetric). In particular, for the attractive δs′ coupling (β<0) and the frequency ω above a certain threshold, we managed to specify the ground state.The second part is devoted to the study of orbital instability of the critical points. We prove spectral instability of the critical points using Grillakis/Jones Instability Theorem. Then orbital instability for p>2 follows from the fact that data-solution mapping associated with the equation is of class C2 in H1(Γ). Moreover, for p>5 we complete and concertize instability results showing strong instability (by blow up in finite time) for the particular critical points.

Critical points bifurcation analysis of high-ℓ bending dynamics in acetylene

The bending dynamics of acetylene with pure vibrational angular momentum excitation and quantum number l not = 0 are analyzed through the method of critical points analysis, used previously [V. Tyng and M. E. Kellman, J. Phys. Chem. B 110, 18859 (2006)] for l = 0 to find new anharmonic modes born in bifurcations of the low-energy normal modes. Critical points in the reduced phase space are computed for continuously varied bend polyad number N(b) = n(4) + n(5) as l = l(4) + l(5) is varied between 0 and 20. It is found that the local L, orthogonal O, precessional P, and counter-rotator CR families persist for all l. In addition, for l > or = 8, there is a fifth family of critical points which, unlike the previous families, has no fixed relative phase ("off great circle" OGC). The concept of the minimum energy path in the polyad space is developed. With restriction to l=0 this is the local mode family L. This has an intuitive relation to the minimum energy path or reaction mode for acetylene-vinylidene isomerization. With l > or = 0 included as a polyad number, the l = 0 minimum energy path forms a troughlike channel in the minimum energy surface in the polyad space, which consists of a complex mosaic of L, O, and OGC critical points. There is a division of the complete set of critical points into layers, the minimum energy surface forming the lowest.

A time-varying Newton algorithm for adaptive subspace tracking

We propose a general framework for tracking the zeros of a time-varying gradient vector field on Riemannian manifolds. Thus, a differential equation, called the time-varying Newton flow, is introduced, whose solutions asymptotically converge to a time-varying family of critical points of the corresponding cost function. A discretization of the differential equation leads to a recursive update scheme for the time-varying critical point. As an application of such techniques we develop new algorithms for computing the principal and minor subspace of a time-varying family of symmetric matrices. Using a convenient local parameterization of the Grassmann manifold, we derive simple expressions for the subspace tracking schemes. Key benefits of the algorithms are (a) the reduced complexity aspects due to efficient parameterizations of the Grassmannian and (b) their guaranteed accuracy during all iterates. Numerical simulations illustrate the feasibility of the approach.

The conformal total tension variational problem in Kaluza-Klein supergravity

We obtain a reduction of variables method for the Willmore-Chen variational problem in conformal structures associated with metrics, on principal fibre bundles, which are obtained via the generalized inverse Kaluza-Klein mechanism. The problem of finding critical points for the conformal total tension functional in those conformal structures is transferred to the search of critical points for certain elastic energy functionals acting on spaces of curves in the base. This technique is applied to construct wide families of critical points for the conformal total tension functional in an ample class of stable vacuum states in the Freund-Rubin spontaneous compactification process of the extra dimensions in d = 11 supergravity.

Bifurcations in reaction-diffusion problems

Publisher Summary This chapter discusses bifurcations in reaction-diffusion problems. The general structure related bifurcation theorems is that from some hypotheses about the linearized version of the problem, conclusions are drawn about the full nonlinear one, valid in an appropriate neighborhood. This is analogous to the implicit function theorem, and indeed proofs of such bifurcation theorems typically involve transformations that convert the problem into a form in which the implicit function theorem can be applied. Another consequence of the hypotheses in the theorem stated above relates to the stability properties of the new family of critical points. When this is adjoined to the mere existence of the critical points, the bifurcation theorem becomes genuinely a result about differential equations. Bifurcations are said to exhibit an exchange of stabilities. To obtain a more satisfactory understanding of the shocks and to derive the conditions characterizing their propagation, one should return to the full reaction-diffusion equations and, at least in certain idealized cases, show that they do have solutions with a shock structure, a region of transition between two different plane wave solutions. This can be done for the case of a shock structure joining two plane-wave solutions with nearly equal wave numbers and frequencies by bifurcation methods used for the weak gas dynamic shock structure problem.