On Artin's braid group and polyconvexity in the calculus of variations

Let Ω ⊂ 2 be a bounded Lipschitz domain and let F : Ω × 2×2 + −→ be a Carathèodory integrand such that F (x, ·) is polyconvex for L2-a.e. x ∈ Ω. Moreover assume that F is bounded from below and satisfies the condition F (x, ξ) ∞ as det ξ 0 for L2-a.e. x ∈ Ω. The paper describes the effect of domain topology on the existence and multiplicity of strong local minimizers of the functional [u] := Ω F (x,∇u (x)) dx, where the map u lies in the Sobolev space W1,p id (Ω,2) with p 2 and satisfies the pointwise condition det ∇u (x) > 0 for L2-a.e. x ∈ Ω. The question is settled by establishing that [·] admits a set of strong local minimizers on W1,p id (Ω,2) that can be indexed by the group n ⊕ n, the direct sum of Artin’s pure braid group on n strings and n copies of the infinite cyclic group. The dependence on the domain topology is through the number of holes n in Ω and the different mechanisms that give rise to such local minimizers are fully exploited by this particular representation

On the existence and multiplicity of topologically twisting incompressible HH-harmonic maps and a structural H-condition

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Self-synchronization of Kerr-nonlinear Optical Parametric Oscillators

We introduce a new, reduced nonlinear oscillator model governing the spontaneous creation of sharp pulses in a damped, driven, cubic nonlinear Schroedinger equation. The reduced model embodies the fundamental connection between mode synchronization and spatiotemporal pulse formation. We identify attracting solutions corresponding to stable cavity solitons and Turing patterns. Viewed in the optical context, our results explain the recently reported $\pi$ and $\pi/2$ steps in the phase spectrum of microresonator-based optical frequency combs

Self-synchronization Phenomena in the Lugiato-Lefever Equation

The damped driven nonlinear Schr\"odinger equation (NLSE) has been used to understand a range of physical phenomena in diverse systems. Studying this equation in the context of optical hyper-parametric oscillators in anomalous-dispersion dissipative cavities, where NLSE is usually referred to as the Lugiato-Lefever equation (LLE), we are led to a new, reduced nonlinear oscillator model which uncovers the essence of the spontaneous creation of sharply peaked pulses in optical resonators. We identify attracting solutions for this model which correspond to stable cavity solitons and Turing patterns, and study their degree of stability. The reduced model embodies the fundamental connection between mode synchronization and spatiotemporal pattern formation, and represents a novel class of self-synchronization processes in which coupling between nonlinear oscillators is governed by energy and momentum conservation.Comment: This manuscript is published in Physical Review A. Copyright 2017 by the American Physical Society. arXiv admin note: text overlap with arXiv:1602.0852

Determination of helminth parasites in abdominal cavity of Alosa caspia (Eichwald, 1838) from the southeast part of the Caspian Sea

Alosa caspia (Eichwald, 1838) belongs to clupeidae family, is considered as one of the main fish in the southern coast of the Caspian Sea. The aim of the present study was to evaluate the helminthic parasite infections in abdominal cavity of A. caspia from southeastern part of the Caspian Sea. In this regard, 30 fish were caught from Bandar-Torkaman and transferred alive to the parasitological laboratory. Then parasites specimens were fixed and transferred to the National Museum of Parasitology, Faculty of Veterinary Medicine at University of Tehran for identification. A total of two parasite species including Anisakis simplex and Pronoprymna ventricosa were isolated from the fish. 100% of the fish were infected with at least one helminthic parasite species. Pronoprymna ventricosa has the highest infection prevalence rate and was isolated from pyloric caeca, intestine and stomach of 93.33% of the fish specimens. Anisakis simplex is found in abdominal cavity of 33.33 % of the studied fish. Intensity of A. simplex and P. ventricosa was calculated as 8.4±5.31 and 91.4±21.46, respectively. Based on the statistical analysis, there were no significant differences in total parasites burden, parasite prevalence and parasite intensity between male and female of the studied fish (P>0.05)

Annular rearrangements, incompressible axi-symmetric whirls and L1-local minimisers of the distortion energy

In this paper we consider a variational problem consisting of an energy functional defined by the integral, F[u, X] = 1/2∫x |∇u|²/|u|² dx, and an associated mapping space, here, the space of incompressible Sobolev mappings of the symmetric annular domain X = {x ∈ R n : a < |x| < b}: Aφ(X) = { u ∈ W1,2 (X, R n ) : det ∇u = 1 a.e. and u|∂X ≡ x } . The goal is then two fold. Firstly to establish and highlight an unexpected difference in the symmetries of the extremiser and local minimisers of F over Aφ(X) in the two special cases n = 2 and n = 3. More specifically, that when n = 3, despite the inherent rotational symmetry in the problem, there are NO non-trivial rotationally symmetric critical points of F over Aφ(X), whereas in sharp contrast, when n = 2, not only that there is an infinitude of rotationally symmetric critical points of the energy but also there is an infinitude of local minimisers of F over Aφ(X) with respect to the L¹ -metric. At the heart of this analysis is an investigation into the rich homotopy structure of the space of self-mappings of annuli. The second aim is to introduce and implement a novel symmetrisation technique in the planar case n = 2 for Sobolev mappings u in Aφ(X) that lowers the energy whilst keeping the homotopy class of u invariant. We finally generalise and extend some of these results to higher dimensions, in particular, we show that only in even dimensions do we have an infinitude of non-trivial rotationally symmetric critical points

Geodesics on SO(n) and a class of spherically symmetric maps as solutions to a nonlinear generalised harmonic map problem

We address questions on existence, multiplicity as well as qualitative features including rotational symmetry for certain classes of geometrically motivated maps serving as solutions to the nonlinear system $$\begin{cases} -\text{\rm div}[ F'(|x|,|\nabla u|^2) \nabla u] = F'(|x|,|\nabla u|^2) |\nabla u|^2 u &\text{in } \mathbb{X}^n,\\ |u| = 1 &\text{in } \mathbb{X}^n ,\\ u = \varphi &\text{on } \partial \mathbb{X}^n. \end{cases}$$% Here \varphi \in \mathscr{C}^\infty(\partial {\mathbb{X}}^n, \mathbb S}^{n-1}) is a suitable boundary map, $F'$ is the derivative of $F$ with respect to the second argument, u \in W^{1,p}(\mathbb{X}^n, \mathbb S}^{n-1}) for a fixed $1< p< \infty$ and ${\mathbb{X}}^n=\{x \in \mathbb R^n : a< |x|< b\}$ is a generalised annulus. Of particular interest are spherical twists and whirls, where following \cite{Taheri2012}, a spherical twist refers to a rotationally symmetric map of the form u\colon x \mapsto \rom{Q}(|x|)x|x|^{-1} with \rom{Q} some suitable path in $\mathscr{C}([a, b], {\rm SO}(n))$ and a whirl has a similar but more complex structure with only $2$-plane symmetries. We establish the existence of an infinite family of such solutions and illustrate an interesting discrepancy between odd and even dimensions