Infinity In Space Research Articles

Multidimensional stability of V-shaped traveling fronts in bistable reaction-diffusion equations with nonlinear convection

This paper is concerned with the multidimensional stability of V-shaped traveling fronts for a reaction-diffusion equation with nonlinear convection term in $\mathbb{R}^n$ ($n\geq3$). We consider two cases for initial perturbations: one is that the initial perturbations decay at space infinity and another one is that the initial perturbations do not necessarily decay at space infinity. In the first case, we show that the V-shaped traveling fronts are asymptotically stable. In the second case, we first show that the V-shaped traveling fronts are also asymptotically stable under some further assumptions. At the same time, we also show that there exists a solution that oscillates permanently between two V-shaped traveling fronts, which means that the traveling fronts are not asymptotically stable under general bounded perturbations.

Inverse scattering transform for the complex short-pulse equation by a Riemann–Hilbert approach

In this paper, we develop the inverse scattering transform (IST) for the complex short-pulse equation (CSP) on the line with zero boundary conditions at space infinity. The work extends to the complex case the Riemann–Hilbert approach to the IST for the real short-pulse equation proposed by A. Boutet de Monvel, D. Shepelsky and L. Zielinski in 2017. As a byproduct of the IST, soliton solutions are also obtained. Unlike the real SPE, in the complex case discrete eigenvalues are not necessarily restricted to the imaginary axis, and, as consequence, smooth 1-soliton solutions exist for any choice of discrete eigenvalue $$k_1\in {\mathbb {C}}$$ with $$\mathop {{\mathrm{Im}}}\nolimits k_1< |\mathop {{\mathrm{Re}}}\nolimits k_1|$$ . The 2-soliton solution is obtained for arbitrary eigenvalues $$k_1,k_2$$ , providing also the breather solution of the real SPE in the special case $$k_2=-k_1^*$$ .

Self-Similar Blow-Up Profiles for a Reaction-Diffusion Equation with Strong Weighted Reaction

Abstract We study the self-similar blow-up profiles associated to the following second-order reaction-diffusion equation with strong weighted reaction and unbounded weight: ∂ t ⁡ u = ∂ x ⁢ x ⁡ ( u m ) + | x | σ ⁢ u p , \partial_{t}u=\partial_{xx}(u^{m})+|x|^{\sigma}u^{p}, posed for x ∈ ℝ {x\in\mathbb{R}} , t ≥ 0 {t\geq 0} , where m &gt; 1 {m&gt;1} , 0 &lt; p &lt; 1 {0&lt;p&lt;1} and σ &gt; 2 ⁢ ( 1 - p ) m - 1 {\sigma&gt;\frac{2(1-p)}{m-1}} . As a first outcome, we show that finite time blow-up solutions in self-similar form exist for m + p &gt; 2 {m+p&gt;2} and σ in the considered range, a fact that is completely new: in the already studied reaction-diffusion equation without weights there is no finite time blow-up when p &lt; 1 {p&lt;1} . We moreover prove that, if the condition m + p &gt; 2 {m+p&gt;2} is fulfilled, all the self-similar blow-up profiles are compactly supported and there exist two different interface behaviors for solutions of the equation, corresponding to two different interface equations. We classify the self-similar blow-up profiles having both types of interfaces and show that in some cases global blow-up occurs, and in some other cases finite time blow-up occurs only at space infinity. We also show that there is no self-similar solution if m + p &lt; 2 {m+p&lt;2} , while the critical range m + p = 2 {m+p=2} with σ &gt; 2 {\sigma&gt;2} is postponed to a different work due to significant technical differences.

Parabolic Higgs bundles and representations of the fundamental group of a punctured surface into a real group

We study parabolic G-Higgs bundles over a compact Riemann surface with fixed punctures, when G is a real reductive Lie group, and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in G with arbitrary holonomy around the punctures. This generalizes Simpson's results for GL(n,C) to arbitrary complex and real reductive Lie groups. Three interesting features are the relation between the parabolic degree and the Tits geometry of the boundary at infinity of the symmetric space, the treatment of the case when the logarithm of the monodromy is on the boundary of a Weyl alcove, and the correspondence of the orbits encoding the singularity via the Kostant–Sekiguchi correspondence. We also describe some special features of the moduli spaces when G is a split real form or a group of Hermitian type.

Particle dynamics in the Newtonian potential sourced by a homogeneous circular ring

Newtonian gravitational potential sourced by a homogeneous circular ring in arbitrary dimensional Euclidean space takes a simple form if the spatial dimension is even. In contrast, if the spatial dimension is odd, it is given in a form that includes complete elliptic integrals. In this paper, we analyze the dynamics of a freely falling massive particle in its Newtonian potential. Focusing on circular orbits on the symmetric plane where the ring is placed, we observe that they are unstable in 4D space and above, while they are stable in 3D space. The sequence of stable circular orbits disappears at $1.6095\cdots$ times the radius of the ring, which corresponds to the innermost stable circular orbit (ISCO). On the axis of symmetry of the ring, there are no circular orbits in 3D space but more than in 4D space. In particular, the circular orbits are stable between the ISCO and infinity in 4D space and between the ISCO and the outermost stable circular orbit in 5D space. There exist no stable circular orbits in 6D space and above.

On the entrance at infinity of Feller processes with no negative jumps

Consider a non-explosive positive Feller process with no negative jumps. It is shown in this note that when infinity is an entrance boundary, in the sense that the entrance times of the process remain bounded when the initial value tends to infinity, the process admits a Feller extension on the compactified state space [0,∞]. Moreover, when started from infinity, the extended Markov process on [0,∞] leaves infinity instantaneously and stays finite, almost-surely. Arguments are adapted from a proof given by Kallenberg (2002) for diffusions. We also show that the process started from x converges weakly towards that started from infinity in the Skorokhod space, when x goes to infinity.

ESSENTIAL SPECTRA OF ORDINARY DIFFERENTIAL OPERATORS II.&#x0D; STABILITY OF SPECTRA

The article contains precise formulas for finding of the essential spectra&#x0D; that are revolted with asymptotically constants on infinity in coefficients with use of&#x0D; rather compact and rather small indignations on infinity in Lebesgue spaces of 𝐿&#x0D; 𝑝&#x0D; .&#x0D; These formulas are new analogs of the classical theorem of Weyl.

Modified mean curvature flow of entire locally Lipschitz radial graphs in hyperbolic space

AbstractThe Asymptotic Plateau Problem asks for the existence of smooth complete hypersurfaces of constant mean curvature with prescribed asymptotic boundary at infinity in the hyperbolic space . The modified mean curvature flow (MMCF) was firstly introduced by Xiao and the second author a few years back in [15], and it provides a tool using geometric flow to find such hypersurfaces with constant mean curvature in . Similar to the usual mean curvature flow, the MMCF is the natural negative L2‐gradient flow of the area‐volume functional associated to a hypersurface Σ. In this paper, we prove that the MMCF starting from an entire locally Lipschitz continuous radial graph exists and stays radially graphic for all time. In general one cannot expect the convergence of the flow as it can be seen from the flow starting from a horosphere (whose asymptotic boundary is degenerate to a point).

Spinning cosmic strings in conformal gravity

We investigate the space–time of a spinning cosmic string in conformal invariant gravity, where the interior consists of a gauged scalar field. We find exact solutions of the exterior of a stationary spinning cosmic string, where we write the metric as [Formula: see text], with [Formula: see text] a dilaton field which contains all the scale dependences. The “unphysical” metric [Formula: see text] is related to the [Formula: see text]-dimensional Kerr space–time. The equation for the angular momentum [Formula: see text] decouples, for the vacuum situation as well as for global strings, from the other field equations and delivers a kind of spin-mass relation. For the most realistic solution, [Formula: see text] falls off as [Formula: see text] and [Formula: see text] close to the core. The space–time is Ricci flat. The formation of closed timelike curves can be pushed to space infinity for suitable values of the parameters and the violation of the weak energy condition can be avoided. For the interior, a numerical solution is found. This solution can easily be matched at the boundary on the exterior exact solution by special choice of the parameters of the string. This example shows the power of conformal invariance to bridge the gap between general relativity and quantum field theory.

ПРОСТРАНСТВО В БАШКИРСКОЙ ЯЗЫКОВОЙ КАРТИНЕ МИРА

This article analyzes the words and idiomatic phrases of the Bashkir language, expressing space and landscape. In the archaic Bashkir linguistic picture of the world, space is associated with the Universe and infinity. The infinity of space is directed both vertically and horizontally. The vertical worldview of the Bashkirs is associated with atmospheric and cosmic phenomena such as clouds, rain, thunderstorms and heavenly bodies, which gave rise to many mythological plots. Such a worldview arose with the desire to explain the existence of underground waters, caves, pits and cracks on the surface of the earth. The horizontal worldview includes four main sides, where the person himself is in the center of the events taking place, and his movements radiate from himself to all sides and are associated with the concepts of «range» and «reach.» Based on stable phrases representing distance from oneself, 4 types of range have been established: yaqyn ‘close’, ergәlә ‘very close’, alïθ, jyraq ‘distant’, bik alïθ ‘very distant’. In addition, it is noted that the types of range can be expressed by somatic vocabulary such as bash osonda ’overhead’, qul osonda ‘at hand’, kut töböndә ‘near the fifth point’, küð aldynda ‘before the eyes’, tanau (athtynda) osonda ‘under the nose, on the tip of the nose’, ajaq athynda ‘under the feet’ etc. Real and surreal space is found in folklore texts. The article also discusses the nomadic space associated with the concepts of pasture, yurt, watering place, etc. According to the author, the ethnic space is formed on the basis of the universal semantic category friend or foe, which divides society into two groups. According to the principle of religious views, two spaces can be distinguished in the Bashkir language: betheŋ donya ‘our world’ and tege donya ‘the afterlife’.

About toroidal soliton-particle of extremal space-time film

Nonlinear field model of extremal space-time film is considered. Its space-localized solution in toroidal coordinates with periodic dependence in time is investigated. A field configuration having a form of the twisted lightlike soliton moving along the ring is considered. An iterative algorithm for obtaining the solution in the form of formal power series in one toroidal variable is proposed. It is discovered that the time dependence in the toroidal solution has a deeply embedded character. Thus the full energy and angular momentum of the solution converge at space infinity.

A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum

<p style='text-indent:20px;'>We deal with the Cauchy problem of nonhomogeneous micropolar fluid equations with zero density at infinity in the entire space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula>. We show that for the initial density allowing vacuum, the strong solution exists globally if a weighted density is bounded from above. It should be noted that our blow-up criterion is independent of micro-rotational velocity.

On unified contribution of pair Coulomb excitations in three-body reactions

SynopsisIn the present work we obtain the unified contribution of pair excited states to the asymptotics (at infinity in configuration space) of the three-body absolutely continuous spectrum eigenfunctions. Note that it is the case that is most interesting and rich from the physical point of view, as it is connected in particular with re-clustering of charged quantum complexes.

SPATIAL VOCABULARY AND ITS ROLE IN THE FORMATION OF THE MORDOVIAN TOPONYMIC SYSTEM

The article is devoted to the comparative analysis of spatial vocabulary in Mordovian (Moksha, Erzya) and Finno-Ugric toponymy in General. The study allows us to determine how spatial representations (reflection of three-dimensionality, limit, infinity of space, etc.) are common among Finno-Ugric ethnic groups and how the process of nomination of geographical objects is represented in related languages. The object of the analysis was spatial lexemes, which have mainly ancient origin and are found in related languages not only as independent words, but also as integral components involved in the formation of geographical names of other Finno-Ugric regions. The sources of the study were etymological and toponymic dictionaries of Finno-Ugric languages, scientific works on toponymy, dialect materials from the authors’ field research projects. The study revealed that the majority of words with spatial meaning, which are part of the toponyms of the languages under consideration, have common structural and semantic features, which indicates similar ways of representing the category of space among Finno-Ugric peoples. At the same time, the lexemes of Finno-Ugric origin, reflecting the spatial coordinates "ahead" - "behind", did not find a place in the toponyms of related languages, they correspond to other locative appellatives.

Conformal scattering of the Maxwell-scalar field system on de Sitter space

We prove small data energy estimates of all orders of differentiability between past null infinity and future null infinity of de Sitter space for the conformally invariant Maxwell-scalar field system. Using these, we construct bounded and invertible, but nonlinear, scattering operators taking past asymptotic data to future asymptotic data. We deduce exponential decay rates for solutions with data having at least two derivatives, and for more regular solutions discover an asymptotic decoupling of the scalar field from the charge. The construction involves a carefully chosen complete gauge fixing condition which allows us to control all components of the Maxwell potential, and a nonlinear Grönwall inequality for higher-order estimates.

Blowup at space infinity for solutions of a system of nonautonomous semilinear heat equations

In this paper, we will see that the global or local existence of solutions to depends on the initial datums and the global or local existence of solutions to We also give some bounds for the maximal existence time of the partial differential system. Moreover, if such existence time is finite and , then we will prove the partial differential system has solutions that blows up at space infinite.

Local strong solutions to the Cauchy problem of two-dimensional nonhomogeneous magneto-micropolar fluid equations with nonnegative density

We study the Cauchy problem of nonhomogeneous magneto-micropolar fluid system with zero density at infinity in the entire space [Formula: see text]. We prove that the system admits a unique local strong solution provided the initial density and the initial magnetic field decay not too slowly at infinity. In particular, there is no need to require any Choe–Kim type compatibility condition for the initial data.

Inverse scattering transform for the defocusing Ablowitz–Ladik system with arbitrarily large nonzero background

AbstractIn this work we develop the inverse scattering transform (IST) for the defocusing Ablowitz–Ladik (AL) equation with arbitrarily a large nonzero background at space infinity. The IST was developed in previous works under the assumption that the amplitude of the background satisfies a “small norm” condition . On the other hand, Ohta and Yang recently showed that the defocusing AL system, which is modulationally stable for , becomes unstable if , and exhibits discrete rogue wave solutions, some of which are regular for all times. Here, we construct the IST for the defocusing AL with , analyze the spectrum, and characterize the soliton and rational solutions from a spectral point of view.We formulate the direct and inverse problems by using a suitable uniformization variable, and pose the inverse problem as an RHP across a simple contour in the complex plane of the uniform variable. As a by‐product of the IST, we also obtain explicit soliton solutions, which are the discrete analog of the celebrated Kuznetsov–Ma, Akhmediev, Peregrine solutions, and which mimic the corresponding solutions for the focusing AL equation. Soliton solutions that are the analog of the dark soliton solutions of the defocusing AL equation in the case are also presented.

Scattering for Stochastic Nonlinear Schrödinger Equations

We study the scattering behavior of global solutions to stochastic nonlinear Schr\"odinger equations with linear multiplicative noise. In the case where the quadratic variation of the noise is globally finite and the nonlinearity is defocusing,we prove that the solutions scatter at infinity in the pseudo-conformal space and in the energy space respectively, including the energy-critical case. Moreover, in the case where the noise is large, non-conservative and has infinite quadratic variation, we show that the solutions scatter at infinity with high probability for all energy-subcritical exponents.

A supercritical scalar field equation with a forcing term

This paper is concerned with the elliptic problem for a scalar field equation with a forcing term −Δu+u=up+κμ in RN, where N≥2, p>1, κ>0 and μ is a Radon measure in RN with a compact support. Under a suitable integrability condition on μ, we give a complete classification of the existence and nonexistence of positive solutions decaying at the space infinity in the case of 1<p<pJL. Here pJL is the so-called Joseph-Lundgren exponent.