Higher Dimensional Field Research Articles

Interaction solutions between lump and other solitons of two classes of nonlinear evolution equations

Interaction solutions are obtained from (2+1)-dimensional BLMP equation and (3+1)-dimensional nonlinear evolution equation by using a direct method. The interaction solutions of the two equations presented that a lump appear from a soliton wave and swallowed by it later, they are all the completely non-elastic interactions that rare to see. It shows that a lump appeared from the one side of the soliton wave (a kinky wave or another solitary wave), and separate slowly from this side of the soliton wave, after reaching the maximum separation at \(t=0,\) the lump gradually walks upon the other side of the soliton wave and finally swallowed by the other side. The dynamical character shows in this work enriches the variety of the dynamics of higher-dimensional nonlinear wave field.

Explicit Form of the Hilbert Symbol on Polynomial Formal Module for Multidimensional Local Field. II

The paper presents an explicit formula for the Hilbert pairing between the Milnor K-group of a higher-dimensional local field and a polynomial formal module. This formula generalizes similar results for the one-dimensional case and the higher-dimensional case of multiplicative group. The case where the field and its first residue field have different characteristics, is considered. Bibliography: 13 titles.

Quasiperiodic waves, solitary waves and asymptotic properties for a generalized (3 + 1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation

Under investigation in this paper is a generalized (3 + 1)-dimensional variable-coefficient BKP equation, which can be used to describe the propagation of nonlinear waves in fluid mechanics and other fields. With the aid of binary Bell’s polynomials, an effective and straightforward method is presented to explicitly construct its bilinear representation with an auxiliary variable. Based on the bilinear formalism, the soliton solutions and multi-periodic wave solutions are well constructed. Furthermore, the tanh method and the tan method are employed to construct more traveling wave solutions of the equation. Finally, the asymptotic properties of the multi-periodic wave solutions are systematically analyzed to reveal the connection between periodic wave solutions and soliton solutions. It is interesting that the periodic waves tend to solitary waves under a limiting procedure. Our results can be used to enrich the dynamical behavior of higher-dimensional nonlinear wave fields.

Dualité pour les groupes de type multiplicatif sur certains corps de fonctions

Let K be the function field of a smooth projective curve X over a higher-dimensional local field k. We define Tate–Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of K coming from a closed point of X. In this note, we state and sketch the proof of an arithmetic duality theorem for Tate–Shafarevich groups of groups of multiplicative type over K (and more generally of some two-term complexes of tori over K).

Dynamical analysis of lump solution for the (2+1)-dimensional Ito equation

Exact kinky breather-wave solution, periodic breather-wave solution, and some lump solutions to the (2+1)-dimensional Ito equation are obtained by using an extended homoclinic test technique and Hirota bi-linear method with a perturbation parameter u0. Furthermore, a new non-linear phenomenon in the lump solution, is investigated and discussed. These interesting non-linear phenomena might provide us with useful information on the dynamics of higher-dimensional non-linear wave field.

Analyticity properties and asymptotic behavior of scattering amplitude in higher dimensional theories

The properties of the high energy behavior of the scattering amplitude of massive, neutral, and spinless particles in higher dimensional field theories are investigated. The axiomatic formulation of Lehmann, Symanzik, and Zimmermann (LSZ) is adopted. The analyticity properties of the causal, the retarded, and the advanced functions associated with the four point elastic amplitudes are studied. The analog of the Lehmann-Jost-Dyson representation is obtained in higher dimensional field theories. The generalized J-L-D representation is utilized to derive the t-plane analyticity property of the amplitude. The existence of an ellipse analogous to the Lehmann ellipse is demonstrated. Thus a fixed-t dispersion relation can be written down with a finite number of subtractions due to the temperedness of the amplitudes. The domain of analyticity of scattering amplitude in s and t variables is extended by imposing unitarity constraints. A generalized version of Martin’s theorem is derived to prove the existence of such a domain in D-dimensional field theories. It is shown that the amplitude can be expanded in a power series in t which converges for t<R, R being s-independent. The positivity properties of absorptive amplitudes are derived to prove the t-plane analyticity of amplitude. In the extended analyticity domain dispersion relations are written with two subtractions. The bound on the total cross section is derived from LSZ axioms without any extra ad hoc assumptions.

Jost–Lehmann–Dyson representation in higher dimensional field theories

Abstract The Jost–Lehmann–Dyson representation is derived for massive scalar field theories in higher spacetime dimensions, D > 4 , for the four point scattering amplitude. The representation is very crucial to investigate the analyticity properties of the amplitude. The axiomatic approach of Lehmann–Symanzik–Zimmermann is adopted to show the existence of such a representation. Consequently, a host of interesting results will follow from derivation of JLD representation such as proof of analyticity properties and asymptotic behavior of the amplitude.

Universal Signatures of Quantum Critical Points from Finite-Size Torus Spectra: A Window into the Operator Content of Higher-Dimensional Conformal Field Theories.

The low-energy spectra of many body systems on a torus, of finite size L, are well understood in magnetically ordered and gapped topological phases. However, the spectra at quantum critical points separating such phases are largely unexplored for (2+1)D systems. Using a combination of analytical and numerical techniques, we accurately calculate and analyze the low-energy torus spectrum at an Ising critical point which provides a universal fingerprint of the underlying quantum field theory, with the energy levels given by universal numbers times 1/L. We highlight the implications of a neighboring topological phase on the spectrum by studying the Ising* transition (i.e. the transition between a Z_{2} topological phase and a trivial paramagnet), in the example of the toric code in a longitudinal field, and advocate a phenomenological picture that provides qualitative insight into the operator content of the critical field theory.

Chow group of 0-cycles with modulus and higher-dimensional class field theory

One of the main results of this paper is a proof of the rank one case of an existence conjecture on lisse l-adic sheaves on a smooth variety over a finite field due to Deligne and Drinfeld. The problem is translated into the language of higher dimensional class field theory over finite fields, which describes the abelian fundamental group by Chow groups of zero cycles with moduli. A key ingredient is the construction of a cycle theoretic avatar of refined Artin conductor in ramification theory originally studied by Kazuya Kato.

The third homology of SL2 of fields with discrete valuation

For a field F with discrete valuation and residue field k we relate the third homology of SL2(F) with half-integral coefficients to the third homology of SL2(k) and a certain refined scissors congruence group of k. As an application, we obtain explicit calculations of the third homology of SL2 of certain higher-dimensional local fields in terms of scissors congruence groups, in the sense of Dupont and Sah, of the residue fields of the associated valuations.

Lump solitons and the interaction phenomena of them for two classes of nonlinear evolution equations

Lump solitons are obtained from (2+1)-dimensional Ito equation and (2+1)-dimensional CDGKS equation by using a direct method and the completely non-elastic interaction between a lump and a stripe of (2+1)-dimensional Ito equation is presented, which shows a lump soliton is drowned or swallowed by a stripe soliton. The direct method is based on Hirota bilinear form of the equations that have special solutions of quadratic function. The general formula of coordinates of the vertex and the amplitude of lump solitons are given. The dynamical character shows in this work enriches the variety of the dynamics of higher dimensional nonlinear wave field.

Abelian-by-central Galois groups of fields II: Definability of inertia/decomposition groups

This paper explores some first-order properties of commuting-liftable pairs in pro-ℓ abelian-by-central Galois groups of fields. The main focus of the paper is to prove that minimized inertia and decomposition groups of many valuations are first-order definable using a predicate for the collection of commuting-liftable pairs. For higher-dimensional function fields over algebraically closed fields, we show that the minimized inertia and decomposition groups of quasi-divisorial valuations are uniformly first-order definable in this language.

Kink degeneracy and rogue potential solution for the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

In this paper, we obtained the exact breather-type kink soliton and breather-type periodic soliton solutions for the (3 + 1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation using the extended homoclinic test technique. Some new nonlinear phenomena, such as kink and periodic degeneracies, are investigated. Using the homoclinic breather limit method, some new rational breather solutions are found as well. Meanwhile, we also obtained the rational potential solution which is found to be just a rogue wave. These results enrich the variety of the dynamics of higher-dimensional nonlinear wave field.

Conformal field theory out of equilibrium: a review

We provide a pedagogical review of the main ideas and results in non-equilibrium conformal field theory and connected subjects. These concern the understanding of quantum transport and its statistics at and near critical points. Starting with phenomenological considerations, we explain the general framework, illustrated by the example of the Heisenberg quantum chain. We then introduce the main concepts underlying conformal field theory (CFT), the emergence of critical ballistic transport, and the CFT scattering construction of non-equilibrium steady states. Using this we review the theory for energy transport in homogeneous one-dimensional critical systems, including the complete description of its large deviations and the resulting (extended) fluctuation relations. We generalize some of these ideas to one-dimensional critical charge transport and to the presence of defects, as well as beyond one-dimensional criticality. We describe non-equilibrium transport in free-particle models, where connections are made with generalized Gibbs ensembles, and in higher-dimensional and non-integrable quantum field theories, where the use of the powerful hydrodynamic ideas for non-equilibrium steady states is explained. We finish with a list of open questions. The review does not assume any advanced prior knowledge of conformal field theory, large-deviation theory or hydrodynamics.

Théorèmes de dualité pour les corps de fonctions sur des corps locaux supérieurs

Let K be the function field of a smooth projective curve X over a higher-dimensional local field k. We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of K coming from a closed point of X. We establish duality theorems between Tate-Shafarevich groups for finite groups schemes and for tori.

Inertial Steady 2D Vector Field Topology

AbstractVector field topology is a powerful and matured tool for the study of the asymptotic behavior of tracer particles in steady flows. Yet, it does not capture the behavior of finite‐sized particles, because they develop inertia and do not move tangential to the flow. In this paper, we use the fact that the trajectories of inertial particles can be described as tangent curves of a higher dimensional vector field. Using this, we conduct a full classification of the first‐order critical points of this higher dimensional flow, and devise a method to their efficient extraction. Further, we interactively visualize the asymptotic behavior of finite‐sized particles by a glyph visualization that encodes the outcome of any initial condition of the governing ODE, i.e., for a varying initial position and/or initial velocity. With this, we present a first approach to extend traditional vector field topology to the inertial case.

Dynamics of kinky wave for ( $$3+1$$ 3 + 1 )-dimensional potential Yu–Toda–Sasa–Fukuyama equation

Exact kinky breather-wave solutions for the (\(3\,{+}\,1\))-dimensional potential Yu–Toda–Sasa–Fukuyama equation are obtained by using extended homoclinic test technique. Based on the kinky breather-wave solution, rational breather-wave solution is generated by homoclinic breather limit method. Some new dynamical features of kinky wave are presented, including kink degeneracy, rational breather wave is drowned or swallowed up by kinky wave in the interaction between rational breather wave and kinky wave. These results enrich the variety of the dynamics of higher dimensional nonlinear wave field.

Defects in cohomological gauge theory and Donaldson–Thomas invariants

Donaldson-Thomas theory on a Calabi-Yau can be described in terms of a certain sixdimensional cohomological gauge theory. We introduce a certain class of defects in this gauge theory which generalize surface defects in four dimensions. These defects are associated with divisors and are defined by prescribing certain boundary conditions for the gauge fields. We discuss generalized instanton moduli spaces when the theory is defined with a defect and propose a generalization of Donaldson-Thomas invariants. These invariants arise by studying torsion free coherent sheaves on Calabi-Yau varieties with a certain parabolic structure along a divisor, determined by the defect. We discuss the case of the affine space as a concrete example. In this case the moduli space of parabolic sheaves admits an alternative description in terms of the representation theory of a certain quiver. The latter can be used to compute the invariants explicitly via equivariant localization. We also briefly discuss extensions of our work to other higher dimensional field theories.

Hasse principles for higher-dimensional fields

For rather general excellent schemes X, K. Kato deflned complexes of GerstenBloch-Ogus type involving the Galois cohomology groups of all residue flelds of X. For arithmetically interesting schemes, i.e., schemes over global or local flelds, or over their rings of integers, he developed a web of conjectures on some of these complexes, generalizing the fundamental exact sequence of Brauer groups for a global fleld. He proved these conjectures for low dimensions. We prove Kato’s conjecture (with inflnite coe‐cients) over number flelds. In particular it gives a Hasse principle for function flelds F over a number fleld K, involving the corresponding function flelds Fv over the completions Kv of K. We get a conditional result over global flelds K of positive characteristic, assuming resolution of singularities. This is unconditional for dim(X) • 3, due to recent results on resolution. There are also applications to other cases considered by Kato.

Reconstructing function fields from rational quotients of mod- $$\ell $$ ℓ Galois groups

In this paper, we develop the main step in the global theory for the mod-$\ell$ analogue of Bogomolov's program in birational anabelian geometry for higher-dimensional function fields over algebraically closed fields. More precisely, we show how to reconstruct a function field $K$ of transcendence degree $\geq 5$ over an algebraically closed field, up-to inseparable extensions, from the mod-$\ell$ abelian-by-central Galois group of $K$ endowed with the collection of mod-$\ell$ rational quotients.