Hierarchy Of Integrable Systems Research Articles

Some more variety of analytical solutions to (2+1)-Bogoyavlensky-Konopelchenko equation

The goal of this study is to get analytical solutions to the Bogoyavlensky-Konopelchenko equation, which describes an interaction of a long-wavelength wave moving along the x-axis and a Riemann wave-form moving along the y-axis. The equation has been widely used in soliton theory, fluid dynamics, optics, biological systems, and differential geometry. It is a member of the Ablowitz-Kaup-Newell-Segur hierarchy of integrable systems. By exploiting the classical Lie symmetry approach, the equation is reduced to an ordinary differential equation. After solving the second similarity reduction, the authors derived a novel class of solutions for each case. Additionally, the derived solutions comprise some parameters, and various functions might be utilized to explore wave profiles like stationary, dark, and bright soliton, parabolic dark and bright soliton, and progressive nature. Solutions are compared with previous researches [–] to show novelty of results. Solutions can help with numerous applications in physics, such as fluid dynamics, plasma physics, and nonlinear optics.

Automorphic Lie algebras and corresponding integrable systems

We study automorphic Lie algebras and their applications to integrable systems. Automorphic Lie algebras are a natural generalisation of celebrated Kac-Moody algebras to the case when the group of automorphisms is not cyclic. They are infinite dimensional and almost graded. We formulate the concept of a graded isomorphism and classify sl(2,C) based automorphic Lie algebras corresponding to all finite reduction groups. We show that hierarchies of integrable systems, their Lax representations and master symmetries can be naturally formulated in terms of automorphic Lie algebras.

Elastic null curve flows, nonlinear C-integrable systems, and geometric realization of Cole-Hopf transformations

Elastic (stretching) flows of null curves are studied in three-dimensional Minkowski space. As a main tool, a natural type of moving frame for null curves is introduced, without use of the pseudo-arclength. This new frame is related to a Frenet null frame by a gauge transformation that belongs to the little group contained in the Lorentz group SO(2, 1) and provides an analog of the Hasimoto transformation (relating a parallel frame to a Frenet frame for curves in Euclidean space). The Cartan structure equations of the transformed frame are shown to encode a hereditary recursion operator giving a two-component generalization of the recursion operator of Burgers equation, as well as a generalization of the Cole-Hopf transformation. Three different hierarchies of integrable systems are obtained from the various symmetries of this recursion operator. The first hierarchy contains two-component Burgers-type and nonlinear Airy-type systems; the second hierarchy contains novel quasilinear Schrödinger-type (NLS) systems; and the third hierarchy contains semilinear wave equations (in two-component system form). Each of these integrable systems is shown to correspond to a geometrical flow of a family of elastic null curves in three-dimensional Minkowski space.

Explicit Lump Solitary Wave of Certain Interesting (3+1)-Dimensional Waves in Physics via Some Recent Traveling Wave Methods.

This study investigates the solitary wave solutions of the nonlinear fractional Jimbo–Miwa (JM) equation by using the conformable fractional derivative and some other distinct analytical techniques. The JM equation describes the certain interesting (3+1)-dimensional waves in physics. Moreover, it is considered as a second equation of the famous Painlev’e hierarchy of integrable systems. The fractional conformable derivatives properties were employed to convert it into an ordinary differential equation with an integer order to obtain many novel exact solutions of this model. The conformable fractional derivative is equivalent to the ordinary derivative for the functions that has continuous derivatives up to some desired order over some domain (smooth functions). The obtained solutions for each technique were characterized and compared to illustrate the similarities and differences between them. Profound solutions were concluded to be powerful, easy and effective on the nonlinear partial differential equation.

Integrable systems with BMS3 Poisson structure and the dynamics of locally flat spacetimes

We construct a hierarchy of integrable systems whose Poisson structure corresponds to the BMS3 algebra, and then discuss its description in terms of the Riemannian geometry of locally flat spacetimes in three dimensions.The analysis is performed in terms of two-dimensional gauge fields for islleft(2,mathrm{mathbb{R}}right) , being isomorphic to the Poincaré algebra in 3D. Although the algebra is not semisimple, the formulation can still be carried out à la Drinfeld-Sokolov because it admits a nondegenerate invariant bilinear metric. The hierarchy turns out to be bi-Hamiltonian, labeled by a nonnegative integer k, and defined through a suitable generalization of the Gelfand-Dikii polynomials. The symmetries of the hierarchy are explicitly found. For k ≥ 1, the corresponding conserved charges span an infinite-dimensional Abelian algebra without central extensions, so that they are in involution; while in the case of k = 0, they generate the BMS3 algebra. In the special case of k = 1, by virtue of a suitable field redefinition and time scaling, the field equations are shown to be equivalent to the ones of a specific type of the Hirota-Satsuma coupled KdV systems. For k ≥ 1, the hierarchy also includes the so-called perturbed KdV equations as a particular case. A wide class of analytic solutions is also explicitly constructed for a generic value of k.Remarkably, the dynamics can be fully geometrized so as to describe the evolution of spacelike surfaces embedded in locally flat spacetimes. Indeed, General Relativity in 3D can be endowed with a suitable set of boundary conditions, so that the Einstein equations precisely reduce to the ones of the hierarchy aforementioned. The symmetries of the integrable systems then arise as diffeomorphisms that preserve the asymptotic form of the spacetime metric, and therefore, they become Noetherian. The infinite set of conserved charges is then recovered from the corresponding surface integrals in the canonical approach.

The partition function of the Bures ensemble as the τ-function of BKP and DKP hierarchies: continuous and discrete

The relationship between matrix integrals and integrable systems was revealed more than 20 years ago. As is known, matrix integrals over a Gaussian ensemble used in random matrix theory could act as the τ-function of several hierarchies of integrable systems. In this article, we will show that the time-dependent partition function of the Bures ensemble, whose measure has many interesting geometric properties, could act as the τ-function of BKP and DKP hierarchies. In addition, if discrete time variables are introduced, then this partition function could act as the τ-function of discrete BKP and DKP hierarchies. In particular, there are some links between the partition function of the Bures ensemble and Toda-type equations.

Paraconformal structures, ordinary differential equations and totally geodesic manifolds

We construct point invariants of ordinary differential equations of arbitrary order that generalise the Tresse and Cartan invariants of equations of order two and three, respectively. The vanishing of the invariants is equivalent to the existence of a totally geodesic paraconformal structure which consists of a paraconformal structure, an adapted G L ( 2 , R ) -connection and a two-parameter family of totally geodesic hypersurfaces on the solution space. The structures coincide with the projective structures in dimension 2 and with the Einstein–Weyl structures of Lorentzian signature in dimension 3. We show that the totally geodesic paraconformal structures in higher dimensions can be described by a natural analogue of the Hitchin twistor construction. We present a general example of Veronese webs that generalise the hyper-CR Einstein–Weyl structures in dimension 3. The Veronese webs are described by a hierarchy of integrable systems.

Bilinear Form and Two Bäcklund Transformations for the (3+1)-Dimensional Jimbo-Miwa Equation

With Bell polynomials and symbolic computation, this paper investigates the (3+1)-dimensional Jimbo-Miwa equation, which is one of the equations in the Kadomtsev-Petviashvili hierarchy of integrable systems. We derive a bilinear form and construct a bilinear Bäcklund transformation (BT) for the (3+1)-dimensional Jimbo-Miwa equation, by virtue of which the soliton solutions are obtained. Bell-polynomial-typed BT is also constructed and cast into the bilinear BT.

Left-Symmetric Algebras in Hydrodynamics

Left-Symmetric algebras are shown to appear naturally in integrable hydrodynamical systems. First, to a data a Left-Symmetric algebra and an operator of strong deformation on it is attached an infinite commuting hierarchy of integrable systems of hydrodynamical type in 1+1−d. Second, this picture (without deformation) is embedded into an infinite-component integrable hydrodynamic chain.

Integrable mappings for noncommutative objects

The method of integrable mappings is generalized to the noncommutative case. Hierarchies of integrable systems corresponding to the noncommutative Darboux-Toda substitution in the two-dimensional spaces and superspaces are constructed.

Integrable mappings and hierarchies in the (2|2) superspace

We propose the supersymmetric generalization of the Darboux transformation in the ( 2 2 ) superspace and construct corresponding to it hierarchy of integrable systems which, at the reduction to the ( 1 2 ) superspace, possess manifest N = 2 supersymmetry.

Two-dimensional superintegrable mappings and integrable hierarchies in the (2|2) superspace

The formalism of integrable mappings is applied to the problem of constructing hierarchies of (1 + 2) dimensional integrable systems in the (2|2) superspace. We find new supersymmetric integrable mappings and corresponding to them new hierarchies of integrable systems which, at the reduction to the (1|2) superspace, possess N = 2 supersymmetry. The general formulae obtained for the hierarchies are used to explicitly derive their first nontrivial equations possessing a manifest N = 2 supersymmetry. New bosonic substitutions and hierarchies are obtained from the supersymmetric counterparts in the bosonic limit.

Moment problem of Hamburger, hierarchies of integrable systems, and the positivity of tau-functions

Moment problem of Hamburger, hierarchies of integrable systems, and the positivity of tau-functions

A simple scheme for generating nonisospectral flows from the zero curvature representation

A simple method is described to construct from the zero curvature representation nonisospectral ( λ t = λ n , n⪖0), λ is the spectral parameter) flows associated with a given isospectral ( λ t =0) hierarchy of integrable systems. Moreover further properties are exhibited. This method consists of searching for an initial vector field determined by a key equation provided that the characteristic operator equation has solutions. We demonstrate the effectiveness of our method by applications to three hierarchies of integrable systems as particular examples.