Journal of Nonlinear Mathematical Physics

The Taub-NUT four-dimensional space-time can be obtained from Euclidean eight-dimensional one through a momentum map construction; the HKLR theorem [9] guarantees the hyperkähler structure of R8 descends to a hyperkähler structure in the Taub-NUT space. Here we present a detailed and fully explicit construction...

We present a theory of compatible differential constraints of a hydrodynamic hierarchy of infinite-dimensional systems. It provides a convenient point of view for studying and formulating integrability properties and it reveals some hidden structures of the theory of integrable systems. Illustrative...

In this paper we study certain aspects of the complete integrability of the Generlized Weierstrass system in the context of the Sinh-Gordon type equation. Using the conditional symmetry approach, we construct the Bäcklund transformation for the Generalized Weierstrass system which is determined by coupled...

Let A be an associative simple (central) superalgebra over C and L an invariant linear functional on it (trace). Let a at be an antiautomorphism of A such that (at )t = (-1)p(a) a, where p(a) is the parity of a, and let L(at ) = L(a). Then A admits a nondegenerate supersymmetric invariant bilinear form...

Considering a complex Lagrange space ([24]), in this paper the complex electromagnetic tensor fields are defined as the sum between the differential of the complex Liouville 1-form and the symplectic 2-form of the space relative to the adapted frames of the Chern–Lagrange complex nonlinear connection....

The paper is concerned with different classes of nonlinear Klein–Gordon and telegraph type equations with variable coefficients c(x)utt+d(x)ut=[a(x)ux]x+b(x)ux+p(x)f(u), where f(u) is an arbitrary function. We seek exact solutions to these equations by the direct method of symmetry reductions using...

We characterize and completely describe some types of separable potentials in twdimensional spaces, S2 [1]2 , of any (positive, zero or negative) constant curvature and either definite or indefinite signature type. The results are formulated in a way which applies at once for the two-dimensional sphere...

The reaction-diffusion system realizing a particular gauge fixing condition of the Jackiw­Teitelboim gravity is represented as a coupled pair of Burgers equations with positive and negative viscosity. For acoustic metric in the Madelung fluid representtion the space-time points where dispersion change...

A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma and B. Fuchssteiner, is proven to pass the Painlevé test for integrability well, and its 4×4 Lax pair with two spectral parameters is found. The results show that the Painlevé classification of coupled KdV equations by A. Karasu...

In this paper, based on the B Ãàacklund transformation for the supersymmetric MKdV equation, we propose a supersymmetric analogy for the second modified KdV equation. We also calculate its one-, two- and three-soliton solutions.

In earlier work, a Hamiltonian describing the classical motion of a particle moving in two dimensions under the combined influence of a perpendicular magnetic field and of a damping force proportional to the particle velocity, was indicated. Here we derive the quantum propagator for the Hamiltonian in...

We revisit an integrable (indeed, superintegrable and solvable) many-body model itroduced almost two decades ago by Gibbons and Hermsen and by Wojciechowski, and we modify it so that its generic solutions are all isochronous (namely, completely periodic with fixed period). We then show how this model...

We present q-discretizations of a second order differential equation in two independent variables that not only go to the differential counterpart as q goes to 1 but admit Moutard-Darboux transformations as well.

The recurrence coefficients of generalized Charlier polynomials satisfy a system of nolinear recurrence relations. We simplify the recurrence relations, show that they are related to certain discrete Painlevé equations, and analyze the asymptotic behaviour.

An analytical method to study the effect of viscosity of a medium and the wave number on sound propagation and sound attenuation numbers in circular ducts has been presented. The method is based on the variation of parameters of the solution corresponding to the case of inviscid acoustic waves in circular...

The aim of this paper is to study the formation of delta standing wave for a scalar conservation law with a linear flux function involving discontinuous coefficients. In order to deal with it, we approximate the discontinuous coefficients by piecewise affine ones and then apply the method of characteristics...

Every smooth second-order scalar ordinary differential equation (ODE) that is solved for the highest derivative has an infinite-dimensional Lie group of contact symmetries. However, symmetries other than point symmetries are generally difficult to find and use. This paper deals with a class of one-parameter...

We demonstrate, through the fourth Painlevé and the modified KdV equations, that the attempt at linearizing the mirror systems (more precisely, the equation satisfied by the new variable introduced in the indicial normalization) near movable poles can naturally lead to the Schlesinger transformations...

Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential equations. For the equations so obtained, first integrals are deduced...

The BBGKY's chain of quantum kinetic equations that describes the system of Bose particles interacting by delta potential is solved by the operator method with the help of nonlinear Schrödinger's equations. The solution of the chain is defined in terms of the Bethe ansatz.

In this article we develop the direct and inverse scattering theory of the Ablowitz-Ladik system with potentials having limits of equal positive modulus at infinity. In particular, we introduce fundamental eigensolutions, Jost solutions, and scattering coefficients, and study their properties.We also...

Quasiclassic method of solving of the Schrödinger equation with quadratic Hamiltonian is used to derive solutions of Klein-Fock equation for the particle in the constant magnetic field and the jumping magnetic field.

The Program SUBMODELS [1] is aimed to exhaust all possibilities derived from the symmetry of differential equations for construction of submodels (i.e., systems of equations of the reduced dimension) which describe classes of exact solutions for initial equations. In the frame of this Program, our paper...

We show that the generalized Yang–Mills system with Hamiltonian H=(p12+p22)/2+V(q1,q2) where V=1/2(aq12+bq22)+(cq14+2eq12q22+dq24)/4 is not completely integrable with Darboux first integrals.

Sine-Gordon (SG) models with position dependent mass or with isolated defects appear in many physical situations, ranging from fluxon or semi-fluxon in nonuniform Josephson junction to spin-waves in quantum spin chain with variable coupling or DNA solitons in the active promoter region. However such...

We study the class of the ordinary differential equations of the form ẍ + a2(t, x)ẋ2 + a1(t, x)ẋ + a0(t, x) = 0, that admit v = ∂x as λ-symmetry for some λ = α(t, x)ẋ + β(t, x). This class coincides with the class of the second-order equations that have first integrals of the form C(t) + 1/(A(t, x)ẋ...

In the calculus of variations, Lepage (n + 1)-forms are closed differential forms, representing Euler–Lagrange equations. They are fundamental for investigation of variational equations by means of exterior differential systems methods, with important applications in Hamilton and Hamilton–Jacobi theory...

A coupled Toda equation and its related equation are derived from 3-coupled bilinear equations. The corresponding Bäcklund transformation and nonlinear superposition formula are presented for the 3-coupled bilinear equations. As an application of the results, solition solutions are derived. Besides,...

The chain of discrete transformation equations is resolved in explicit form. The new found form of solution alow to solve the problem of interrupting of the chain in the most strigtforward way. More other this form of solution give a guess to its generalization on the case of arbitrary semisimple algebra...

In this article, we focus on left-invariant pseudo-Einstein metrics on Lie groups. To begin with, we give some examples of pseudo-Einstein metrics on Lie groups. Also we calculate the Levi-civita connection, and then Ricci tensor associated with left-invariant pseudo-Riemannian metrics on the unimodular...

We consider a long­range homogeneous chain where the local variables are the geerators of the direct sum of N e(3) interacting Lagrange tops. We call this classical integrable model rational "Lagrange chain" showing how one can obtain it starting from su(2) rational Gaudin models. Moreover we construct...

We present a scenario concerning the existence of a large class of reflectionless seladjoint analytic difference operators. In order to exemplify this scenario, we summarize our results on reflectionless self-adjoint difference operators of relativistic CalogerMoser type.

Hybrid Ermakov-Painlevé II-IV systems are introduced here in a unified manner. Their admitted Ermakov invariants together with associated canonical Painlevé equations are used to establish integrability properties.

In this paper, we mainly investigate an equivalent form of the constrained modified KP hierarchy: the bilinear identities. By introducing two auxiliary functions ρ and σ, the corresponding identities are written into the Hirota forms. Also, we give the explicit solution forms of ρ and σ.

We investigate weakly coupled semilinear parabolic systems in unbounded domains in R2 or R3 with polynomial nonlinearities. Three sufficient conditions are presented to ensure the stability of the zero solution with respect to non-negative H2 -perturbations.

It is shown that in the case of multicomponent integrable systems connected with algebras An, the discrete transformation T possesses the fine structure and can be represented in the form T = Tli i , where Ti are n commuting basis discrete transfomations and li are arbitrary natural numbers. All the...

We consider an important class of deformations of the genus zero bihamiltonian struture defined on the loop space of semisimple Frobenius manifolds, and present results on such deformations at the genus one and genus two approximations.

We study a potential introduced by Darboux to describe conjugate nets, which within the modern theory of integrable systems can be interpreted as a τ-function. We investigate the potential using the nonlocal ∂-dressing method of Manakov and Zakharov and we show that it can be interpreted as...

A counter-intuitive result of Gauss (formulae (1.6), (1.7) below) is made less mysterious by virtue of being generalized through the introduction of an additional parameter.

This paper studies the canonical symmetric connection ∇ associated to any Lie group G. The salient properties of ∇ are stated and proved. The Lie symmetries of the geodesic system of a general linear connection are formulated. The results are then applied to ∇ in the special case where the Lie algebra...

Formalism of differential forms is developed for a variety of Quantum and noncommutative situations.

We draw attention to the connections recently established by others between the classical integrable KdV and KP hierarchies in 1+1 and 2+1 dimensions respectively and the matrix models which relate to the partition functions of 2-dimensional (1 + 1 dimensional) quantum gravity. The symmetries of the...

In this paper, we give the classification of (n + 2)-dimensional metric n-Lie algebras in terms of some facts about n-Lie algebras.

Sklyanin’s formula provides a set of canonical spectral coordinates on the standard Calogero-Moser space associated with the quiver consisting of a vertex and a loop. We generalize this result to Calogero-Moser spaces attached to cyclic quivers by constructing rational functions that relate spectral...

In this paper we present the full classification of symmetry-invariant solutions for the Gibbons–Tsarev equation. Then we use these solutions to construct explicit expressions for two-component reductions of Benney’s moments equations, to get solutions of Pavlov’s equation, and to find integrable reductions...

If we are given a smooth differential operator in the variable x R/2Z, its normal form, as is well known, is the simplest form obtainable by means of the Diff(S1 )-group action on the space of all such operators. A versal deformation of this operator is a normal form for some parametric infinitesimal...

We consider a representation of canonical commutation relations (CCR) appearing in a (non-Abelian) gauge theory on a non-simply connected region in the two-dimensional Euclidean space. A necessary and sufficient condition for the representation to be equivalent to the Schrödinger representation of CCR...

We introduce the concept of natural Poisson bivectors, which generalizes the Benenti approach to construction of natural integrable systems on Riemannian manifolds and allows us to consider almost the whole known zoo of integrable systems in framework of bi-hamiltonian geometry.

In this paper, using the properties of hyperelliptic σ- and ℘- functions, ℘μν : = ∂μ∂ν log σ, we propose an algorithm to obtain particular solutions of the coupled nonlinear differential equations, such as a general (2+1)- dimensional breaking soliton equation and static Veselov-Novikov(SVN) equation,...

The Lie algebra gl(λ) with λ∈ℂ, introduced by B L Feigin, can be embedded into the Lie algebra of differential operators on the real line (see [7]). We give an explicit formula of the embedding of gl(λ) into the algebra Dλ of differential operators on the space of tensor...

We use Lax equations to define a scattering problem on an infinite elbow shaped line of the (x, t) plane. The evolution of scattering coefficients when the elbow is translated in the plane shows how convenient scannings may reconstruct the solution V (x, t) of the nonlinear equation associated to the...

We characterize the Liouvillian and analytic first integrals for the polynomial differential systems of the form x′ = a − (b + 1)x + x2y, y′ = bx − x2y, with a, b ∈ ℝ, called the Brusselator differential systems.

A class of symmetry transformations of a type originally introduced in a nonlinear optics context is used here to isolate an integrable Ermakov-Painlevé II reduction of a resonant NLS equation which encapsulates a nonlinear system in cold plasma physics descriptive of the uni-axial propagation of magneto-acoustic...

In this paper, we investigate some relations between Bernoulli numbers and Frobenius- Euler numbers, and we study the values for p-adic l -function.

The classical quantization of a Liénard-type nonlinear oscillator is achieved by a quantization scheme (M. C. Nucci. Theor. Math. Phys., 168:994–1001, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger equation. This method straightforwardly...

In this paper, we bring out the Lie symmetries and associated similarity reductions of the recently proposed (2+1) dimensional long dispersive wave equation. We point out that the integrable system admits an infinite-dimensional symmetry algebra along with Kac-Moody-Virasoro-type subalgebras. We also...

We define the deformation quantization in the Fedosov sense for a limit model of Taubes in white noise analysis.

We briefly derive the infinite set of multi-point correlation equations based on the Navier–Stokes equations for an incompressible fluid. From this we reconsider the previously derived set of Lie symmetries, i.e. those directly induced by the ones from classical mechanics and also new symmetries. The...

A simple algoritm for the inverse scattering approach to the Camassa-Holm equation is presented.

Let M be an odd-dimensional Euclidean space endowed with a contact 1-form ?. We investigate the space of symmetric contravariant tensor fields over M as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up of those vector fields that preserve the contact structure...

We summarize the most featured items characterizing the semi-discrete nonlinear Schrödinger system with background-controlled inter-site resonant coupling. The system is shown to be integrable in the Lax sense that make it possible to obtain its soliton solutions in the framework of properly parameterized...

For positive parameters a+ and a- the commuting difference operators exp(ia±d/dz) + exp(2z/a), acting on meromorphic functions f(z), z = x + iy, are formally self-adjoint on the Hilbert space H = L2 (R, dx). Volkov showed that they admit joint eigenfunctions. We prove that the joint eigenfunctions for...

We study two-dimensional triangular systems of Newton equations (acceleration = velocity-independent force) admitting three functionally independent quadratic intgrals of motion. The main idea is to exploit the fact that the first component M1(q1) of a triangular force depends on one variable only. By...

We compute the spectrum of the trigonometric Sutherland spin model of BCN type in the presence of a constant magnetic field. Using Polychronakos's freezing trick, we derive an exact formula for the partition function of its associated Haldane­Shastry spin chain.

Cartan described some of the finite dimensional simple Lie algebras and three of the four series of simple infinite dimensional vectorial Lie algebras with polynomial coefficients as prolongs, which now bear his name. The rest of the simple Lie algebras of these two types (finite dimensional and vectorial)...

We propose a simple procedure to identify the collective coordinate Q which is used to generate the isochronous Hamiltonian. The new isochronous Hamiltonian generates more and more isochronous oscillators, recursively.

Novikov superalgebras are related to the quadratic conformal superalgebras which correspond to the Hamiltonian pairs and play fundamental role in the completely integrable systems. In this note, we divide Novikov superalgebras into two types: N and S. Then we show that the Novikov superalgebras of dimension...

In this talk we introduce generalised Calogero­Moser models and demonstrate their integrability by constructing universal Lax pair operators. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H3, H4, and the dihedral group I2(m),...

We study local conservation laws and corresponding boundary conditions for the ptential Zabolotskaya-Khokhlov equation in (3+1)-dimensional case. We analyze an infinite Lie point symmetry group of the equation, and generate a finite number of conserved quantities corresponding to infinite symmetries...

It is shown that the system of two coupled Korteweg-de Vries equations passes the Painlevé test for integrability in nine distinct cases of its coefficients. The integrability of eight cases is verified by direct construction of Lax pairs, whereas for one case it remains unknown.

We consider the propagation of TE-polarized electromagnetic waves in cylindrical dielectric waveguides of circular cross section filled with lossless, nonmagnetic, and isotropic medium exhibiting a local Kerr-type dielectric nonlinearity. We look for axially-symmetric solutions and reduce the problem...

Stationary structures in a classical isotropic two-dimensional continuous Heisenberg ferromagnetic spin system are studied in the framework of the (2 + 1)-dimensional Landau­Lifshitz model. It is established that in the case of S(r, t) = S(r - vt) the Landau­Lifshitz equation is closely related to the...

The phenomenon of an implicit function which solves a large set of second order partial differential equations obtainable from a variational principle is explicated by the introduction of a class of universal solutions to the equations derivable from an arbitrary Lagrangian which is homogeneous of weight...

This is the second part of a series of papers dealing with an extensive class of anlytic difference operators admitting reflectionless eigenfunctions. In the first part, the pertinent difference operators and their reflectionless eigenfunctions are constructed from given "spectral data", in analogy with...

In this paper, we give the Lichnerowicz type formulas for statistical de Rham Hodge operators. Moreover, Kastler-Kalau-Walze type theorems for statistical de Rham Hodge operators on compact manifolds with (respectively without) boundary are proved.

We apply the averaging method to analyze spatio-temportal structures in nonlinear Schrödinger equations and thereby study the dynamics of quasi-one-dimensional collisionally inhomogeneous Bose–Einstein condensates with the scattering length varying periodically in space and crossing zero. Infinitely...

Here, we give the existence of analytical Cartesian solutions of the multi-component Camassa-Holm (MCCH) equations. Such solutions can be explicitly expressed, in which the velocity function is given by u = b(t) + A(t)x and no extra constraint on the dimension N is required. The advantage of our method...

The q-discrete two-dimensional Toda lattice equation with self-consistent sources is presented through the source generalization procedure. In addition, the Gramm- type determinant solutions of the system are obtained. Besides, a bilinear B ̈acklund transformation (BT) for the system is given.

Integrable systems are derived from inelastic flows of timelike, spacelike, and null curves in 2– and 3– dimensional Minkowski space. The derivation uses a Lorentzian version of a geometrical moving frame method which is known to yield the modified Korteveg-de Vries (mKdV) equation and the nonlinear...

Let be the space of Schrödinger operators in (1 + 1)-dimensions with periodic time-dependent potential. The action on 𝒮lin of a large infinite-dimensional reparametrization group SV with Lie algebra [8, 10], called the Schrödinger–Virasoro group and containing the Virasoro group, is proved to...

One of the most powerful methods for finding and solving integrable nonlinear partial differential equations is Hirota's bilinear method. The idea behind it is to make first a nonlinear change in the dependent variables after which multisoliton solutions of integrable systems can be expressed as polynomials...

Relying on recent advances in the theory of entropy solutions for nonlinear (strongly) degenerate parabolic equations, we present a direct proof of an L1 error estimate for viscous approximate solutions of the initial value problem for tw + div V (x)f(w) = A(w), where V = V (x) is a vector field, f =...

We consider the following spectral problem y1 y2 x = -w u v w y1 y2 M y1 y2 , (1) where u, v, w are smooth functions. It produces a hierarchy of evolution equations with an arbitrary function Am-1. This hierarchy includes the WKI [8] and Heiseberg [7] hierarchies by properly selecting the special function...

I analyze the one-dimensional, cubic Schrödinger equation with a nonlinearity constructed from the current density rather than, as is usual, from the charge density. A soliton solution is found, where the soliton moves only in one direction. Relation to the higher-dimensional Chern­Simons theory is indicated....

The article deals with local symmetries of the infinite-order jet space of C∞-smooth curves in ℝm+1 (m ≥ 1). Transformations under consideration are the most general possible: they need not preserve the distinction between dependent and independent variables and the order of derivatives may be arbitrarily...

Starting from known solutions of the functional Yang-Baxter equations, we construct a series of nonautonomous integrable recurrences, “median graphs”, and give their explicit solution.

So far, there are described in the literature two ways to superize the Liouville equation: for a scalar field (for N 4) and for a vector-valued field (analogs of the Leznov­ Saveliev equations) for N = 1. Both superizations are performed with the help of Neveu­Schwarz superalgebra. We consider another...

Explicit algebraic relations between the quantum integrals of the elliptic Calogero­ Moser quantum problems related to the root systems A2 and B2 are found.

We investigate the structure of certain types of subalgebras of Galilei algebras and the relationship between the conjugacies of these subalgebras under different groups of automorphisms.

We study a class of piecewise linear solutions to the inviscid Burgers equation driven by a linear forcing term. Inspired by the analogy with peakons, we think of these solutions as being made up of solitons situated at the breakpoints. We derive and solve ODEs governing the soliton dynamics, first for...

We discuss the interrelations between symmetry of an Ito stochastic differential equations (or systems thereof) and its integrability, extending in party results by R. Kozlov [J. Phys. A 43 (2010) & 44 (2011)]. Together with integrability, we also consider the relations between symmetries and reducibility...

Fundamental questions posed in classical genetics since early 20th century are still fundamental in today post genomic age. What has changed is the availability of huge amount of molecular genetics information on a broad spectrum of species and a more powerful and rich methodological approach, particularly...

The Singular Manifold Method is presented as an excellent tool to study a 2 + 1 dimensional equation in despite of the fact that the same method presents several problems when applied to 1 + 1 reductions of the same equation. Nevertheless these problems are solved when the number of dimensions of the...

In the paper we analyze the Kaldor­Kalecki model of business cycle. The time dlay is introduced to the capital accumulation equation according to Kalecki's idea of delay in investment processes. The dynamics of this model is represented in terms of time delay differential equation system. In the special...

The reduction by symmetry of the linear system of the self-dual Yang-Mills equations in four-dimensions under representatives of the conjugacy classes of subgroups of the connected part to the identity of the corresponding Euclidean group under itself is carried out. Only subgroups leading to systems...

Coupled KdV equations are deduced by considering the homogeneous manifold corresponding to the homogeneous Heisenberg subalgebra of the Loop group (L(S1 , SL(2, C)). Utilisation of Birkhoff decomposition and further subalgebra consideration leads to a new generalised form of Miura map and two sets of...

In this paper a three-dimensional system with five parameters is considered. For some particular values of these parameters, one finds known dynamical systems. The purpose of this work is to study some symmetries of the considered system, such as Lie-point symmetries, conformal symmetries, master symmetries...

In this article we give sufficient conditions on the scattering data of a defocusing or focusing Zakharov-Shabat system in order that its potential is square integrable. For a dense subset of integrable as well as square integrable potentials, we show that the scattering data actually satisfy these sufficient...

We determine the solutions of a nonlinear Hamilton-Jacobi-Bellman equation which arises in the modelling of mean-variance hedging subject to a terminal condition. Firstly we establish those forms of the equation which admit the maximal number of Lie point symmetries and then examine each in turn. We...