Journal of Nonlinear Mathematical Physics

We attempt to reconstruct the irreducible unitary representations of the Banach Lie group U0(H) of all unitary operators U on a separable Hilbert space H for which U - I is compact, originally found by Kirillov and Ol'shanskii, through constrained quantization of its coadjoint orbits. For this purpose...

A relativistic two-particle system with time-asymmetric scalar and vector interactions in the two-dimensional space-time is considered within the frame of the front form of dynamics using the dynamical symmetry approach. The mass-shell equation may be represented in terms of the nonlinear canonical realization...

We describe some recent results for a class of nonlinear hydrodynamical approximation models where the geometric approach gives insight into a variety of aspects. The main contribution concerns analytical results for Euler equations on the diffeomorphism group of the circle for which the inertia operator...

In this paper we are interested in developments of the elliptic functions of Jacobi. In particular a trigonometric expansion of the classical theta functions introduced by the author (Algebraic methods and q-special functions, C.R.M. Proceedings and Lectures Notes, A.M.S., vol 22, Providence, 1999, 53-57)...

Heisenberg motion equations in Quantum mechanics can be put into the Hamilton form. The difference between the commutator and its principal part, the Poisson bracket, can be accounted for exactly. Canonical transformations in Quantum mechanics are not, or at least not what they appear to be; their properties...

A two component vector generalization of the Sch ?afer-Wayne short pulse equation is derived. It describes propagation of ultra-short pulses in optical fibers with Kerr nonlinearity beyond the slowly varying envelope approximation and takes into account the effects of anisotropy and polarization. We...

In this paper, we give a unified construction of the recursion operators from the Lax representation for three integrable hierarchies: Kadomtsev–Petviashvili (KP), modified Kadomtsev–Petviashvili (mKP) and Harry Dym under n-reduction. This shows a new inherent relationship between them. To illustrate...

In this paper, we study the geodesic completeness of nondegenerate submanifolds in semi-Euclidean spaces by extending the study of Beem and Ehrlich [1] to semi-Euclidean spaces. From the physical point of view, this extend may have a significance that a semi-Euclidean space contains more variety of Lorentzian...

We classify (up to an isomorphism in the category of affine groups) the complex crystallographic groups generated by reflections and such that d, its linear part, is a Coxeter group, i.e., d is generated by "real" reflections of order 2.

An application of approximate transformation groups to study dynamics of a system with distinct time scales is discussed. The utilization of the Krylov–Bogoliubov–Mitropolsky method of averaging to find solutions of the Lie equations is considered. Physical illustrations from the plasma kinetic theory...

For all almost affine (hyperbolic) Lie superalgebras, the defining relations are computed in terms of their Chevalley generators.

We answered an old question: does there exist a mechanical system with 3 degrees of freedom, except for the Coulomb system, which has 6 first integrals generating the Lie algebra 𝔬(4) by means of the Poisson brackets? A system which is not centrally symmetric, but has 6 first integrals generating Lie...

The Singular Manifold Method (SMM) is applied to an equation in 2 + 1 dimensions [13] that can be considered as a generalization of the sine-Gordon equation. SMM is useful to prove that the equation has two Painlevé branches and, therefore, it can be considered as the modified version of an equation...

We introduce the notion of a ghost characteristic for nonlocal differential equations. Ghosts are essential for maintaining the validity of the Jacobi identity for the charateristics of nonlocal vector fields.

The paper presents a survey of some new results concerning the approach to construction of explicit solutions for nonlinear evolution equations du/dt = F[u], proposed in [1, 2].

Effects of geometric constraints on a steady flow potential are described by an elliptic- hyperbolic generalization of the harmonic map equations. Sufficient conditions are given for global triviality.

We construct all partial Noether operators corresponding to a partial agrangian for a system with two degrees of freedom. Then all the first integrals are obtained explicitly by utilizing a Noether-like theorem with the help of the partial Noether operators. We show how the first integrals can be constructed...

By the use of geometric methods for linearizing systems of second-order cubically semi-linear ordinary differential equations and the conditional linearizability of third-order quintically semi-linear ordinary differential equations, we extend to the fourth-order by differentiating the third-order conditionally...

The methods of [vdP-Sa, vdP1, vdP2] are applied to the fourth Painlevéequation. One obtains a Riemann–Hilbert correspondence between moduli spaces of rank two connections on ℙ1 and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto–Painlevé varieties...

We reduce planar measure-preserving rational maps over finite fields, and study their discrete dynamics. We show that application to the orbit analysis over these fields of the Hasse-Weil bound for the number of points on an algebraic curve gives a strong indication of the existence of an integral for...

An integrable discretization of an inhomogeneous Volterra lattice is introduced. Some aspects of dynamics of a soliton affected by a random force are discussed.

In this paper we apply truncated Painleve expansions to the Lax pair of a PDE to derive gauge Backlund transformations of this equation. It allows us to construct an algorithmic method to derive solutions by starting from the simplest one. Actually, we use this method to obtain an infinite set of lump...

This article summarizes a new, direct approach to the determination of the expansion into spherical harmonics of the exponential ei(x|y) with x, y Rd . It is elementary in the sense that it is based on direct computations with the canonical decomposition of homogeneous polynomials into harmonic components...

We consider the discretization of Darboux integrable equations. For each of the integrals of a Laine equation we constructed either a semi-discrete equation which has that integral as an n-integral, or we proved that such an equation does not exist. It is also shown that all constructed semi-discrete...

In the famous 1910 “cinq variables” paper Cartan showed in particular that for maximally nonholonomic rank 2 distributions in ℝ5 with non-zero covariant binary biquadratic form the dimension of the pseudo-group of local symmetries does not exceed 7 and among such distributions he described the one-parametric...

A problem of finding point symmetries of controlled systems is discussed, basic theorems and algorithms are formulated. The application to some problems of flight dynamics is suggested.

We present a method devised by Jacobi to derive Lagrangians of any second-order differential equation: it consists in finding a Jacobi Last Multiplier. We illustrate the easiness and the power of Jacobi's method by applying it to several equations, including a class of equations recently studied...

We present a 3×3 Riemann-Hilbert problem formalism for the initial-boundary value problem of the two-component nonlinear Schrödinger (2-NLS) equation on the half-line. And we also get the Dirichlet to Neuemann map through analysising the global relation in this paper.

It is shown that the 2-discrete dimensional Lotka-Volterra lattice, the two dmensional Toda lattice equation and the recent 2-discrete dimensional Toda lattice equation of Santini et al can be obtained from a 2-discrete 2-continuous dimensional Lotka-Volterra lattice.

The discrete group generated by reflections of the sphere, or the Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite...

A huge family of separable potential perturbations of integrable billiard systems and the Jacobi problem for geodesics on an ellipsoid is given through the Appell hypegeometric functions F4 of two variables, leading to an interesting connection between two classical theories: separability in Hamilton­Jacobi...

In this paper, we establish the connection between the quantized W-algebra of sl(2, 1) and quantum parafermions of Uq(^sl(2)) that a shifted product of the two quantum parafermions of Uq(^sl(2)) generates the quantized W-algebra of sl(2, 1).

This report is devoted to generalization of the equivalence transformations. Let a system of differential equations be given. Almost all systems of differential equations have arbitrary elements: arbitrary functions or arbitrary constants.

In this paper, we first obtain Wronskian solutions to the Bäcklund transformation of the Leznov lattice and then derive the coupled system for the Bäcklund transformation through Pfaffianization. It is shown the coupled system is nothing but the Bäcklund transformation for the coupled Leznov lattice...

We give a complete description of the Darboux and Liouville integrability of a general Rayleigh-Duffing oscillator through the characterization of its polynomial first integrals, Darboux polynomials and exponential factors.

We classify quadratic Poisson structures on so (4) and e(3), which have the same foliations by symplectic leaves as canonical Lie-Poisson tensors. The separated variables for some of the corresponding bi-integrable systems are constructed.

In a recent study of Landau-Ginzburg model of string field theory by Gaiotto, Moore and Witten, there appears a type of perturbed Cauchy-Riemann equation, i.e. the ζ-instanton equation. Solutions of ζ-instanton equation have degenerate asymptotics. This degeneracy is a severe restriction for obtaining...

It is known that the 6 models of Bianchi class A have no periodic solutions. In this article we provide a new, direct, unified and easier proof of this result.

An approach to the Painlevé analysis of fourth-order ordinary differential equations is presented. Some fourth-order ordinary differential equations which pass the Painlevé test are found.

We ssuggest an effective method for reducing Yang-Mills equations to systems of ordinary differential equations. With the use of this method, we construct wide families of new exact solutions of the Yang-Mills equations. Analysis of the solutions obtained shows that they correspond to conditional symmetry...

We provide closed form solutions for an equation which describes the transport of turbulent kinetic energy in the framework of a turbulence model with a single equation.

Based on the Lax triple (Bm, Bn, L) of the BKP and CKP hierarchies, we derive the nonlinear evolution equations from the generalized Lax equation. The solutions of some evolution equations are presented, such as soliton and rational solutions.

Materials with memory are here considered. The introduction of the dependence on time not only via the present, but also, via the past time represents a way, alterntive to the introduction of possible non linearities, when the physical problem under investigation cannot be suitably described by any linear...

In this article we present a Lagrangian representation for evolutionary systems with a Hamiltonian structure determined by a differential-geometric Poisson bracket of the first order associated with metrics of constant curvature. Kaup-Boussinesq system has three local Hamiltonian structures and one nonlocal...

We find a numerically-analytical solution of a boundary problem for the third-order partial differential equation, which describes the mass and heat transfer in active media.

We construct a deformation quantization for two cases of configuration spaces: the multiplicative group of positive real numbers R+ and the circle S1 . In these cases we define the momenta using the Fourier transform. Using the identification of symbols of quantum observables -- real functions on the...

The dependence on time of the moments of the one-soliton KdV solutions is given by Bernoulli polynomials. Namely, we prove the formula R x n sech 2 (x − t) dx = 2 π n (−i) n Bn ( 1 2 + t π i) , expressing the moments of the one-soliton function sech 2 (x−t) in terms of the Bernoulli polynomials...

We present a simple and general method for calculation of finite­gap elliptic solutions of the KdV equation as an isospectral deformation of Schrödinger potential based on their representation by rational functions of the elliptic Weierstrass functions.

We describe the problem of finding a harmonic map between noncompact manifold. Given some sufficient conditions on the domain, the target and the initial map, we prove the existence of a harmonic map that deforms the given map.

Lie point symmetry group classification of a scalar stochastic differential equation (SDE) with one-dimensional Brownian motion is presented. First we prove that the admitted symmetry group is at most three-dimensional. Then the classification is carried out with the help of Lie algebra realizations...

Separable coordinate systems are introduced in complex and real four-dimensional flat spaces. We use maximal Abelian subgroups to generate coordinate systems with a maximal number of ignorable variables. The results are presented (also graphically) in terms of subgroup chains. Finally the explicit solutions...

The inhomogeneity of the media or the external forces usually destroy the integrability of a system. We propose a systematic construction of a class of quantum models, which retains their exact integrability inspite of their explicit inhomogeneity. Such models include variable mass sine-Gordon model,...

We consider the rational potentials of the one-dimensional mechanical systems, which have a family of periodic solutions with the same period (isochronous potentials). We prove that up to a shift and adding a constant all such potentials have the form U(x) = 1 2 2 x2 or U(x) = 1 8 2 x2 + c2 x-2 .

We give a short review of recent results on L2 -cohomology of infinite configuration spaces equipped with Poisson measures.

By computing certain cohomology of Vect(M ) of smooth vector fields we prove that on 1-dimensional manifolds M there is no quantization map intertwining the action of non-pro jective embeddings of the Lie algebra sl(2) into the Lie algebra Vect(M ). Contrariwise, for pro jective embeddings sl(2)-equivariant...

The similarity solution to Prandtl’s boundary layer equations for two-dimensional and radial flows with vanishing or constant mainstream velocity gives rise to a third-order ordinary differential equation which depends on a parameter ?. For special values of ? the third-order ordinary differential equation...

An overview is presented of recent progress on the relation between the pressure at the bottom of the flat water bed and the elevation of the free water boundary within the context of the one-dimensional, irrotational water wave problem. We present five different approaches to this problem. All are compared...

The purpose of this work is to perform group classification of a coupled system of partial differential equations (PDEs) modeling a flow in collapsible tubes. This system of PDEs contains unknown functions of the dependent variables whose forms are specified via the classification with respect to subalgebras...

In a recent paper we introduced a new 2 + 1-dimensional non-isospectral extension of the Volterra lattice hierarchy, along with its corresponding hierarchy of underlying linear problems. Here we consider reductions of this lattice hierarchy to hierarchies of discrete equations, which we obtain once again...

This paper investigates the nature of particle collisions for three-soliton solutions of the Korteweg-de Vries (KdV) equation by describing mathematically the interaction of soliton particles and generation of ghost particle radiation. In particular, it is proven that a collision between any two soliton...

From the action principle, the quantum dynamical equation is obtained both relativistically and gauge invariant, which is analogous to the Dirac equation and describes behaviour of an arbitrary number of self-acting charged particles. It is noted that solutions of this equation are indicative of the...

This paper investigates symmetries of autonomous ordinary stochastic differential equations. Change of time includes the stochastic process itself, and is uniquely determined by the transformation of the spatial variable. As a particular feature, the time change by an admitted Lie symmetry group may...

Explicit formulas for the Gauss decomposition of elliptic Cauchy type matrices are derived in a very simple way. The elliptic Cauchy identity is an immediate corollary.

By using gauge transformations, we manage to obtain new solutions of (2 + 1)-dimensional Kadomtsev–Petviashvili (KP), Kaup–Kuperschmidt (KK) and Sawada–Kotera (SK) equations from nonzero seeds. For each of the preceding equations, a Galilean type transformation between these solutions u2 and the previously...

In this paper, a mixed Kuper-CH-HS equation by a Kupershmidt deformation is introduced and its integrable properties are studied. Moreover, that the equation can be viewed as a constraint Hamiltonian flow on the coadjoint orbit of Neveu-Schwarz superalgebra is shown.

A systematic investigation to derive nonlinear lattice equations governed by partial difference equations (PΔΔE) admitting specific Lax representation is presented. Further it is shown that for a specific value of the parameter the derived nonlinear PΔΔE's can be transformed into a linear PΔΔE's...

Negatons are a solution class with the following characteristic properties: They consist of solitons which are organized in groups. Solitons belonging to the same group are coupled in the sense that they drift apart from each other only logarithmically. The groups themselves rather behave like particles....

In the previous two parts of this series of papers, we introduced and studied a large class of analytic difference operators admitting reflectionless eigenfunctions, focusing on algebraic and function-theoretic features in the first part, and on connections with solitons in the second one. In this third...

Exactly integrable systems connected to semisimple algebras of second rank with an arbitrary choice of grading are presented in explicit form. General solutions of these systems are expressed in terms of matrix elements of two fundamental representations of the corresponding semisimple groups.

Under the Neumann constraints, each equation of the KdV hierarchy is decomposed into two finite dimensional systems, including the well-known Neumann model. Like in the case of the Bargmann constraint, the explicit Lax representations are deduced from the adjoint representation of the auxiliary spectral...

The generalization of parasupersymmetric quantum mechanics generated by an arbitrary number of parasupercharges and characterized by an arbitrary order of paraquantization is given. The relations for parasuperpotentials are obtained. It is shown that parasuperpotentials can be explicitly expressed via...

By using the extended Harrison and Estabrook's differential forms approach, in this paper, we investigate the Lie symmetries of the continuous and discrete dispersive long waves system, respectively. Based on this method, two closed ideals written in terms of a set of differential forms are constructed...

A simple method for calculating finite-gap elliptic potentials and corresponding spectra of the one-dimensional Schrödinger operator which is based on a general representation of the potentials in a form of rational functions of the Weierstrass function and trace formulae is proposed. It is shown that...

An N = (2|2) superfield formulation of the N = (1|1) supersymmetric Toda lattice hierarchy is proposed, and its five real forms are presented.

Renormalization group flow equations for scalar 4 are generated using smooth smearing functions. Numerical results for the critical exponent in d = 3 are caculated by polynomial truncation of the blocked potential. It is shown that the covergence of with the order of truncation can be improved by fine...

In this paper we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of integrals of motion. The proposed algorithm is simple, straightforward and...

The note provides the relation between symmetries and first integrals of Itô stochastic differential equations and symmetries of the associated Kolmogorov Backward Equation (KBE). Relation between the symmetries of the KBE and the symmetries of the Kolmogorov forward equation is also given.

We discuss a method of solving nth order scalar ordinary differential equations by extending the ideas based on the Prelle-Singer (PS) procedure for second order ordnary differential equations. We also introduce a novel way of generating additional integrals of motion from a single integral. We illustrate...

Reductions for systems of ODEs integrable via the standard factorization method (the Adler-Kostant-Symes scheme) or the generalized factorization method, developed by the authors earlier, are considered. Relationships between such reductions, operator Yang-Baxter equations, and some kinds of non-associative...

It is well known that the celebrated Lorenz system has an attractor such that every orbit ends inside a certain ellipsoid in forward time. We complement this result by a new phenomenon and by a new interpretation. We show that “infinity” is a global repeller for a set of parameters wider than that usually...

We present an explicit representation of an N-fold Darboux transformation T̃N for the short pulse equation, by the determinants of the eigenfunctions of its Lax pair. In the course of the derivation of T̃N, we show that the quasi-determinant is avoidable, and it is contrast to a recent paper (J. Phys....

We study the Hamiltonian structure of the gauge symmetry breaking and enhancment. After giving a general discussion of these phenomena in terms of the constrained phase space, we perform a canonical analysis of the Grassmannian nonlinear sigma model coupled with Chern-Simons term, which contains a free...

In this paper we consider a class of equations that model diffusion. For some of these equations nonclassical potential symmetries are derived by using a modified system approach. These symmetries allow us to increase the number of exact known solutions. These solutions are unobtainable from classical...

It is well-known that symmetry properties are extremely important for choosing differential equations which can be suitable for description of real physical processes. We present functional bases of second-order differential invariants for various representations of some extensions of the Poincaré group...

We analyze travelling solitary wave solutions in the Barbi-Cocco-Peyrard and in a simplified version of the Cocco-Monasson models of nonlinear DNA dynamics. We identify conditions to be satisfied by parameters for such solutions to exist, and provide first order ODEs whose solutions give the required...

We consider the general Liénard-type equation u¨=∑k=0nfku˙k for n ≥ 4. This equation naturally admits the Lie symmetry ∂∂t. We completely characterize when this equation admits another Lie symmetry, and give an easily verifiable condition for this on the functions f0,..., fn. Moreover, we give an equivalent...

The simple model of the non-linear DNA dynamics [4] is pursued in order to study the local untwisting of DNA double helix. It is shown how the advancing RNA polymerase may force the motion of the torsional solitary wave along DNA.

In the framework of a multidimensional superposition principle involving an analytical approach to nonlinear PDEs, a numerical technique for the analysis of soliton invari- ant manifolds is developed. This experimental methodology is based on the use of computer simulation data of soliton–perturbation...

We consider a constrained system of four rigid bodies located in axisymmetric potential and gyroscopic force fields and interacting by means of angular velocities. We describe an integrable case (not in Liouville sence!) when 12-dimensional phase space of the above system is fibered by the coisotropic...

We study conditions for a sequence of Appell polynomials to be a basis on a sequence space.

The notion of solvable structure is generalized in order to exploit the presence of an 𝔰𝔩(2, ℝ) algebra of symmetries for a kth-order ordinary differential equation ℰ with k > 3. In this setting, the knowledge of a generalized solvable structure for ℰ allows us to reduce ℰ to a family of second-order...

We study nonlocal symmetries of Plebański’s second heavenly equation in an infinite-dimensional covering associated to a Lax pair with a non-removable spectral parameter. We show that all local symmetries of the equation admit lifts to full-fledged nonlocal symmetries in the infinite-dimensional covering....

We associate Hamiltonian homological evolutionary vector fields – which are the non-Abelian variational Lie algebroids’ differentials – with Lie algebra-valued zero-curvature representations for partial differential equations.

In this paper we study Lie symmetries, Kac-Moody-Virasoro algebras, similarity reductions and particular solutions of two different recently introduced (2+1)-dimensional nonlinear evolution equations, namely (i) (2+1)-dimensional breaking soliton equation and (ii) (2+1)-dimensional nonlinear Schrödinger...

Solutions of the Schrödinger equation for a particle with a tensor-like mass are considered. It is shown that the problem of determination of the coherent states in this case is reduced to integration of the nonlinear system of ordinary differential equations.

We study exponentiability of the infinite polynomials with maximal degree 2 of cration and annihilation operators, which give a Fock Space-representation of the coplexification of the affine symplectic group.

Novel classes of dynamical systems are introduced, including many-body problems characterized by nonlinear equations of motion of Newtonian type (“acceleration equals forces”) which determine the motion of points in the complex plane. These models are solvable, namely their configuration at any time can...