Journal of Nonlinear Mathematical Physics

In this contribution, we describe the simplest, classical problem in water waves, and use this as a vehicle to outline the techniques that we adopt to analyse this particular approach to the derivation of soliton-type equations. The surprise, perhaps, is that such an apparently transparent set of equations...

We study the inverse problem for the so-called operators with energy depending potentials. In particular, we study spectral operators with quadratic dependence on the spectral parameter. The corresponding hierarchy of integrable equations includes the Kaup–Boussinesq equation. We formulate the inverse...

An analog of Gelfand-Levitan-Marchenko integral equations for multi- dimensional Delsarte transmutation operators is constructed by means of studying their differentiageometric structure based on the classical Lagrange identity for a formally conjugated pair of differential operators. An extension of...

Cheb-Terrab and Roche (J. Sym. Comp. 27 (1999), 501­519) presented what they termed a systematic algorithm for the construction of integrating factors for second order ordinary differential equations. They showed that there were instances of odinary differential equations without Lie point symmetries...

For basic discrete probability distributions, - Bernoulli, Pascal, Poisson, hypergemetric, contagious, and uniform, - q-analogs are proposed.

The subalgebras of the invariance algebra of equation 2u+(u2

The nonlinear integral equation P(x) = dyw(y)P(y)P(x + y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions Pn(x). For polynomial solutions, this nonlinear integral equation reduces to a finite set of coupled linear algebraic equations...

We consider some well-known partial differential equations that arise in Financial Mathematics, namely the Black–Scholes–Merton, Longstaff, Vasicek, Cox–Ingersoll–Ross and Heath equations. Our central aim is to discover any underlying connections taking into account the Lie remarkability property of...

We construct a family of integrable equations of the form vt = f(v; vx; vxx; vxxx) such that f is a transcendental function in v; vx; vxx. This family is related to the Krichever-Novikov equation by a differential substitution. Our construction of integrable equations and the corresponding differential...

We give a new derivation and characterisation of the generalised elliptic genus of Krichever-Höhn by means of a functional equation.

This paper concerns linear standing gravity water waves on finite depth. We obtain qualitative and quantitative understanding of the particle paths within the wave.

In the paper, we study real forms of the complex generic Neumann system. We prove that the real forms are completely integrable Hamiltonian systems. The complex Neumann system is an example of the more general Mumford system. The Mumford system is characterized by the Lax pair (Lℂ(λ), Mℂ(λ)) of 2 × 2...

A classical Korteweg capillarity system with a Karman-Tsien type (κ, ρ) constitutive relation is shown, via a Madelung transformation and use of invariants of motion, to admit integrable Hamiltonian subsystems.

Conditions necessary for the existence of local higher order generalized symmetries and conservation laws are derived for a class of dynamical lattice equations with explicit dependence on the spatial discrete variable and on time. We explain how to use the obtained conditions for checking a given equation....

A series of rational solutions are presented for an extended Lotka-Volterra eqution. These rational solutions are obtained by using Hirota's bilinear formalism and Bäcklund transformation. The crucial step is the use of nonlinear superposition fomula. The so-called extended Lotka-Volterra equation is...

Ansatzes for the Navier-Stokes field are described. These ansatzes reduce the Navier-Stokes equations to system of differential equations in three, two, and one independent variables. The large sets of exact solutions of the Navier-Stokes equations are constructed.

The λ-symmetry approach is applied to a family of second-order ODEs whose algebra of Lie point symmetries is insufficient to integrate them. The general solution and two functionally independent first integrals of a subclass of the studied equations can be expressed in terms of a fundamental set of solutions...

We present a new method for the derivation of mappings of HKY type. These are second-order mappings which do not have a biquadratic invariant like the QRT mappings, but rather an invariant of degree higher than two in at least one of the variables. Our method is based on folding transformations which...

This paper explores the quasi-deformation scheme devised in [1, 3] as applied to the simple Lie algebra sl2(F) for specific choices of the involved parameters and underlying algebras. One of the main points of this method is that the quasi-deformed algebra comes endowed with a canonical twisted Jacobi...

The Riemann problem for a simplified chromatography system is considered and the global Riemann solutions are constructed in all kinds of situations. In particular, the zero rarefaction wave, the zero shock wave and the zero delta shock wave are discovered in the Riemann solutions in some limit situations,...

We show that the m-dimensional Euler­Manakov top on so (m) can be represented as a Poisson reduction of an integrable Hamiltonian system on a symplectic extended Stiefel variety ¯V(k, m), and present its Lax representation with a rational parameter. We also describe an integrable two-valued symplectic...

The Lie algebra L(h) of symmetries of a discrete analogue of the non-linear Schrödinger equation (NLS) is studied. A five-dimensional subspace of L(h), generated by both point and generalized symmetries, transforms into the five-dimensional point symmtry algebra L(0) of the NLS equation. We use the lowest...

We present a numerical study of the nonlinear system of 4 0 equations of motion. The solution is obtained iteratively, starting from a precise point-sequence of the appropriate Banach space, for small values of the coupling constant. The numerical results are in perfect agreement with the main theoretical...

The influence of a magnetic field on the flow of an incompressible third grade electrically conducting fluid bounded by a rigid plate is investigated. The flow is induced due by the motion of a plate in its own plane with an arbitrary velocity. The solution of the equations of conservation of mass and...

We propose a method to identify and classify evolution equations and systems that can be multipotentialised in given target equations or target systems. We refer to this as the converse problem. Although we mainly study a method for (1 + 1)-dimensional equations/system, we do also propose an extension...

Lie group analysis is applied to a mathematical model for thin liquid films, namely a nonlinear fourth order partial differential equation in two independent variables. A three-dimensional Lie symmetry algebra is found and reductions to fourth order ordinary differential equations are obtained by using...

Reducible representations of CAR and CCR are applied to second quantization of Dirac and Maxwell fields. The resulting field operators are indeed operators and not operator-valued distributions. Examples show that the formalism may lead to a finite quantum field theory.

We consider the Calogero­Degasperis­Ibragimov­Shabat (CDIS) equation and find the complete set of its nonlocal symmetries depending on the local variables and on the integral of the only local conserved density of the equation in question. The Lie algebra of these symmetries turns out to be a central...

The reduction of two nonlinear equations of the type □u+F(u, u1)u0 = 0 with respect to all rank three subalgebras of a subdirect sum of the extended Euclidean algebras A ~E(1) and A ~E(3) is carried out. Some new invariant exact solutions of these equations are obtained.

In this paper we develop a straightforward procedure to construct higher dimensional isochronous Hamiltonian systems. We first show that a class of singular Hamiltonian systems obtained through the Ω-modified procedure is equivalent to constrained Newtonian systems. Even though such systems admit isochronous...

We consider the following question: Suppose part of the boundary of a cavity contaiing a gas is set into oscillation, the damping in the boundary being small. What is the nature of the oscillations in the gas? We treat the low-frequency limit (wavelength much greater than dimensions of the cavity). Experiment...

In this paper, we construct the noncommutative B and C type KP hierarchies using pseudo-differential operators and reducing conditions. Further a series of additional flows of the noncommutative B and C type KP hierarchies will be defined and the additional symmetries constitute the B and C type infinite...

The generalized dressing method is extended to variable-coefficient AKNS equations, including a variable-coefficient coupled nonlinear Schr¨odinger equation and a variablcoefficient coupled mKdV equation. A general variable-coefficient KP equation is proposed and decomposed into the two 1+1 dimensional...

In this paper, the family of BBM equation with strong nonlinear dispersive B(m,n) is considered. We apply the classical Lie method of infinitesimals. The symmetry reductions are derived from the optimal system of subalgebras and lead to systems of ordinary differential equations. We obtain for special...

We develop a Nambu bracket formulation for a wide class of nonlinear biochemical reactions by exploiting previous work that focused on elementary biochemical mass action reactions. To this end, we consider general reaction mechanisms including for example enzyme kinetics. Furthermore, we go beyond elementary...

We study statistical properties of inhomogeneous Burgers lattices which are solved by the discrete Cole­Hopf transformation. Using exact solutions we investigate effect of various kinds of noise on the dynamics of solutions.

This paper is an attempt to present and discuss at some length the Singular Manifold Method. This Method is based upon the Painlevé Property systematically used as a tool for obtaining clear cut answers to almost all the questions related with Nonlinear Partial Differential Equations: Lax pairs, Miura,...

Legendre transformations provide a natural symmetry on the space of solutions to the WDVV equations, and more specifically, between different Frobenius manifolds. In this paper a twisted Legendre transformation is constructed between solutions which define the corresponding dual Frobenius manifolds....

A class of involutive Wick algebras (called anyonic-type Wick algebras) is selected and some its elementary properties are described. In particular, the Fock representtions of the selected anyonic-type commutation relations are described. For the class of so-called r-yonic systems the question of the...

Computer-aided symbolic and graphic computation allows to make significantly easier both theoretical and applied symmetry analysis of PDE. This idea is illustrated by applying a special "Mathematica" package for obtaining conditional symmetries of the nonlinear wave equation ut = (u ux)x invariant or...

We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painlevé equation. For the general solution, explicit recursive formulae for the Taylor expansion of the tau-function around a zero are given, which are natural extensions...

We obtain lower bounds on the ground state energy, in one and three dimensions, for the spinless Salpeter equation (Schrödinger equation with a relativistic kinetic energy operator) applicable to potentials for which the attractive parts are in Lp (Rn ) for some p > n (n = 1 or 3). An extension to confining...

A q-difference analogue of the Painlevé III equation is considered. Its derivations, affine Weyl group symmetry, and two kinds of special function type solutions are discussed.

We construct new topological invariants of three-dimensional manifolds which can, in particular, distinguish homotopy equivalent lens spaces L(7, 1) and L(7, 2). The invariants are built on the base of a classical (not quantum) solution of pentagon equation, i.e. algebraic relation corresponding to a...

Rosenhain's famous formula expresses the periods of first kind integrals of genus two hyperelliptic curves in terms of θ-constants. In this paper we generalize the Rosenhain formula to higher genera hyperelliptic curves by means of the second Thomae formula for derivative non-singular θ-constants.

Deformation equation of a non-associative deformation in operad is proposed. Its integrability condition (the Bianchi identity) is considered. Algebraic meaning of the latter is explained.

In this paper, we obtain some exact solutions of Derivative Reaction-Diffusion (DRD) system and, as by-products, we also show some exact solutions of DNLS via Hirota bilinearization method. At first, we review some results about two by two AKNS-ZS system, then introduce Hirota bilinearization method...

We present a simple approach showing that Gerstner’s flow is dynamically possible: each particle moves on a circle, but the particles never collide and fill out the entire region below the surface wave.

We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms, and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated...

We show that an ultradiscrete analogue of the third Painlevé equation admits discrete Riccati type solutions. We derive these solutions by considering a framework in which the ultradiscretization process arises as a restriction of a non-archimedean valuation over a field. Using this framework we may...

There is a well known principle in classical mechanic stating that a variational problem independent of a configuration space variable w (so called cyclic variable), but dependent on its velocity w′ can be expressed without both w and w′. This principle is known as the Routh reduction. In this paper,...

We study a model describing some aspects of the dynamics of biopolymers. The models involve either one or two finite chains with a number N of sites that represent the "units" of a biophysical system. The mechanical degrees of freedom of these chains are coupled to the internal degrees of freedom through...

The asymptotic lattices and their transformations are included into the theory of quadrilateral lattices.

Symmetries for variational problems are considered as symmetries of vector-valued exterior differential systems. This approach is applied to equations for the classical spinning particle.

In this paper we prove that solutions of the f-plane approximation for equatorial geophysical deep water waves, which have the property that the pressure is constant along the streamlines and do not possess stagnation points, are Gerstner-type waves. Furthermore, for waves traveling over a flat bed,...

This work addresses a construction of a dual pair of nonlinear coherent states (NCS) in the context of changes of bases in the underlying Hilbert space for a model pertaining to an electron-phonon model in the condensed matter physics, obeying a f-deformed Heisenberg algebra. The existence and properties...

The automation of the traditional Painlev´e test in Mathematica is discussed. The package PainleveTest.m allows for the testing of polynomial systems of nonlinear ordinary and partial differential equations which may be parameterized by arbitrary functions (or constants). Except where limited by memory,...

We search for hidden symmetries of two-particle equations with oscillator-equivalent potential proposed by Moshinsky with collaborators. We proved that these equations admit hidden symmetries and parasupersymmetries which enable easily to find the Hamiltonian spectra using algebraic methods.

Operators of non­local symmetry are used to construct exact solutions of nonlinear heat equations. A method for finding of new classes of ansatzes reducing nonlinear wave equations to a systems of ordinary differential equations was suggested in [1]. This approach is based on non­local symmetry of differential...

In the deterministic realm, both differential equations and symmetry generators are geometrical objects, and behave properly under changes of coordinates; actually this property is essential to make symmetry analysis independent of the choice of coordinates and applicable. When trying to extend symmetry...

The paper is devoted to the Lie group properties of the one-dimensional Green-Naghdi equations describing the behavior of fluid flow over uneven bottom topography. The bottom topography is incorporated into the Green-Naghdi equations in two ways: in the classical Green-Naghdi form and in the approximated...

This investigation deals with the mechanism of peristaltic transport of a non-Newtonian, incompressible and electrically conducting fluid with variable viscosity and an endoscope effects. The magnetic Reynolds number is taken to be small. The mechanical properties of the material are represented by the...

We extend the traditional formulation of Gauge Field Theory by incorporating the (non- Abelian) gauge group parameters (traditionally simple spectators) as new dynamical (nonlinearsigma- model-type) fields. These new fields interact with the usual Yang-Mills fields through a generalized minimal coupling...

In terms of a 3-Lie algebra and the classical Poisson bracket {Bn,L} of the dKP hierarchy, a special 3-bracket {Bm,Bn,L} is proposed. When m = 0 or m = 1, the 3-lax equation ∂L∂t={Bm,Bn,L} is the dKP hierarchy and the corresponding proof is given. Meanwhile, for the generalized case (m,n), the generalized...

A group invariant solution for a steady two-dimensional jet is derived by considering a linear combination of the Lie point symmetries of Prandtl's boundary layer equations for the jet. Only two Lie point symmetries contribute to the solution and the ratio of the constants in the linear combination is...

A regular gradient-holonomic approach to studying the Lax type integrability of the Ablowitz–Ladik hierarchy of nonlinear Lax type integrable discrete dynamical systems in the vertex operator representation is presented. The relationship to the Lie-algebraic integrability scheme is analyzed and the connection...

It is proved that among the finite groups of order less than 32 only the tetrahedral group and the binary tetrahedral group are not determined by their endomorphism semigroups in the class of all groups.

It is proven that the essential spectrum of any self-adjoint operator associated with the matrix differential expression

A method of quantization of classical soliton cellular automata (QSCA) is put forward that provides a description of their time evolution operator by means of quantum cicuits that involve quantum gates from which the associated Hamiltonian describing a quantum chain model is constructed. The intrinsic...

We derive solution-formulae for the Krichever–Novikov equation by a systematic multipotentialisation of the equation. The formulae are achieved due to the connections of the Krichever–Novikov equations to certain symmetry-integrable 3rd-order evolution equations which admit autopotentialisations.

Nicodemi and Prisco recently proposed a model for X-chromosome inactivation in mammals, explaining this phenomenon in terms of a spontaneous symmetry-breaking mechanism [Phys. Rev. Lett. 99 (2007) 108104]. Here we provide a mean-field version of their model.

Nonlocal symmetries of the (1+1)-dimensional Sinh-Gordon (ShG) equation are obtained by requiring it, together with its Bäcklund transformation (BT), to be form invariant under the infinitesimal transformation. Naturally, the spectrum parameter in the BT enters the nonlocal symmetries, and thus through...

We give a review of some recent work on generalization of the Bethe ansatz in the case of 2 + 1-dimensional models of quantum field theory. As such a model, we consider one associated with the tetrahedron equation, i.e. the 2+1-dimensional generalization of the famous Yang­Baxter equation. We construct...

The (2+1)-dimensional Davey-Stewartson-like equations with variable coefficients have the applications in the ultra-relativistic degenerate dense plasmas and Bose-Einstein condensates. Via the Bell polynomials and symbolic computation, the bilinear form, Bäcklund transformation and Lax pair for such...

We prove bispectral duality for the generalized Calogero­Moser­Sutherland systems related to configurations An,2(m), Cn(l, m). The trigonometric axiomatics of the Baker­Akhiezer function is modified, the dual difference operators of rational Madonald type and the Baker­Akhiezer functions related to both...

This paper is devoted to an extension of Burchnall-Chaundy theory on the inteplay between algebraic geometry and commuting differential operators to the case of q-difference operators.

We study the complex Riccati tensor equation DcG + GCG - R = 0 on a geodesic c on a Riemannian 3-manifold. This non-linear equation appears in the study of Gaussian beams. Gaussian beams are asymptotic solutions to hyperbolic equations that at each time instant are concentrated around one point in space....

We study the discrete Painlevé equations associated to the E7(1) affine Weyl group which can be obtained by the implementation of a special limits of E8(1)-associated equations. This study is motivated by the existence of two E7(1)-associated discrete both having a double ternary dependence in their...

The Einstein field equations for several cosmological models reduce to polynomial systems of ordinary differential equations. In this paper we shall concentrate our attention to the spatially homogeneous diagonal G2 cosmologies. By using Darboux's theory in order to study ordinary differential equations...

Properties of approximate symmetries of equations with a small parameter are discussed. It turns out that approximate symmetries form an approximate Lie algebra. A concept of approximate invariants is introduced and the algorithm of their calculating is proposed.

A new approach to the analysis of wave-breaking solutions to the generalized Camassa-Holm equation is presented in this paper. Introduction of a set of variables allows for solving the singularities. A continuous semigroup of dissipative solutions is also built. The solutions have non-increasing H1 energy...

We introduce an extension of the factorization-decomposition technique that allows us to manufacture new solvable nonlinear matrix evolution equations. Several examples of such equations are reported.

Reduction of a nonlinear system of differential equations for spinor field is studied. The ansatzes obtained are shown to correspond to operators of conditional symmetry of these equations.

Here, a recently introduced nine-body problem is shown to be decomposable via a novel class of reciprocal transformations into a set of integrable Ermakov systems. This Ermakov decomposition is exploited to construct more general integrable nine-body systems in which the canonical nine-body system is...

In this paper we construct small amplitude periodic internal waves traveling at the boundary region between two rotational and homogeneous fluids with different densities. Within a period, the waves we obtain have the property that the gradient of the stream function associated to the fluid beneath the...

We investigate linear stability of solitary waves of a Hamiltonian system. Unlike weakly nonlinear water wave models, the physical system considered here is nonlinearly dispersive, and contains nonlinearity in its highest derivative term. This results in more detailed asymptotic analysis of the eigenvalue...

A simple trick is illustrated, whereby nonlinear evolution equations can be modified so that they feature a lot ­ or, in some cases, only ­ periodic solutions. Several examples (ODEs and PDEs) are exhibited.

We study the time and temperature dependent correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundary conditions (x1, 0) (x2, t) ±,T . We derive the Fredholm determinant formulae for the correlation functions, by means of the Bethe Ansatz. For the special case x1 = 0, we express...

Brownian motion on a smash line algebra (a smash or braided version of the algebra resulting by tensoring the real line and the generalized paragrassmann line algebras), is constructed by means of its Hopf algebraic structure. Further, statistical moments, non stationary generalizations and its diffusion...

We investigate the discrete Painlevé equations (dPII and qPII) over finite fields. We first show that they are well defined by extending the domain according to the theory of the space of initial conditions. Then we treat them over local fields and observe that they have a property that is similar to...

Total differentiation operators as linear vector fields, their flows, invariants and symmetries form the geometry of jet space. In the jet space the dragging of tensor fields obeys the exponential law. The composition of smooth maps induces a composition of jets in corresponding jet spaces. The prolonged...

The (Hamiltonian, rotation- and translation-invariant) "goldfish" N-body problem in the plane is characterized by the Newtonian equations of motion ¨zn - i zn = 2 N m=1,m=n an,m zn zm (zn - zm) -1 , written here in their complex version, entailing the identification of the real "physical" plane with...

The method of multiple scales is used to introduce a small-time scale into the nolinear diffusion equation modelling the spreading of a thin liquid drop under gravity. The Lie group method is used to analyse the resulting system. An approximate group invariant solution and an approximation to the waiting-time...

Reduction of multidimensional Poincaré-invariant equations to ordinary differential equations and 2-dimensional equations is considered.

In recent years T.A. Osborn and his coworkers at the University of Manitoba have extensively developed the well known connected graph expansion and applied it to a wide variety of problems in semiclassical approximation to quantum dynamics [2, 5, 7, 19, 21, 22, 26, 27]. The work I am reporting on attempts...

In this work we discuss the complete synchronization of two identical double-well Duffing oscillators unidirectionally coupled, from the point of view of symbolic dynamics. Working with Poincaré cross-sections and the return maps associated, the synchronization of the two oscillators, in terms of the...

In the paper by F Calogero and the author [Commun. Math. Phys. 59 (1978), 109­