Journal of Nonlinear Mathematical Physics

+ Advanced search
902 articles
Research Article

New Geometrical Applications of the Elliptic Integrals: The Mylar Balloon

Ivaïlo M. MLADENOV
Pages: 55 - 65
An explicit parameterization in terms of elliptic integrals (functions) for the Mylar balloon is found which then is used to calculate various geometric quantities as well as to study all kinds of geodesics on this surface.
Research Article

On Exact Solution of a Classical 3D Integrable Model

S.M. SERGEEV
Pages: 57 - 72
We investigate some classical evolution model in the discrete 2+1 space-time. A map, giving an one-step time evolution, may be derived as the compatibility condition for some systems of linear equations for a set of auxiliary linear variables. Dynamical variables for the evolution model are the coefficients...
Research Article

Solitons in Yakushevich-like models of DNA dynamics with improved intrapair potential

Giuseppe GAETA
Pages: 57 - 81
The Yakushevich model provides a very simple pictures of DNA torsion dynamics, yet yields remarkably correct predictions on certain physical characteristics of the dynamics. In the standard Yakushevich model, the interaction between bases of a pair is modelled by a harmonic potential, which becomes anharmonic...
Research Article

On the Transformations of the Sixth Painlevé Equation

Valery I GROMAK, Galina FILIPUK
Pages: 57 - 68
In this paper we investigate relations between different transformations of the slutions of the sixth Painlevé equation. We obtain nonlinear superposition formulas linking solutions by means of the Bäcklund transformation. Algebraic solutions are also studied with the help of the Bäcklund transformation.
Research Article

Soliton Asymptotics of Rear Part of Non-Localized Solutions of the Kadomtsev-Petviashvili Equation

Anne BOUTET de MONVEL, Eugene KHRUSLOV
Pages: 58 - 76
We construct non-localized, real global solutions of the Kadomtsev-Petviashvili-I eqution which vanish for x - and study their large time asymptotic behavior. We prove that such solutions eject (for t ) a train of curved asymptotic solitons which move behind the basic wave packet.
Research Article

Algebraic Linearization of Hyperbolic Ruijsenaars­Schneider Systems

R CASEIRO, J P FRANCOISE
Pages: 58 - 61
In this article, we present an explicit linearization of dynamical systems of RuijsenaarSchneider (RS) type and of the perturbations introduced by F Calogero [2] of these systems with all orbits periodic of the same period. The existence of this linearization and its algebraic nature relies on the dynamical...
Research Article

On the Non-Dimensionalisation, Scaling and Resulting Interpretation of the Classical Governing Equations for Water Waves

Adrian Constantin, Robin Stanley Johnson
Pages: 58 - 73
In this note we describe the underlying principles — and pitfalls — of the process of non-dimensionalising and scaling the equations that model the classical problem in water waves. In particular, we introduce the two fundamental parameters (associated with amplitude and with wave length) and show how...
Research Article

Superanalogs of the Calogero Operators and Jack Polynomials

A SERGEEV
Pages: 59 - 64
A depending on a complex parameter k superanalog SL of Calogero operator is costructed; it is related with the root system of the Lie superalgebra gl(n|m). For m = 0 we obtain the usual Calogero operator; for m = 1 we obtain, up to a change of indterminates and parameter k the operator constructed by...
Research Article

Some Special Integrable Surfaces

M G ÜRSES
Pages: 59 - 66
We consider surfaces arising from integrable partial differential equations and from their deformations. Symmetries of the equation, gauge transformation of the corrsponding Lax pair and spectral parameter transformations are the deformations which lead infinitely many integrable surfaces. We also study...
Research Article

A Basis of Conservation Laws for Partial Differential Equations

A H KARA, F M MAHOMED
Pages: 60 - 72
The classical generation theorem of conservation laws from known ones for a system of differential equations which uses the action of a canonical Lie­Bäcklund generator is extended to include any Lie­Bäcklund generator. Also, it is shown that the Lie algebra of Lie­Bäcklund symmetries of a conserved...
Research Article

On the Origins of Symmetries of Partial Differential Equations: the Example of the Korteweg-de Vries Equation

Keshlan S. Govinder, Barbara Abraham-Shrauner
Pages: 60 - 68
Type II hidden symmetries of partial differential equations () are extra symme- tries in addition to the inherited symmetries of the differential equations which arise when the number of independent and dependent variables is reduced by a Lie point symmetry. (Type I hidden symmetries arise in the increase...
Research Article

Antireduction and exact solutions of nonlinear heat equations

WILHELM FUSHCHYCH, RENAT ZHDANOV
Pages: 60 - 64
We construct a number of ansatzes that reduce one-dimensional nonlinear heat equations to systems of ordinary differential equations. Integrating these, we obtain new exact solution of nonlinear heat equations with various nonlinearities.
Research Article

On Huygens' Principle for Dirac Operators and Nonlinear Evolution Equations

Fabio A C C CHALUB, Jorge P ZUBELLI
Pages: 62 - 68
We exhibit a class of Dirac operators that possess Huygens' property, i.e., the support of their fundamental solutions is precisely the light cone. This class is obtained by considering the rational solutions of the modified Korteweg-de Vries hierarchy.
Research Article

The Derivative Nonlinear Schrödinger Equation in Analytic Classes

Zoran GRUJIC, Henrik KALISCH
Pages: 62 - 71
The derivative nonlinear Schrödinger equation is shown to be locally well-posed in a class of functions analytic on a strip around the real axis. The main feature of the result is that the width of the strip does not shrink in time. To overcome the derivative loss, Kato-type smoothing results and space-time...
Research Article

Conditional and Nonlocal Symmetry of Nonlinear Heat Equation

Mykola I. SEROV
Pages: 63 - 67
Conditional symmetry We investigate conditional symmetry in three directions. The first direction is a research of the Q-conditional symmetry. The second direction is studying conditional symmetry when an algebra of invariance is known and an additional condition is unknown. The third direction is the...
Research Article

Orthogonal matrix polynomials satisfying first order differential equations: a collection of instructive examples

Mirta M CASTRO, F Alberto GRUNBAUM
Pages: 63 - 76
We describe a few families of orthogonal matrix polynomials of size N × N satisfying first order differential equations. This problem differs from the recent efforts reported for instance in [7] (Orthogonal matrix polynomials satisfying second order differential equations, Internat. Math. Research Notices,...
Research Article

Vortex Line Representation for the Hydrodynamic Type Equations

E A KUZNETSOV
Pages: 64 - 80
In this paper we give a brief review of the recent results obtained by the author and his co-authors for description of three-dimensional vortical incompressible flows in the hydrodynamic type systems. For such flows we introduce a new mixed LagrangiaEulerian description - the so called vortex line representation...
Research Article

On Dispersionless BKP Hierarchy and its Reductions

L V BOGDANOV, B G KONOPELCHENKO
Pages: 64 - 73
Integrable dispersionless Kadomtsev-Petviashvili (KP) hierarchy of B type is consiered. Addition formula for the -function and conformally invariant equations for the dispersionless BKP (dBKP) hierarchy are derived. Symmetry constraints for the dBKP hierarchy are studied.
Research Article

Nonlinear Schrödinger, Infinite Dimensional Tori and Neighboring Tori

M SCHWARZ Jr
Pages: 65 - 77
In this work, we explain in what sense the generic level set of the constants of motion for the periodic nonlinear Schrödinger equation is an infinite dimensional torus on which each generalized nonlinear Schrödinger flow is reduced to straight line almost periodic motion, and describe how neighboring...
Research Article

Hard Loss of Stability in Painlevé-2 Equation

O M KISELEV
Pages: 65 - 95
A special asymptotic solution of the Painlevé-2 equation with small parameter is stdied. This solution has a critical point t corresponding to a bifurcation phenomenon. When t < t the constructed solution varies slowly and when t > t the solution oscillates very fast. We investigate the transitional...
Research Article

A completely integrable system associated with the Harry-Dym hierarchy

ZHIJUN QIAO
Pages: 65 - 74
By use of nonlinearization method about spectral problem, a classical completely integrable system associated with the Harry-Dym (HD) hierarchy is obtained. Furthermore, the involutive solution of each equation in the HD hierarchy is presented, in particular, the involutive solution of the well-known...
Research Article

Existence of Dark Soliton Solutions of the Cubic Nonlinear Schrödinger Equation with Periodic Inhomogeneous Nonlinearity

Juan Belmonte-Beitia, Pedro J Torres
Pages: 65 - 72
In this paper, we give a proof of the existence of stationary dark soliton solutions of the cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity, together with an analytical example of a dark soliton.
Research Article

von Neumann Quantization of Aharonov-Bohm Operator with Interaction: Scattering Theory, Spectral and Resonance Properties

Gilbert HONNOUVO, Mahouton Norbert HOUNKONNOU, Gabriel Yves Hugues
Pages: 66 - 71
Using the theory of self-adjoint extensions, we study the interaction model formally given by the Hamiltonian H + V (r), where H is the Aharonov-Bohm Hamiltonian and V (r) is the -type interaction potential on the cylinder of radius R . We give the mathematical definition of the model, the self-adjointness...
Research Article

Leading order integrability conditions for differential-difference equations

Mark S HICKMAN
Pages: 66 - 86
A necessary condition for the existence of conserved densities and fluxes of a differential-difference equation which depend on q shifts, for q sufficiently large, is presented. This condition depends on the eigenvalues of the leading terms in the differential-difference equation. It also gives, explicitly,...
Research Article

Symmetries of a Class of Nonlinear Fourth Order Partial Differential Equations

Peter A. CLARKSON, Thomas J. PRIESTLEY
Pages: 66 - 98
In this paper we study symmetry reductions of a class of nonlinear fourth order partial differential equations utt = u + u2 xx + uuxxxx + µuxxtt + uxuxxx + u2 xx, (1) where , , , and µ are arbitrary constants. This equation may be thought of as a fourth order analogue of a generalization of the Camassa-Holm...
Research Article

Rewriting in Operads and PROPs

Lars HELLSTROM
Pages: 66 - 75
This paper is an informal collection of observations on how established rewriting techniques can be applied to or need to be adapted for use in non-associative algebras, operads, and PROPs.
Research Article

Is My ODE a Painlevé Equation in Disguise?

Jarmo HIETARINTA, Valery DRYUMA
Pages: 67 - 74
Painlevé equations belong to the class y +a1 y 3 +3a2 y 2 +3a3 y +a4 = 0, where ai = ai(x, y). This class of equations is invariant under the general point transformation x = (X, Y ), y = (X, Y ) and it is therefore very difficult to find out whether two equations in this class are related. We describe...
Research Article

Lie Symmetries of Einstein's Vacuum Equations in N Dimensions

Louis MARCHILDON
Pages: 68 - 81
We investigate Lie symmetries of Einstein's vacuum equations in N dimensions, with a cosmological term. For this purpose, we first write down the second prolongation of the symmetry generating vector fields, and compute its action on Einstein's equations. Instead of setting to zero the coefficients of...
Research Article

Moyal Deformation of 2D Euler Equation and Discretization

Partha GUHA
Pages: 69 - 76
In this paper we discuss the Moyal deformed 2D Euler flows and its Lax pairs. This in turn yields the semi-discrete version of 2D Euler equation.
Research Article

Alternate Derivation of the Critical Value of the Frank-Kamenetskii Parameter in Cylindrical Geometry

Charis Harely, Ebrahim Momoniat
Pages: 69 - 76
Noether’s theorem is used to determine first integrals admitted by a generalised Lane-Emden equation of the second kind modelling a thermal explosion. These first integrals exist for rectangular and cylindrical geometry. For rectangular geometry the first integrals show the symmetry of the temperature...
Research Article

Variational Symmetry in Non-integrable Hamiltonian Systems

Umeno KEN
Pages: 69 - 77
We consider the variational symmetry from the viewpoint of the non-integrability criterion towards dynamical systems. That variational symmetry can reduce complexity in determining non-integrability of general dynamical systems is illustrated here by a new non-integrability result about Hamiltonian systems...
Research Article

Calogero­Moser Systems and Super Yang­Mills with Adjoint Matter

Eric D'HOKER, D H PHONG
Pages: 69 - 78
We review the construction of Lax pairs with spectral parameter for twisted and utwisted elliptic Calogero-Moser systems defined by a general simple Lie algebra G, and the corresponding solution of N = 2 SUSY G Yang-Mills theories with a hypermultplet in the adjoint representation of G.
Research Article

The Classical Problem of Water Waves: a Reservoir of Integrable and Nearly-Integrable Equations

Robin S JOHNSON
Pages: 72 - 92
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...
Research Article

Green function for Klein-Gordon-Dirac equation

Vasyl KOVALCHUK
Pages: 72 - 77
The Green function for Klein-Gordon-Dirac equation is obtained. The case with the dominating Klein-Gordon term is considered. There seems to be a formal analogy between our problem and a certain problem for a 4-dimensional particle moving in the external field. The explicit relations between the wave...
Research Article

q-Probability: I. Basic Discrete Distributions

Boris A. KUPERSHMIDT
Pages: 73 - 93
For basic discrete probability distributions, - Bernoulli, Pascal, Poisson, hypergemetric, contagious, and uniform, - q-analogs are proposed.
Research Article

Symmetry Reduction for Equation 2u + (u2 1 + u2 2 + u2 3)1/2 u0 = 0

L.F. BARANNYK, H.O. LAHNO
Pages: 73 - 89
The subalgebras of the invariance algebra of equation 2u+(u2
Research Article

Nonlinear-Integral-Equation Construction of Orthogonal Polynomials

Carl M. Bender, E. Ben-Naim
Pages: 73 - 80
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...
Research Article

Symmetries and Integrating Factors

P G L LEACH, S É BOUQUET
Pages: 73 - 91
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...
Research Article

The Structure of Gelfand-Levitan-Marhenko Type Equations for Delsarte Transmutation Operators of Linear Multidimensional Differential Operators and Operator Pencils. Part 1.

Jolanta GOLENIA, Anatoliy K PRYKARPATSKY, Yarema A PRYKARPATSKY
Pages: 73 - 87
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...
Research Article

On the Fluid Motion in Standing Waves

Mats Ehrnstrom, Erik Wahlen
Pages: 74 - 86
This paper concerns linear standing gravity water waves on finite depth. We obtain qualitative and quantitative understanding of the particle paths within the wave.
Research Article

Functional Equations and the Generalised Elliptic Genus

H W BRADEN, K E FELDMAN
Pages: 74 - 85
We give a new derivation and characterisation of the generalised elliptic genus of Krichever-Höhn by means of a functional equation.
Research Article

Rational Solutions of an Extended Lotka-Volterra Equation

X B HU, P A CLARKSON
Pages: 75 - 83
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...
Research Article

Symmetry reduction and exact solutions of the Navier-Stokes equations. I

WILHELM FUSHCHYCH, ROMAN POPOWYCH
Pages: 75 - 113
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.
Research Article

Integrability Conditions for n and t Dependent Dynamical Lattice Equations

R YAMILOV, D LEVI
Pages: 75 - 101
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....
Research Article

On Some Almost Quadratic Algebras Coming from Twisted Derivations

Daniel LARSSON, Gunnar SIGURDSSON, Sergei D SILVESTROV
Pages: 76 - 87
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...
Research Article

The Discrete Nonlinear Schrödinger Equation and its Lie Symmetry Reductions

R HERNÁNDEZ HEREDERO, D LEVI
Pages: 77 - 94
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...
Research Article

Integrable flows and Bäcklund transformations on extended Stiefel varieties with application to the Euler top on the Lie group SO(3)

Yuri N FEDOROV
Pages: 77 - 94
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...
Research Article

The Numerical Study of the Solution of the 4 0 Model

S GLADKOFF, A ALAIE, Y SANSONNET, M MANOLESSOU
Pages: 77 - 85
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...