Journal of Nonlinear Mathematical Physics

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905 articles

Nonlinearized Perturbation Theories

Miloslav ZNOJIL
Pages: 51 - 62
A brief review is presented of the two recent perturbation algorithms. Their common idea lies in a not quite usual treatment of linear Schrödinger equations via nonlinear mathematical means. The first approach (let us call it a quasi-exact perturbation theory, QEPT) tries to get the very zero-order approximants...

Contact Symmetry of Time-Dependent Schrödinger Equation for a Two-Particle System: Symmetry Classification of Two-Body Central Potentials

Pages: 51 - 65
Symmetry classification of two-body central potentials in a two-particle Schrödinger equation in terms of contact transformations of the equation has been investigated. Explicit calculation has shown that they are of the same four different classes as for the point transformations. Thus in this problem...

The Poincaré­Nekhoroshev Map

Giuseppe GAETA
Pages: 51 - 64
We study a generalization of the familiar Poincaré map, first implicitely introduced by N N Nekhoroshev in his study of persistence of invariant tori in hamiltonian systems, and discuss some of its properties and applications. In particular, we apply it to study persistence and bifurcation of invariant...

A Hidden Hierarchy for GD4

Pages: 53 - 57
A reduction process to construct hidden hierarchies corresponding to the Gelfand­ Dickey ones is outlined in a specific example, not yet treated in the literature.

An L2 Norm Trajectory-Based Local Linearization for Low Order Systems

Pages: 53 - 72
This paper presents a linear transformation for low order nonlinear autonomous diferential equations. The procedure consists of a trajectory-based local linearization, which approximates the nonlinear system in the neighborhood of its equilibria. The approximation is possible even in the non-hyperbolic...

The Motion of a Gyrostat in a Central Gravitational Field: Phase Portraits of an Integrable Case

M. C. Balsas, E. S. Jiménez, J. A. Vera
Pages: 53 - 64
In this paper we describe the Hamiltonian dynamics, in some invariant manifolds of the mo- tion of a gyrostat in Newtonian interaction with a spherical rigid body. Considering a first integrable approximation of this roto-translatory problem, by means of Liouville-Arnold the- orem and some specifics...

Orthogonalization of Graded Sets of Vectors

Pages: 54 - 58
I propose an orthogonalization procedure preserving the grading of the initial graded set of linearly independent vectors. In particular, this procedure is applicable for orthonormalization of any countable set of polynomials in several (finitely many) ideterminates.

The Integrability of Lie-invariant Geometric Objects Generated by Ideals in the Grassmann Algebra

Pages: 54 - 67
We investigate closed ideals in the Grassmann algebra serving as bases of Lie-invariant geometric objects studied before by E.Cartan. Especially, the E.Cartan theory is enlarged for Lax integrable nonlinear dynamical systems to be treated in the frame work of the Wahlquist Estabrook prolongation structures...

Description of a Class of 2-Groups

Pages: 55 - 65
Let n be an integer such that n 3 and Cm denote a cyclic group of order m . It is proved that there exist exactly 17 non-isomorphic groups of order 22n+1 which can be represented as a semidirect product (C2n × C2n ) C2. These groups are given by generators and defining relations.

New Geometrical Applications of the Elliptic Integrals: The Mylar Balloon

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.

On Exact Solution of a Classical 3D Integrable Model

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...

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...

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

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.

Algebraic Linearization of Hyperbolic Ruijsenaars­Schneider Systems

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...

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...

Superanalogs of the Calogero Operators and Jack Polynomials

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...

Some Special Integrable Surfaces

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...

A Basis of Conservation Laws for Partial Differential Equations

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...

Antireduction and exact solutions of nonlinear heat equations

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.

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...

On Huygens' Principle for Dirac Operators and Nonlinear Evolution Equations

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.

The Derivative Nonlinear Schrödinger Equation in Analytic Classes

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...

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...

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

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,...

Similarity Solutions for a Nonlinear Model of the Heat Equation

Effat A. SAIED, Magdy M. HUSSEIN
Pages: 63 - 67
We apply the similarity method based on a Lie group to a nonlinear model of the heat equation and find its Lie algebra.The optimal system of the model is contructed from the Lie algebra. New classes of similarity solutions are obtained.

Vortex Line Representation for the Hydrodynamic Type Equations

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...

On Dispersionless BKP Hierarchy and its Reductions

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.

Nonlinear Schrödinger, Infinite Dimensional Tori and Neighboring Tori

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...

Hard Loss of Stability in Painlevé-2 Equation

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...

A completely integrable system associated with the Harry-Dym hierarchy

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...

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.

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...

Symmetries of a Class of Nonlinear Fourth Order Partial Differential Equations

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...

Leading order integrability conditions for differential-difference equations

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,...

Rewriting in Operads and PROPs

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.

Is My ODE a Painlevé Equation in Disguise?

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...

Lie Symmetries of Einstein's Vacuum Equations in N Dimensions

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...

Fuzzy Relational Fixed Point Clustering

Roelof K. Brouwer
Pages: 69 - 82
The proposed relational fuzzy clustering method, called FRFP ( fuzzy relational fixed point), is based on determining a fixed point of a function of the desired membership matrix. The ethod is compared to other relational clustering methods. Simulations show the method to be very effective and less computationally...

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.

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...

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...

Calogero­Moser Systems and Super Yang­Mills with Adjoint Matter

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.

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

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...

Green function for Klein-Gordon-Dirac equation

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...

q-Probability: I. Basic Discrete Distributions

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

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

Pages: 73 - 89
The subalgebras of the invariance algebra of equation 2u+(u2

Symmetries and Integrating Factors

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...