Journal of Nonlinear Mathematical Physics

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902 articles
Research Article

Integrable 1D Toda cellular automata

Mariusz BIALECKI
Pages: 28 - 35
First, we recall the algebro-geometric method of construction of finite field valued solutions of the discrete KP equation, and next we perform a reduction of dKP to the discrete 1D Toda equation.
Research Article

The Initial-Boundary Value Poblem for the Korteweg-de Vries Equation on the Positive Quarter-Plane

Pham Loi VU
Pages: 28 - 43
The paper deals with a problem of developing an inverse-scattering transform for solving the initial-boundary value problem (IBVP) for the Korteweg-de Vries equation on the positive quarter-plane: pt - 6ppx + pxxx = 0, x 0, t 0, (a) with the given initial and boundary conditions: p(x, 0) = p(x), p(x)...
Research Article

Symmetries of Separating Nonlinear Schrödinger Equations

George SVETLICHNY
Pages: 28 - 35
We review here the main properties of symmetries of separating hierarchies of nonlinear Schrödinger equations and discuss the obstruction to symmetry liftings from (n)particles to a higher number. We argue that for particles with internal degrees of freedom, new multiparticle effects must appear at each...
Research Article

The Intermediate Surface Diffusion Flow on Spheres

Joachim ESCHER
Pages: 28 - 46
It is shown that solutions to the intermediate surface diffusion flow are real analytic in space and time, provided the initial surface is real diffeomorphic to a Euclidean sphere.
Research Article

Bäcklund Transformations on Coadjoint Orbits of the Loop Algebra ~gl(r)

Yuri FEDOROV
Pages: 29 - 46
There is a wide class of integrable Hamiltonian systems on finite-dimensional coadjoint orbits of the loop algebra ~gl(r) which are represented by r × r Lax equations with a rational spectral parameter. A reduced complex phase space is foliated with open subsets of Jacobians of regularized spectral curves....
Research Article

Existence of Dual Equations by Means of Strong Necessary Conditions - Analysis of Integrability of Partial Differential Nonlinear Equations

K SOKALSKI, T WIETECHA, D SOKALSKA
Pages: 31 - 52
A concept of strong necessary conditions for extremum of functional has been aplied for analysis an existence of dual equations for a system of two nonlinear Partial Differential Equations (PDE) in 1+1 dimensions. We consider two types of the dual equations: the Bäcklund transformations and the Bogomolny...
Research Article

On Bilinear Invariant Differential Operators Acting on Tensor Fields on the Symplectic Manifold

Pavel GROZMAN
Pages: 31 - 37
Let M be an n-dimensional manifold, V the space of a representation : GL(n)GL(V ). Locally, let T(V ) be the space of sections of the tensor bundle with fiber V over a sufficiently small open set U M, in other words, T(V ) is the space of tensor fields of type V on M on which the group Diff(M) of diffeomorphisms...
Research Article

Some Group Theoretical Aspects of Nonlinear Quantal Oscillators

K ANDRIOPOULOS, P G L LEACH
Pages: 32 - 42
We investigate the algebraic properties of the time-dependent Schrödinger equations of certain nonlinear oscillators introduced by Calogero and Graffi (Calogero F & Graffi S, On the quantisation of a nonlinear Hamiltonian oscillator Physics Letters A 313 (2003) 356-362; Calogero F, On the quantisation...
Research Article

Solutions of Adler's Lattice Equation Associated with 2-Cycles of the Bäcklund Transformation

James Atkinson, Frank Nijhoff
Pages: 34 - 42
The Bäcklund transformation (BT) of Adler's lattice equation is inherent in the equation itself by virtue of its multidimensional consistency. We refer to a solution of the equation that is related to itself by the composition of two BTs (with different Bäcklund parameters) as a 2-cycle of the BT. In...
Research Article

On the Structure of the Bäcklund Transformations for the Relativistic Lattices

Vsevolod E. ADLER
Pages: 34 - 56
The Bäcklund transformations for the relativistic lattices of the Toda type and their discrete analogues can be obtained as the composition of two duality transformations. The condition of invariance under this composition allows to distinguish effectively the integrable cases. Iterations of the Bäcklund...
Research Article

Variational Methods for Solving Nonlinear Boundary Problems of Statics of Hyper-Elastic Membranes

V.A. TROTSENKO
Pages: 35 - 50
A number of important results of studying large deformations of hyper-elastic shells are obtained using discrete methods of mathematical physics [1]­[6]. In the present paper, using the variational method for solving nonlinear boundary problems of statics of hyper-elastic membranes under the regular...
Research Article

Threshold Behavior for Nonlinear Wave Equations

Piotr BIZON
Pages: 35 - 41
In this brief contribution, which is based on my talk at the conference, I discuss the dynamics of solutions of nonlinear wave equations near the threshold of singularity formation. The heuristic picture of threshold behavior is first presented in a general setting and then illustrated with three examples.
Research Article

Conservations laws for critical Kohn-Laplace equations on the Heisenberg group

Yuri Dimitrov BOZHKOV, Igor Leite FREIRE
Pages: 35 - 47
Using the complete group classification of semilinear differential equations on the three-dimensional Heisenberg group H, carried out in a preceding work, we estab- lish the conservation laws for the critical Kohn-Laplace equations via the Noether’s Theorem.
Research Article

Singularity Analysis and a Function Unifying

the Painlevé, the Psi Series
Pages: 36 - 48
The classical (ARS) algorithm used in the Painlevé test picks up only those functions analytic in the complex plane. We complement it with an iterative algorithm giving the leading order and the next terms in all cases. This algorithm works both for an ascending series (about a singularity at finite...
Research Article

Second Order Dynamic Inclusions

Martin BOHNER, Christopher C TISDELL
Pages: 36 - 45
The theory of dynamic inclusions on a time scale is introduced, hence accommodating the special cases of differential inclusions and difference inclusions. Fixed point theory for set-valued upper semicontinuous maps, Green's functions, and upper and lower solutions are used to establish existence results...
Research Article

On the Self-Similar Solutions of Generalized Hydrodynamics Equations and Nonlinear Wave Patterns

V.A. DANYLENKO, V.A. VLADIMIROV
Pages: 36 - 43
Solutions of the system of dynamical equations of state and equations of the balance of mass and momentum are studied. The system possesses families of periodic, quasiperiodic and soliton-like invariant solutions. Self-similar solutions of this generalized hydrodynamic system are studied. Various complicated...
Research Article

Symmetries and Differential Forms

A. H. Davison, A. H. Kara
Pages: 36 - 43
The method for writing a differential equation or system of differential equations in terms of differential forms and finding their symmetries was devised by Harrison and Estabrook (1971). A modification to the method is demonstrated on a wave equation with variable speed, and the modified method is...
Research Article

SO(2) and Hamilton-Dirac mechanics

Cestmir BURDIK, Eugen PAAL, Juri VIRKEPU
Pages: 37 - 43
Canonical formalism for plane rotations is developed. This group can be seen as a toy model of the Hamilton-Dirac mechanics with constraints. The Lagrangian and Hamiltonian are explicitly constructed and their physical interpretation are given. The Euler-Lagrange and Hamiltonian canonical equations coincide...
Research Article

On Pauli graded contractions of sl(3, C)

Miloslav HAVLICEK, Jiri PATERA, Edita PELANTOVA, Jiri TOLAR
Pages: 37 - 42
We consider a special fine grading of sl(3, C), where the grading subspaces are geerated by 3 × 3 generalized Pauli matrices. This fine grading decomposes sl(3, C) into eight one­dimensional subspaces. Our aim is to find all contractions of sl(3, C) which preserve this grading. We have found that the...
Research Article

Fractal and Chaotic Solutions of the Discrete Nonlinear Schrödinger Equation in Classical and Quantum Systems

H S DHILLON, F V KUSMARTSEV, K E KÜRTEN
Pages: 38 - 49
We discuss stationary solutions of the discrete nonlinear Schrödinger equation (DNSE) with a potential of the 4 type which is generically applicable to several quantum spin, electron and classical lattice systems. We show that there may arise chaotic spatial structures in the form of incommensurate or...
Research Article

On Asymptotic Methods in the Theory of Differential Equations of Mathematical Physics

Mykola SHKIL
Pages: 40 - 50
In 1807 Fourier suggested a original method of solving partial differential equations. The method is known to lead to ordinary differential equations containing some arbitrary parameter...
Research Article

Symmetries, Lagrangian Formalism and Integration of Second Order Ordinary Difference Equations

Vladimir DORODNITSYN, Roman KOZLOV, Pavel WINTERNITZ
Pages: 41 - 56
An integration technique for difference schemes possessing Lie point symmetries is proposed. The method consists of determining an invariant Lagrangian and using a discrete version of Noether's theorem to obtain first integrals. This lowers the order of the invariant difference scheme.
Research Article

Progressive Internal Gravity Waves With Bounded Upper Surface Climbing a Triangular Obstacle

Mina B. ABD-EL-MALEK, Malak N. MAKAR
Pages: 41 - 53
In this paper we discuss a theoretical model for the interfacial profiles of progressive non-linear waves which result from introducing a triangular obstacle, of finite height, attached to the bottom below the flow of a stratified, ideal, two layer fluid, bounded from above by a rigid boundary. The derived...
Research Article

Sp(2) Quantization of Solitonic Theories

V CALIAN
Pages: 42 - 47
The gauge-field theoretical formulation of solitonic theories is quantized by using an extended version of the BRST Sp(2) symmetric formalism. The proposed method is based on a modified triplectic geometry which allows us to incorporate the linear and/or nonlinear global symmetries of the model and to...
Research Article

On Weak Convergence of Locally Periodic Functions

Dag LUKKASSEN, Peter WALL
Pages: 42 - 57
We prove a generalization of the fact that periodic functions converge weakly to the mean value as the oscillation increases. Some convergence questions connected to locally periodic nonlinear boandary value problems are also considered.
Research Article

Singular Scattering Matrices

David ATKINSON
Pages: 43 - 49
A nonlinear integrodifferential equation is solved by the methods of S-matrix theory. The technique is shown to be applicable to situations in which the effective potential is singular.
Research Article

Superintegrable Anharmonic Oscillators on N-dimensional Curved Spaces

Angel Ballesteros, Alberto Enciso, Francisco José Herranz, Orlando Ragnisco
Pages: 43 - 52
The maximal superintegrability of the intrinsic harmonic oscillator potential on N-dimensional spaces with constant curvature is revisited from the point of view of sl(2)-Poisson coalgebra symmetry. It is shown how this algebraic approach leads to a straightforward definition of a new large family of...
Research Article

On Poisson Realizations of Transitive Lie Algebroids

Yuri VOROBIEV
Pages: 43 - 48
We show that every transitive Lie algebroid over a connected symplectic manifold comes from an intrinsic Lie algebroid of a symplectic leaf of a certain Poisson structure. The reconstruction of the corresponding Poisson structures from the Lie algebroid is given in terms of coupling tensors.
Research Article

New approach to the complete sum of products of the twisted (h, q)-Bernoulli numbers and polynomials

Yilmaz SIMSEK, Veli KURT, Daeyeoul KIM
Pages: 44 - 56
In this paper, by using q-Volkenborn integral[10], the first author[25] constructed new generating functions of the new twisted (h, q)-Bernoulli polynomials and numbers. We define higher-order twisted (h, q)-Bernoulli polynomials and numbers. Using these numbers and polynomials, we obtain new approach...
Research Article

The graphical calculus for ribbon categories: Algebras, modules, Nakayama automorphisms

Jurgen FUCHS
Pages: 44 - 54
The graphical description of morphisms in rigid monoidal categories, in particular in ribbon categories, is summarized. It is illustrated with various examples of algebraic structures in such categories, like algebras, (weak) bi-algebras, Frobenius algebras, and modules and bimodules. Nakayama automorphisms...
Research Article

On New Galilei- and Poincare-Invariant Nonlinear Equations for Electromagnetic Field

Wilhelm FUSHCHYCH, Ivan TSYFRA
Pages: 44 - 48
Nonlinear systems of differential equations for E and H which are compatible with the Galilei relativity principle are proposed. It is proved that the Schrödinger equation together with the nonlinear equation of hydrodynamic type for E and H are invariant with respect to the Galilei algebra. New Poincare-invariant...
Research Article

A Formal Approach for Handling Lie Point Symmetries of Scalar First-Order Ito Stochastic Ordinary Differential Equations

Ebrahim Fredericks, Fazal M Mahomed
Pages: 44 - 59
Many methods of deriving Lie point symmetries for Itˆo stochastic ordinary differential equations (SODEs) have surfaced. In the Itˆo calculus context both the formal and intuitive understanding of how to construct these symmetries has led to seemingly disparate results. The impact of Lie point symmetries...
Short Communication

Geometrical symmetries of the Universal equation

V. DERJAGIN, A. LEZNOV
Pages: 46 - 50
It is shown that the group of geometrical symmetries of the Universal equation of D-dimensional space coincides with SL(D + 1, R).
Research Article

Bilinear recurrences and addition formulae for hyperelliptic sigma functions

Harry W BRADEN, Victor Z ENOLSKII, Andrew N W HONE
Pages: 46 - 62
The Somos 4 sequences are a family of sequences satisfying a fourth order bilinear recurrence relation. In recent work, one of us has proved that the general term in such sequences can be expressed in terms of the Weierstrass sigma function for an associated elliptic curve. Here we derive the analogous...
Research Article

Ehrenpreis Type Representations and Their Riemann­Hilbert Nonlinearisation

Athanassios S FOKAS
Pages: 47 - 61
We review a new method for studying boundary value problems for evolution PDEs. This method yields explicit results for a large class of evolution equations which iclude: (a) Linear equations with constant coefficients, (b) certain classes of linear equations with variable coefficients, and (c) integrable...
Research Article

New Symmetry Reductions for some Ordinary Differential Equations

M L GANDARIAS, E MEDINA, C MURIEL
Pages: 47 - 58
In this work we derive potential symmetries for ordinary differential equations. By using these potential symmetries we find that the order of the ODE can be reduced even if this equation does not admit point symmetries. Moreover, in the case for which the ODE admits a group of point symmetries, we find...
Research Article

On Conditionally Invariant Solutions of Magnetohydrodynamic Equations. Multiple Waves.

A M GRUNDLAND, P PICARD
Pages: 47 - 74
We present a version of the conditional symmetry method in order to obtain multiple wave solutions expressed in terms of Riemann invariants. We construct an abelian distribution of vector fields which are symmetries of the original system of PDEs subjected to certain first order differential constraints....
Research Article

On the (2+1)-Dimensional Extension of 1-Dimensional Toda Lattice Hierarchy

Yuji OGAWA
Pages: 48 - 65
We introduce a (2+1)-dimensional extension of the 1-dimensional Toda lattice hierar- chy. The hierarchy is given by two different formulations. For the first formulation, we obtain the bilinear identity for the ? -functions and construct explicit solutions ex- pressed by Wronski determinants. For the...
Research Article

Bilinear Approach to Supersymmetric KdV Equation

A S CARSTEA
Pages: 48 - 52
Extending the gauge-invariance principle for functions of the standard bilinear fomalism to the supersymmetric case, we define N = 1 supersymmetric Hirota operators. Using them, we bilinearize SUSY KdV equation. The solution for multiple collisions of super-solitons is given.
Research Article

On Integration of the Nonlinear d'Alembert-Eikonal System and Conditional Symmetry of Nonlinear Wave Equations

Renat Z. ZHDANOV
Pages: 49 - 61
We study integrability of a system of nonlinear partial differential equations consisting of the nonlinear d'Alembert equation 2u = F(u) and nonlinear eikonal equation uxµ uxµ = G(u) in the complex Minkowski space R(1, 3). A method suggested makes it possible to establish necessary and sufficient compatibility...
Research Article

Deformation quantization for almost-Kähler manifolds

Martin SCHLICHENMAIER
Pages: 49 - 54
On an arbitrary almost-Kähler manifold, starting from a natural affine connection with nontrivial torsion which respects the almost-Kähler structure, in joint work with A. Karabegov a Fedosov-type deformation quantization on this manifold was costructed. This contribution reports on the result and supplies...
Research Article

Basis of Joint Invariants for (1 + 1) Linear Hyperbolic Equations

I K, F M MAHOMED, C WAFO SOH
Pages: 49 - 59
We obtain a basis of joint or proper differential invariants for the scalar linear hperbolic partial differential equation in two independent variables by the infinitesimal method. The joint invariants of the hyperbolic equation consist of combinations of the coefficients of the equation and their derivatives...
Research Article

Tunnelling in Nonlocal Evolution Equations

Giovanni BELLETTINI, Anna DE MASI, Errico PRESUTTI
Pages: 50 - 63
We study "tunnelling" in a one-dimensional, nonlocal evolution equation by assigning a penalty functional to orbits which deviate from solutions of the evolution equation. We discuss the variational problem of computing the minimal penalty for orbits which connect two stable, stationary solutions.
Research Article

On the Recovery of the Free Surface from the Pressure within Periodic Traveling Water Waves

Joachim Escher, Torsten Schlurmann
Pages: 50 - 57
We present a consistent derivation of the pressure transfer function for small amplitude waves within the framework of linear wave theory and discuss some nonlinear aspects.
Short Communication

On Algebraic Integrability of the Deformed Elliptic Calogero­Moser Problem

L A KHODARINOVA, I A PRIKHODSKY
Pages: 50 - 53
Algebraic integrability of the elliptic Calogero­Moser quantum problem related to the deformed root systems A2(2) is proved. Explicit formulae for integrals are found.
Research Article

Construction of Special Solutions for Nonintegrable Systems

Sergey Yu VERNOV
Pages: 50 - 63
The Painlev´e test is very useful to construct not only the Laurent series solutions of systems of nonlinear ordinary differential equations but also the elliptic and trigonmetric ones. The standard methods for constructing the elliptic solutions consist of two independent steps: transformation of a...
Research Article

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...
Research Article

Symmetry Reduction and Exact Solutions of the Yang-Mills Equations

Victor LAHNO, Renat ZHDANOV, Wilhelm FUSHCHYCH
Pages: 51 - 72