Journal of Nonlinear Mathematical Physics

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905 articles

Book Reviews by F Calogero

F Calogero
Pages: 0 - 0
Four books published by Birkhäuser are reviewed.

Book Reviews by F Calogero

F Calogero
Pages: 0 - 0
Three books are reviewed, namely Masao Nagasawa: Schroedinger Equations and Diffusion Theory. Birkhaeuser, Basel Boston Berlin, 1993. 332 pages. David Wick (with a mathematical appendix by William Farris): The Infamous Bounary - Seven Decades of Controversy in Quantum Physics. Birkhaeuser. Boston Basel...


Adrian Constantin, Lotfi A. Zadeh
Pages: 0 - 0

Book Reviews by F Calogero

F Calogero
Pages: 0 - 0
Seven books published by Birkhäuser are reviewed.

Book Review by P A Clarkson

P A Clarkson
Pages: 0 - 0
Peter E Hydon: Symmetry Methods for Differential Equations: A Beginner's Guide Cambridge Texts in Applied Mathematics, Cambridge University Press, 2000.

Book Review by B A Kupershmidt

B A Kupershmidt
Pages: 0 - 0
Five books are reviewed, namely Bruce C Berndt: Ramanujan's Notebooks. Part I. (With a foreword by S Chadrasekhar). Springer-Verlag, New York Berlin, 1985. 357 pages. --: Ramanujan's Notebooks. Part II. Springer-Verlag, New York Berlin, 1989. 359 pages. --: Ramanujan's Notebooks. Part III. Springer-Verlag,...

Book Review by P G L Leach

P G L Leach
Pages: 0 - 0
N H Ibragimov: Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley, New York, 1999, 347 pages.

A Chorin-Type Formula for Solutions to a Closure Model for the von K´arm´an­Howarth Equation 1

Pages: 1 - 9
The article is devoted to studying the Millionshtchikov closure model (a particular case of a model introduced by Oberlack [14]) for isotropic turbulence dynamics which appears in the context of the theory of the von K´arm´an-Howarth equation. We write the model in an abstract form that enables us to...

UV Manifold and Integrable Systems in Spaces of Arbitrary Dimension

Pages: 1 - 7
The 2n dimensional manifold with two mutually commutative operators of differetiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general solution of them is presented in explicit form.

Lax Pairs, Painlevé Properties and Exact Solutions of the Calogero Korteweg-de Vries Equation and a New (2 + 1)-Dimensional Equation

Song-Ju YU, Kouichi TODA
Pages: 1 - 13
We prove the existence of a Lax pair for the Calogero Korteweg-de Vries (CKdV) equation. Moreover, we modify the T operator in the the Lax pair of the CKdV equation, in the search of a (2 + 1)-dimensional case and thereby propose a new equation in (2+1) dimensions. We named this the (2+1)-dimensional...

Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves

Pages: 1 - 7
We show that within the framework of linear theory the particle paths in a periodic gravity-capillary or pure capillary deep-water wave are not closed.

Algebraic Discretization of the Camassa-Holm and Hunter-Saxton Equations

Rossen I Ivanov
Pages: 1 - 12
The Camassa-Holm (CH) and Hunter-Saxton (HS) equations have an interpretation as geodesic flow equations on the group of diffeomorphisms, preserving the H 1 and H 1 right-invariant metrics correspondingly. There is an analogy to the Euler equations in hydrodynamics, which describe geodesic flow for a...

Kovalevski Exponents and Integrability Properties in Class A Homogeneous Cosmological Models

Pages: 1 - 10
Qualitative approach to homogeneous anisotropic Bianchi class A models in terms of dynamical systems reveals a hierarchy of invariant manifolds. By calculating the Kovalevski Exponents according to Adler - van Moerbecke method we discuss how algebraic integrability property is distributed in this class...

Complex Angle Variables for Constrained Integrable Hamiltonian Systems

Pages: 1 - 4
We propose Dirac formalism for constraint Hamiltonian systems as an useful tool for the algebro-geometrical and dynamical characterizations of a class of integrable systems, the so called hyperelliptically separable systems. As a model example, we apply it to the classical geodesic flow on an ellipsoid.

Hidden Symmetries, First Integralsvand Reduction of Order of Nonlinear Ordinary Differential Equations

Pages: 1 - 9
The reduction of nonlinear ordinary differential equations by a combination of first integrals and Lie group symmetries is investigated. The retention, loss or even gain in symmetries in the integration of a nonlinear ordinary differential equation to a first integral are studied for several examples....

On a "Quasi" Integrable Discrete Eckhaus Equation

Pages: 1 - 12
In this paper, a discrete version of the Eckhaus equation is introduced. The discretiztion is obtained by considering a discrete analog of the transformation taking the cotinuous Eckhaus equation to the continuous linear, free Schrödinger equation. The resulting discrete Eckhaus equation is a nonlinear...

From Bi-Hamiltonian Geometry to Separation of Variables: Stationary Harry-Dym and the KdV Dressing Chain

Pages: 1 - 13
Separability theory of one-Casimir Poisson pencils, written down in arbitrary coordnates, is presented. Separation of variables for stationary Harry-Dym and the KdV dressing chain illustrates the theory.

On a graded q-differential algebra

Pages: 1 - 8
Given an associative unital ZN -graded algebra over the complex numbers we construct the graded q-differential algebra by means of a graded q-commutator, where q is a primitive N-th root of unity. The N-differential d of the graded q-differential algebra is a homogeneous endomorphism of degree 1 satisfying...

Noncentral Extensions as Anomalies in Classical Dynamical Systems

Pages: 1 - 9
A two cocycle is associated to any action of a Lie group on a symplectic manifold. This allows to enlarge the concept of anomaly in classical dynamical systems considered by F Toppan in [J. Nonlinear Math. Phys. 8, Nr. 3 (2001), 518­533] so as to encompass some extensions of Lie algebras related to noncanonical...

On the Quantization of Yet Another Two Nonlinear Harmonic Oscillators

Francesco CALOGERO
Pages: 1 - 6
In two previous papers the quantization was discussed of three one-degree-of-freedom Hamiltonians featuring a constant c, the value of which does not influence at all the corresponding classical dynamics (which is characterized by isochronous solutions, all of them periodic with period 2: "nonlinear...

A Note on Surface Profiles for Symmetric Gravity Waves with Vorticity

Pages: 1 - 8
We consider a nontrivial symmetric periodic gravity wave on a current with nondcreasing vorticity. It is shown that if the surface profile is monotone between trough and crest, it is in fact strictly monotone. The result is valid for both finite and infinite depth.

Asymptotic behavior of discrete holomorphic maps zc and log(z)

Pages: 1 - 14
It is shown that discrete analogs of zc and log(z), defined via particular "integrable" circle patterns, have the same asymptotic behavior as their smooth counterparts. These discrete maps are described in terms of special solutions of discrete Painlevé-II equations, asymptotics of these solutions providing...

Solutions of WDVV Equations in Seiberg-Witten Theory from Root Systems

Pages: 1 - 4
We present a complete proof that solutions of the WDVV equations in Seiberg-Witten theory may be constructed from root systems. A generalization to weight systems is proposed.

Novel Robust Stability Criteria for Uncertain Stochastic Neural Networks with Time-Varying Delay

Yonggang Chen, Yunrui Guo, Wenlin Li
Pages: 1 - 9
This paper considers the robust stability analysis problem for a class of uncertain stochastic neural net- works with time-varying delay. Based on the Lyapunov functional method, and by resorting to the new technique for estimating the upper bound of the stochastic derivative of Lyapunov functionals,...

On the Moyal Quantized BKP Type Hierarchies

Dolan Chapa SEN, A. Roy CHOWDHURY
Pages: 1 - 7
Quantization of BKP type equations are done through the Moyal bracket and the formalism of pseudo-differential operators. It is shown that a variant of the dressing operator can also be constructed for such quantized systems.

So. . . what was the question?

Gérard G EMCH
Pages: 1 - 8
An overview of the lectures at the 2002 Bialowiea Workshop is presented. The symbol* after a proper name indicates that a copy of the corresponding contribution to the proceedings was communicated to the author of this summary.

Comparisons Between Vector and Matrix Padé Approximants

Pages: 1 - 12
In this paper, we compare the degrees and the orders of approximation of vector and matrix Padé approximants for series with matrix coefficients. It is shown that, in this respect, vector Padé approximants have better properties. Then, matrix­vector Padé approximants are defined and constructed. Finally,...

Integrability of the Perturbed KdV Equation for Convecting Fluids: Symmetry Analysis and Solutions

J.M. CERVER´O, O. Zurr'on
Pages: 1 - 23
As an example of how to deal with nonintegrable systems, the nonlinear partial differential equation which describes the evolution of long surface waves in a convecting fluid ut + (uxxx + 6uux) + 5uux + (uxxx + 6uux)x = 0, is fully analyzed, including symmetries (nonclassical and contact transformatons),...

Navier­Stokes Equations with Nonhomogeneous Dirichlet Data

Herbert AMANN
Pages: 1 - 11
We discuss the solvability of the time-dependent incompressible Navier­Stokes equtions with nonhomogeneous Dirichlet data in spaces of low regularity.

Nonlocal Extensions of Similarity Methods

George Bluman
Pages: 1 - 24
Similarity methods include the calculation and use of symmetries and conservation laws for a given partial differential equation (PDE). There exists a variety of software to calculate and use local symmetries and local conservation laws. However, it is often the case that a given PDE admits no useful...

Quantum Integrability of the Dynamics on a Group Manifold

V Aldaya, M Calixto, J Guerrero, F F Lopez-Ruiz
Pages: 1 - 12
We study the dynamics of a particle moving on the SU(2) group manifold. An exact quantization of this system is accomplished by finding the unitary and irreducible representations of a finite-dimensional Lie subalgebra of the whole Poisson algebra in phase space. In fact, the basic position and momentum...

Non-linear Schrödinger Equations, Separation and Symmetry

Pages: 2 - 26
We investigate the symmetry properties of hierarchies of non-linear Schrödinger equations, introduced in [2], which describe non-interacting systems in which tensor product wave-functions evolve by independent evolution of the factors (the separation property). We show that there are obstructions to...

Neumann and Bargmann Systems Associated with an Extension of the Coupled KdV Hierarchy

Zhimin JIANG
Pages: 5 - 12
An eigenvalue problem with a reference function and the corresponding hierarchy of nonlinear evolution equations are proposed. The bi-Hamiltonian structure of the hierarchy is established by using the trace identity. The isospectral problem is nonlinearized as to be finite-dimensional completely integrable...

Taming Spatiotemporal Chaos by Impurities in the Parametrically Driven Damped Nonlinear Schrödinger Equation

Pages: 5 - 12
Solitons of the parametrically driven, damped nonlinear Schrödinger equation become unstable and seed spatiotemporal chaos for sufficiently large driving amplitudes. We show that the chaos can be suppressed by introducing localized inhomogeneities in the parameters of the equation. The pinning of the...