1487 articles

Sandra Carillo

Pages: i - iii

Dimitry Leites

Pages: v - viii

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,...

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.

F. Calogero

Pages: 0 - 0

Four books published by Birkhäuser are reviewed.

F. Calogero

Pages: 0 - 0

Seven books published by Birkhäuser are reviewed.

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...

Jinbing Chen, Rong Tong

The Hirota equation is reduced to a pair of complex Finite-dimensional Hamiltonian Systems (FDHSs) with real-valued Hamiltonians, which are proven to be completely integrable in the Liouville sense. It turns out that involutive solutions of the complex FDHSs yield finite parametric solutions of the Hirota...

Gh. Haghighatdoost, S. Abdolhadi-Zangakani, J. Abedi-Fardad

In this work, we give a method to construct compatible Poisson structures on Lie groups by means of structure constants of their Lie algebras and some vector field. In this way we calculate some compatible Poisson structures on low-dimensional Lie groups. Then, using a theorem by Magri and Morosi, we...

Matthew Randall

We show that the solutions to the second-order differential equation associated to the generalised Chazy equation with parameters k = 2 and k = 3 naturally show up in the conformal rescaling that takes a representative metric in Nurowski’s conformal class associated to a maximally symmetric (2,3,5)-distribution...

Siqi Jian, Jipeng Cheng

In this paper, we first construct the squared eigenfunction symmetries for the q-deformed Kadomtsev–Petviashvili (KP) and q-deformed modified KP hierarchies, including the unconstrained and constrained cases. Then the Miura links of the squared eigenfunction symmetries are investigated. At last, we also...

Roman Kozlov

The note provides the relation between symmetries and first integrals of Itô stochastic differential equations and symmetries of the associated Kolmogorov Backward Equation (KBE). Relation between the symmetries of the KBE and the symmetries of the Kolmogorov forward equation is also given.

Robert Conte

We provide closed form solutions for an equation which describes the transport of turbulent kinetic energy in the framework of a turbulence model with a single equation.

Arezoo Zohrabi, Pasha Zusmanovich

We prove simplicity of algebras in the title, and compute their δ-derivations and symmetric associative forms.

I.A. Batalin, S.E. Konstein, I.V. Tyutin

The algebra
≔H1,ν(I2(2m+1))
of observables of the Calogero model based on the root system I2(2m + 1) has an m-dimensional space of traces and an (m + 1)-dimensional space of supertraces. In the preceding paper we found all values of the parameter ν for which either the space of traces contains...

Hongli An, Engui Fan, Manwai Yuen

In this paper, by introducing a fractional transformation, we obtain a bilinear formulation for the (N + 1)-dimensional Burgers equation. Such a formulation constitutes a bilinear extension to the (N + 1)-dimensional Cole-Hopf transformation between the (N + 1)-dimensional Burgers system and generalized...

Juan Hu, Jia-Liang Ji, Guo-Fu Yu

In this paper, we study the correspondence between the Coupled Dispersionless (CD)-type equations and the Short Pulse (SP)-type equations. From the real and complex modified CD equations, we construct the real and complex Modified Short Pulse (mSP) equations geometrically and algebraically. From the...

Yi Zhao, Engui Fan

In this article, we focus on the inverse scattering transformation for the Fokas–Lenells (FL) equation with nonzero boundary conditions via the Riemann–Hilbert (RH) approach. Based on the Lax pair of the FL equation, the analyticity, symmetry and asymptotic behavior of the Jost solutions and scattering...

V.A. Dorodnitsyn, E.I. Kaptsov, S.V. Meleshko

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...

Gülden Gün Polat, Teoman Özer

This study focuses on the analysis of Ramsey dynamical model with current Hamiltonian defining an optimal control problem in a neoclassical growth model by utilizing Lie group theory. Lie point symmetries of coupled nonlinear first-order ordinary differential equations corresponding to first-order conditions...

Javier Pérez Álvare

In this article, given a regular Lagrangian system L on the phase space TM of the configuration manifold M and a 1-parameter group G of transformations of M whose lifting to TM preserve the canonical symplectic dynamics associated to L, we determine conditions so that its infinitesimal generator produces...

Özlem Orhan, Teoman Özer

The analytical solutions of a nonlinear fin problem with variable thermal conductivity and heat transfer coefficients are investigated by considering theory of Lie groups and its relations with λ-symmetries and Prelle-Singer procedure. Additionally, the classification problem with respect to different...

Cornelis van der Mee

In this article we derive the triangular representations of the fundamental eigensolutions of the focusing 1 + 1 AKNS system with symmetric nonvanishing boundary conditions. Its continuous spectrum equals
∪[-iμ,iμ]
, where μ is the absolute value of the AKNS solution at spatial infinity. We...

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.

Yong Wang

In this paper, we classify affine Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections and perturbed canonical connections and perturbed Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.

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...

Norbert Euler, Enrique G. Reyes

Pages: 1 - 2

Nalini Joshi, Masatoshi Noumi, Hidetaka Sakai, Claude M. Viallet

Pages: 1 - 2

M. Bruschi, F. Calogero, F. Leyvraz, M. Sommacal

Pages: 1 - 31

Several applications of an explicitly invertible transformation are reported. This transformation is elementary and therefore all the results obtained via it might be considered trivial; yet the findings highlighted in this paper are generally far from appearing trivial until the way they are obtained...

Alexander Dynin

Pages: 1 - 13

A boson-fermion correspondence allows an analytic definition of functional super derivative, in particular, and a bosonic functional calculus, in general, on Bargmann Gelfand triples for the second super quantization. A Feynman integral for the super transformation matrix elements in terms of bosonic...

Kostis Andriopoulos, Tassos Bountis, K. Van Der Weele, Liana Tsigaridi

Pages: 1 - 12

We study the solitary wave solutions of a non-integrable generalized KdV equation proposed by Fokas [A. S. Fokas, Physica D87, 145 (1995)], aiming to describe unidirectional waves in shallow water with greater accuracy than the standard KdV equation. This generalized equation includes higher-order terms...

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.

R. Martini, P.K.H. Gragert

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.

J.M. Cerveró, O. Zurrón

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),...

A.N. Leznov

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.

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...

David Henry

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.

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...

S. Ianuş, R. Mazzocco, G.E. Vilcu

Pages: 1 - 8

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...

S. Abenda, Yu Fedorov

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.

Maciej Blaszak

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.

Viktor Abramov

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...

Barbara Abraham-Shrauner

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....

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...

Claude Brezinski

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,...

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.

Jorge E. Solomin, Marcela Zuccalli

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...

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.

M.J. Ablowitz, C.D. Ahrens

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...

Sergey I. Agafonov

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...

Mats Ehrnström

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.

Marek Szydłowski, Marek Biesiada

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...

V.N. Grebenev, M. Oberlack

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...

Francesco Calogero

Pages: 1 - 6

A simple application of a neat formula relating the time evolution of the N zeros of a (monic) time-dependent polynomial of degree N in the complex variable w to the time evolution of its N coefficients allows to identify integrable Hamiltonian N-body problems in the plane featuring N arbitrary functions,...

Ali Zaghouani

Pages: 1 - 20

Starting from an operator given as a product of q-exponential functions in irreducible representations of the positive discrete series of the q-deformed algebra suq(1, 1), we express the associated matrix elements in terms of d-orthogonal polynomials. An algebraic setting allows to establish some properties...

Marcus Kardell

Pages: 1 - 16

In this article we study a new kind of unbounded solutions to the Novikov equation, found via a Lie symmetry analysis. These solutions exhibit peakon creation, i.e., these solutions are smooth up until a certain finite time, at which a peak is created. We show that the functions are still weak solutions...

Sergi Simon

Pages: 1 - 16

This is an example of application of Ziglin-Morales-Ramis algebraic studies in Hamiltonian integrability, more specifically the result by Morales, Ramis and Simó on higher-order variational equations, to the well-known Friedmann-Robertson-Walker cosmological model. A previous paper by the author formalises...

Claudia Valls

Pages: 1 - 8

In this work we consider the modified Michaelis-Menten equation in biochemistry
x˙=-a(E-y)x+by, y˙=a(E-y)x-(b+r)y, z˙=ry.
It models the enzyme kinetics. We contribute to the understanding of its global dynamics, or more precisely, to the topological structure of its orbits by studying the integrability...

Metin Gürses, Atalay Karasu, Refik Turhan

Pages: 1 - 6

We construct the recursion operators for the non-commutative Burgers equations using their Lax operators. We investigate the existence of any integrable mixed version of left- and right-handed Burgers equations on higher symmetry grounds.

Martin Kohlmann

Pages: 1 - 8

In this paper, we study the μ-variant of the periodic b-equation and show that this equation can be realized as a metric Euler equation on the Lie group Diff∞(������) if and only if b = 2 (for which it becomes the μ-Camassa–Holm equation). In this case, the inertia operator generating the metric on Diff∞(������)...

A. M. Perelomov

Pages: 1 - 5

In this note we give the conditions for the existence of algebraic geodesics on some two-dimensional quadrics, namely, on hyperbolic paraboloids and elliptic paraboloids. It appears that in some cases, such geodesics are the rational space curves.

Askold M. Perelomov

Pages: 1 - 6

By considering a relation between Euler’s trinomial problem and the problem of decomposing tensor powers of the adjoint 𝔰𝔩(2)-module I derive some new results for both problems, as announced in arXiv:1902.08065.

Xianguo Geng, Yunyun Zhai, Bo Xue, Jiao Wei

Pages: 1 - 23

Based on the Lenard recursion relation and the zero-curvature equation, we derive a hierarchy of long wave-short wave type equations associated with the 3 × 3 matrix spectral problem with three potentials. Resorting to the characteristic polynomial of the Lax matrix, a trigonal curve is defined, on which...

Denis Blackmore, Anatoliy K. Prykarpatsky

Pages: 1 - 15

A novel approach — based upon vertex operator representation — is devised to study the AKNS hierarchy. It is shown that this method reveals the remarkable properties of the AKNS hierarchy in relatively simple, rather natural and particularly effective ways. In addition, the connection of this vertex...

Alexander A. Alexeyev

Pages: 1 - 33

This article continues the series of the works of 1998–2007 years devoted to the Multidimensional Superposition Principle, the concept easily explaining both classical soliton and more complex wave interactions in nonlinear PDEs and allowing one, in particular, to construct the general Superposition...

George Svetlichny

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...

Serge E. Bouquet, Robert Conte, Vincent Kelsch, Fabien Louvet

Pages: 3 - 17

We perform the analytic study of the buoyancy-drag equation with a time-dependent acceleration γ(t) by two methods. We first determine its equivalence class under the point transformations of Roger Liouville, and thus for some values of γ(t) define a time-dependent Hamiltonian from which the buoyancy-drag...

Johan van de Leur, Henrik Aratyn

Pages: 3 - 16

We introduce certain Bäcklund transformations for rational solutions of the Painlevé VI equation. These transformations act on a family of Painlevé VI tau functions. They are obtained from reducing the Hirota bilinear equations that describe the relation between certain points in the 3 component polynomial...

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...

W. Sarlet, F. Cantrijn, E. Martínez

Pages: 5 - 24

N.V. Alexeeva, I.V. Barashenkov, G.P. Tsironis

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...

Gerald A. Goldin

Pages: 6 - 11

An enlarged gauge group acts nonlinearly on the class of nonlinear Schrödinger equations introduced by the author in joint work with Doebner. Here the equations and the group action are displayed in the presence of an external electromagnetic field. All the gauge-invariants are listed for the coupled...

Mohamed Kachkachi

Pages: 7 - 12

At the quantum level of a bidimensional conformal model, the conformal symmtry is broken by the diffeomorphism anomaly and the conformal covariance is not maintained. Here we interpret geometrically this conformal covariance as an exact holomorphy condition on a two-dimensional Riemann surface on which...

Jipeng Cheng

Pages: 7 - 19

The Miura and anti-Miura transformations between the q-deformed KP and the q-deformed modified KP hierarchies are investigated in this paper. Then the auto-Backlund transformations for the q-deformed KP and q-deformed modified KP hierarchies are constructed through the combinations of the Miura and anti-Miura...

Marcus Wunsch

Pages: 7 - 11

We show that the geodesic flow on the infinite-dimensional group of diffeomorphisms of the circle, endowed with a fractional Sobolev metric at the identity, is described by the generalized Constantin–Lax–Majda equation with parameter a=−12.

Yohei Tutiya

Pages: 7 - 23

A previously unknown bright N-soliton solution for an intermediate nonlinear Schrödinger equation of focusing type is presented. This equation is constructed as a reduction of an integrable system related to a Sato equation of a 2-component KP hierarchy for certain differential-difference dispersion...

S.E. Konstein, I.V. Tyutin

Pages: 7 - 11

If G is a finite Coxeter group, then symplectic reflection algebra H := H1,η (G) has Lie algebra 𝔰𝔩2 of inner derivations and can be decomposed under spin: H = H0 ⊕ H1/2 ⊕ H1 ⊕ H3/2 ⊕ ... We show that if the ideals ℐi (i = 1,2) of all the vectors from the kernel of degenerate bilinear forms Bi(x,y)...

Johan Byström

Pages: 8 - 30

In this paper we study homogenization of quasi-linear partial differential equations of the form -div (a (x, x/h, Duh)) = fh on with Dirichlet boundary conditions. Here the sequence (h) tends to 0 as h and the map a (x, y, ) is periodic in y, monotone in and satisfies suitable continuity conditions....

Ismail Naci Cangül, Veli Kurt, Yilmaz Simsek, Hong Kyung Pak, Seog-Hoon Rim

Pages: 8 - 14

The purpose of this paper is to consider q-Euler numbers and polynomials which are q-extensions of ordinary Euler numbers and polynomials by the computations of the p-adic q-integrals due to T. Kim, cf. [1, 3, 6, 12], and to derive the "complete sums for q-Euler polynomials" which are evaluated by using...

M.L. Gandarias, M.S. Bruzon

Pages: 8 - 12

We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions f(u) for which this equation admit either the classical or the...