# Journal of Nonlinear Mathematical Physics

1499 articles

**Research Article**

## Composition of Lie Group Elements from Basis Lie Algebra Elements

George W. Bluman, Omar Mrani-Zentar, Deshin Finlay

Pages: 528 - 557

It is shown explicitly how one can obtain elements of Lie groups as compositions of products of other elements based on the commutator properties of associated Lie algebras. Problems of this kind can arise naturally in control theory. Suppose an apparatus has mechanisms for moving in a limited number...

**Research Article**

## Lp - Lq Decay Estimates for Wave Equations with Time-Dependent Coefficients

Michael Reissig

Pages: 534 - 548

The goal of this survey article is to explain the up-to-date state of the theory of Lp - Lq decay estimates for wave equations with time-dependent coefficients. We explain the influence of oscillations in the coefficients by using a precise classification. Moreover, we will see how mass and dissipation...

**Research Article**

## Integrable Discretizations of Some Cases of the Rigid Body Dynamics

Yuri B. Suris

Pages: 534 - 560

A heavy top with a fixed point and a rigid body in an ideal fluid are important examples of Hamiltonian systems on a dual to the semidirect product Lie algebra e(n) = so(n) Rn . We give a Lagrangian derivation of the corresponding equations of motion, and introduce discrete time analogs of two integrable...

**Research Article**

## Mapping between the dynamic and mechanical properties of the relativistic oscillator and Euler free rigid body

Alberto Molgado, Adan Rodríguez

Pages: 534 - 547

In this work we investigate a formal mapping between the dynamical properties of the unidimensional relativistic oscillator and the asymmetrical rigid top at a clas- sical level. We study the relativistic oscillator within Yamaleevâ€™s interpretation of Nambu mechanics. Such interpretation is based on...

**Research Article**

## Minimal surfaces associated with orthogonal polynomials

Vincent Chalifour, Alfred Michel Grundland

Pages: 529 - 549

This paper is devoted to a study of the connection between the immersion functions of two-dimensional surfaces in Euclidean or hyperbolic spaces and classical orthogonal polynomials. After a brief description of the soliton surfaces approach defined by the Enneper-Weierstrass formula for immersion and...

**Research Article**

## The Virasoro action on the tau function for the constrained discrete KP hierarchy

Jipeng Cheng, Maohua Li, Jingsong He

Pages: 529 - 538

With the help of the squared eigenfunction potential, the action of the Virasoro symmetry on the tau function of the constrained discrete KP hierarchy is derived.

**Research Article**

## What an Effective Criterion of Separability says about the Calogero Type Systems

Stefan Rauch-Wojciechowski, Claes Waksjö

Pages: 535 - 547

In [15] we have proved a 1-1 correspondence between all separable coordinates on Rn (according to Kalnins and Miller [9]) and systems of linear PDEs for separable potetials V (q). These PDEs, after introducing parameters reflecting the freedom of choice of Euclidean reference frame, serve as an effective...

**Research Article**

## Analytic Behaviour of Competition among Three Species

P.G.L. Leach, J. Miritzis

Pages: 535 - 548

We analyse the classical model of competition between three species studied by May and Leonard (SIAM J Appl Math 29 (1975) 243-256) with the approaches of singlarity analysis and symmetry analysis to identify values of the parameters for which the system is integrable. We observe some striking relations...

**Research Article**

## Generalized Conditional Symmetries, Related Solutions and Conservation Laws of the Grad-Shafranov Equation with Arbitrary Flow

Rodica Cimpoiasu

Pages: 531 - 544

The generalized conditional symmetry (GCS) method is applied to the case of a generalized Grad-Shafranov equation (GGSE) with incompressible flow of arbitrary direction. We investigate the conditions which yield the GGSE that admits a special class of second-order GCSs. Three GCS generators and the associated...

**Research Article**

## On the dynamics of internal waves interacting with the equatorial undercurrent

Alan Compelli, Rossen Ivanov

Pages: 531 - 539

The interaction of the nonlinear internal waves with a nonuniform current with a specific form, characteristic for the equatorial undercurrent, is studied. The current has no vorticity in the layer, where the internal wave motion takes place. We show that the nonzero vorticity that might be occuring...

**Research Article**

## The gauge transformation of the *q*-deformed modified KP hierarchy

Jipeng Cheng, Jinzheng Wang, Xingyong Zhang

Pages: 533 - 542

In this paper, we mainly study three types of gauge transformation operators for the q-mKP hierarchy. The successive applications of these gauge transformation operators are derived. And the corresponding communities between them are also investigated.

**Research Article**

## On A Group Of Automorphisms Of The Noncommutative Burgers Hierarchy

Boris A. Kupershmidt

Pages: 539 - 549

Bäcklund transformations are constructed for the noncommutative Burgers hierarchy, generalizing the commutative ones of Weiss, Tabor, Carnevale, and Pickering. These transformations are shown to be invertible and form a group.

**Research Article**

## A Variational Principle for Volume-Preserving Dynamics

Giuseppe Gaeta, Paola Morando

Pages: 539 - 554

We provide a variational description of any Liouville, i.e. volume preserving, autnomous vector field on a smooth manifold. This is obtained via a "maximal degree" variational principle; critical sections for this are integral manifolds for the Liouville vector field. We work in coordinates and provide...

**Research Article**

## On Slant Magnetic Curves in *S*-manifolds

Güvenç Şaban, Cihan Özgür

Pages: 536 - 554

We consider slant normal magnetic curves in (2n + 1)-dimensional S-manifolds. We prove that γ is a slant normal magnetic curve in an S-manifold (M2m+s, φ, ξα, ηα, g) if and only if it belongs to a list of slant φ-curves satisfying some special curvature equations. This list consists of some specific...

**Research Article**

## Octahedral Structure of the Hirota–Miwa Equation

Satoru Saito

Pages: 539 - 550

The Hirota–Miwa equation is studied from the view point of derived category.

**Research Article**

## Existence of Natural and Conformally Invariant Quantizations of Arbitrary Symbols

P. Mathonet, F. Radoux

Pages: 539 - 556

A quantization can be seen as a way to construct a differential operator with prescribed principal symbol. The map from the space of symbols to the space of differential operators is moreover required to be a linear bijection.
In general, there is no natural quantization procedure, that is, spaces of...

**Research Article**

## Equilibria of a solvable *N*-body problem and related properties of the *N* numbers *x*_{n} at which the Jacobi polynomial of order *N* has the same value

_{n}

Oksana Bihun, Francesco Calogero

Pages: 539 - 551

The class of solvable N-body problems of “goldfish” type has been recently extended by including (the additional presence of) three-body forces. In this paper we show that the equilibria of some of these systems are simply related to the N roots xn of the polynomial equation
PN(α,β)
(x)= w, where...

**Research Article**

## On the Existence of Benthic Storms

Ronald Quirchmayr

Pages: 540 - 544

We study a model for the wind-induced current field of the Pacific ocean in order to demonstrate that currents in the surface layer are carried down to the deepest regions above the abyssal sea floor, which indicates the existence of the phenomenon of comparably strong currents in bottom regions as a...

**Research Article**

## On a Periodic 2-Component Camassa–Holm Equation with Vorticity

Qiaoyi Hu

Pages: 541 - 556

We consider a periodic 2-component Camassa–Holm equation with vorticity in the paper. We first give the local well-posedness and the blow-up criterion for strong solutions to the equation in the Sobolev space Hs,
s>32
. We then present a global existence result for strong solutions to the equation....

**Research Article**

## Vortex Trains in Super-Alfvénic Magnetogasdynamics. Application of Reciprocal-Bäcklund Transformations

C. Rogers, W.K. Schief

Pages: 548 - 564

A multi-parameter class of reciprocal transformations is coupled with the action of a Bäcklund transformation to construct periodic solutions of breather-type in plane, aligned, super-Alfvénic magnetogasdynamics. The constitutive law adopts a genealised K´arm´an-Tsien form.

**Research Article**

## On the (3, N ) Maurer-Cartan equation

Mauricio Angel, Jaime Camacaro, Rafael Díaz

Pages: 548 - 569

Deformations of the 3-differential of 3-differential graded algebras are controlled by the (3, N ) Maurer-Cartan equation. We find explicit formulae for the coefficients appearing in that equation, introduce new geometric examples of N -differential graded algebras, and use these results to study N Lie...

**Research Article**

## On deformation and classification of ∨-systems

V. Schreiber, A.P. Veselov

Pages: 543 - 583

The ∨-systems are special finite sets of covectors which appeared in the theory of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. Several families of ∨-systems are known, but their classification is an open problem. We derive the relations describing the infinitesimal deformations...

**Research Article**

## Vect(S1) Action on Pseudodifferential Symbols on S1 and (Noncommutative) Hydrodynamic Type Systems

Partha Guha

Pages: 549 - 565

The standard embedding of the Lie algebra V ect(S1 ) of smooth vector fields on the circle V ect(S1 ) into the Lie algebra D(S1 ) of pseudodifferential symbols on S1 identifies vector field f(x) x V ect(S1 ) and its dual as (f(x) x ) = f(x) (u(x)dx2 ) = u(x)-2 . The space of symbols can be viewed as...

**Research Article**

## Uniqueness for Autonomous Planar Differential Equations and the Lagrangian Formulation of Water Flows with Vorticity

Erik Wahlén

Pages: 549 - 555

We prove a uniqueness result for autonomous divergence-free systems of ODE's in the plane and give an application to the study of water flows with vorticity.

**Research Article**

## A discrete negative AKNS equation: generalized Cauchy matrix approach

Song-lin Zhao

Pages: 544 - 562

Generalized Cauchy matrix approach is used to investigate a discrete negative Ablowitz–Kaup–Newell–Segur (AKNS) equation. Several kinds of solutions more than multi-soliton solutions to this equation are derived by solving determining equation set. Furthermore, applying an appropriate continuum limit...

**Research Article**

## Asymptotic Scaling in a Model Class of Anomalous Reaction-Diffusion Equations

Giuseppe Gaeta, Rosaria Mancinelli

Pages: 550 - 566

We analyze asymptotic scaling properties of a model class of anomalous reactiodiffusion (ARD) equations. Numerical experiments show that solutions to these have, for large t, well defined scaling properties. We suggest a general framework to anlyze asymptotic symmetry properties; this provides an analytical...

**Research Article**

## The peculiar (monic) polynomials, the zeros of which equal their coefficients

F. Calogero, F. Leyvraz

Pages: 545 - 555

We evaluate the number of complex monic polynomials, of arbitrary degree N, the zeros of which are equal to their coefficients. In the following, we call polynomials with this property peculiar polynomials. We further show that the problem of determining the peculiar polynomials of degree N simplifies...

**Research Article**

## Symmetric waves are traveling waves for a shallow water equation modeling surface waves of moderate amplitude

Anna Geyer

Pages: 545 - 551

Following a general principle introduced by Ehrnström, Holden and Raynaud in 2009, we prove that for an equation modeling the free surface evolution of moderate amplitude waves in shallow water, all symmetric waves are traveling waves.

**Research Article**

## Toda Equations and -Functions of Genera One and Two

Shigeki Matsutani

Pages: 555 - 569

We study the Toda equations in the continuous level, discrete level and ultradiscrete level in terms of elliptic and hyperelliptic and functions of genera one and two. The ultradiscrete Toda equation appears as a discrete-valuation of recursion relations of functions.

**Research Article**

## Study on geometric structures on Lie algebroids with optimal control applications

Esmaeil Peyghan, Liviu Popescu

Pages: 550 - 580

We construct ρ£-covariant derivatives in π*π as the generalization of covariant derivative in π*π to £πE. Moreover, we introduce Berwald and Yano derivatives as two important classes of ρ£-covariant derivatives in π*π and we study properties of them. Finally, we solve an optimal control problem using...

**Research Article**

## Landau Levels in a Two-Dimensional Noncommutative Space: Matrix and Quaternionic Vector Coherent States

Mahouton Norbert Hounkonnou, Isiaka Aremua

Pages: 551 - 579

The behavior of an electron in an external uniform electromagnetic background coupled to an harmonic potential, with noncommuting space coordinates, is considered in this work. The thermodynamics of the system is studied. Matrix vector coherent states (MVCS) as well as quaternionic vector coherent states...

**Research Article**

## Miura-reciprocal transformations for non-isospectral Camassa-Holm hierarchies in 2 + 1 dimensions

P. G. Estévez, C. Sardón

Pages: 552 - 564

We present two hierarchies of partial differential equations in 2 + 1 dimensions. Since there exist reciprocal transformations that connect these hierarchies to the Calogero-Bogoyavlenski-Schiff equation and its modified version, we can prove that one of the hierarchies can be considered as a modified...

**Research Article**

## A simple-looking relative of the Novikov, Hirota-Satsuma and Sawada-Kotera equations

Alexander G. Rasin, Jeremy Schiff

Pages: 555 - 568

We study the simple-looking scalar integrable equation fxxt − 3(fx ft − 1) = 0, which is related (in different ways) to the Novikov, Hirota-Satsuma and Sawada-Kotera equations. For this equation we present a Lax pair, a Bäcklund transformation, soliton and merging soliton solutions (some exhibiting instabilities),...

**Research Article**

## On Testing Integrability

Peter H. van der Kamp, Jan A. Sanders

Pages: 561 - 574

We demonstrate, using the symbolic method together with p-adic and resultant methods, the existence of systems with exactly one or two generalized symmetries. Since the existence of one or two symmetries is often taken as a sure sign (or as the definition) of integrability, that is, the existence of...

**Research Article**

## Einstein-like Lorentzian Lie groups of dimension four

Amirhesam Zaeim

Pages: 556 - 570

Einstein-like examples of four-dimensional Lorentzian Lie groups are listed and geometric properties of each class have been investigated.

**Research Article**

## Binary Darboux Transformation for the Supersymmetric Principal Chiral Field Model

Bushra Haider, M. Hassan

Pages: 557 - 581

The standard binary Darboux transformation is investigated and is used to obtain quasideterminant multisoliton solutions of the supersymmetric chiral field model in two dimensions.

**Research Article**

## Propagation of Twist Solitons in Fully Inhomogeneous DNA Chains

Mariano Cadoni, Roberto de Leo, Sergio Demelio, Giuseppe Gaeta

Pages: 557 - 569

In the framework of a recently introduced model of DNA torsional dynamics, we argued — on the basis of perturbative considerations — that an inhomogeneous DNA chain could support long-lived soliton-type excitations due to the peculiar geometric structure of DNA and the effect of this on nonlinear torsional...

**Research Article**

## Singular Hartree equation in fractional perturbed Sobolev spaces

Alessandro Michelangeli, Alessandro Olgiati, Raffaele Scandone

Pages: 558 - 588

We establish the local and global theory for the Cauchy problem of the singular Hartree equation in three dimensions, that is, the modification of the non-linear Schrödinger equation with Hartree non-linearity, where the linear part is now given by the Hamiltonian of point interaction. The latter is...

**Research Article**

## Isometric Reflectionless Eigenfunction Transforms for Higher-order AOs

S.N.M. Ruijsenaars

Pages: 565 - 598

In a previous paper (Regular and Chaotic Dynamics 7 (2002), 351391, Ref. [1]), we obtained various results concerning reflectionless Hilbert space transforms arising from a general Cauchy system. Here we extend these results, proving in particular an isometry property conjectured in Ref. [1]. Crucial...

**Research Article**

## Distribution of positive type in Quantum Calculus

Akram Nemri

Pages: 566 - 583

In this paper, we study some remarkable spaces of Sq,(Rq,+) space of the q-tempered distribution introduced by M.A. Olshanetsky and V.B.K. Rogov [14], namely the q-analogue of the pseudo-measure FqL (Rq,+), the q-function of the positive type FqM , and we give a q-version of the Bochner-Shwartz theorem...

**Research Article**

## Classification of Fully Nonlinear Integrable Evolution Equations of Third Order

Rafael Hernández Heredero

Pages: 567 - 585

A fully nonlinear family of evolution equations is classified. Nine new integrable equtions are found, and all of them admit a differential substitution into the Korteweg-de Vries or Krichever-Novikov equations. One of the equations contains hyperelliptic functions, but it is transformable into the Krichever-Novikov...

**Research Article**

## Lax Integrability of the Modified Camassa-Holm Equation and the Concept of Peakons

Xiangke Chang, Jacek Szmigielski

Pages: 563 - 572

In this Letter we propose that for Lax integrable nonlinear partial differential equations the natural concept of weak solutions is implied by the compatibility condition for the respective distributional Lax pairs. We illustrate our proposal by comparing two concepts of weak solutions of the modified...

**Research Article**

## On a construction of self-dual gauge fields in seven dimensions

E.K. Loginov, A.N. Grishkov

Pages: 570 - 577

We consider gauge fields associated with a semisimple Malcev algebra. We construct a gauge-invariant Lagrangian and found a solution of modified Yang-Mills equations in seven dimensions.

**Research Article**

## Contiguity relations for linearisable systems of Gambier type

A. Ramani, B. Grammaticos, P. Guha

Pages: 565 - 576

We introduce the Schlesinger transformations for the Gambier, linearisable, equation and by combining the former construct the contiguity relations of the solutions of the latter. We extend the approach to the discrete domain obtaining thus the Schlesinger transformations and the contiguity relations...

**Research Article**

## On the global dynamics of the Newell–Whitehead system

Claudia Valls

Pages: 569 - 578

In this paper by using the Poincaré compactification in ℝ3 we make a global analysis of the model x′ = z, y′ = b(x−dy), z′ = x(x2 −1)+y+cz with b ∈ ℝ and c, d ∈ ℝ+, here known as the three-dimensional Newell–Whitehead system. We give the complete description of its dynamics on the sphere at infinity....

**Research Article**

## Optimal Solution for the Viscous Modified Camassa–Holm Equation

Anna Gao, Chunyu Shen

Pages: 571 - 589

In this paper, we study the optimal control problem for the viscous modified Camassa–Holm equation. We first prove the existence and uniqueness of a weak solution to this equation in a short interval by using the Galerkin method. Furthermore, the existence of an optimal solution to the viscous modified...

**Research Article**

## Note on algebro-geometric solutions to triangular Schlesinger systems

Vladimir Dragović, Vasilisa Shramchenko

Pages: 571 - 583

We construct algebro-geometric upper triangular solutions of rank two Schlesinger systems. Using these solutions we derive two families of solutions to the sixth Painlevé equation with parameters (1/8, ‒1/8, 1/8, 3=8) expressed in simple forms using periods of differentials on elliptic curves. Similarly...

**Research Article**

## Some spherical solutions of ideal magnetohydrodynamic equations

P.Y. Picard

Pages: 578 - 588

Some spherical solutions of the ideal magnetohydrodynamic (MHD) equations are obtained from the method of the weak transversality method (WTM), which is based on Lie group theory. This analytical method makes use of the symmetry group of the MHD system in situations where the â€œclassicalâ€ Lie approach...

**Research Article**

## Generalized negative flows in hierarchies of integrable evolution equations

Stephen C. Anco, Shahid Mohammad, Thomas Wolf, Chunrong Zhu

Pages: 573 - 606

A one-parameter generalization of the hierarchy of negative flows is introduced for integrable hierarchies of evolution equations, which yields a wider (new) class of non-evolutionary integrable nonlinear wave equations. As main results, several integrability properties of these generalized negative...

**Research Article**

## On Bianchi permutability of Bäcklund transformations for asymmetric quad-equations

Raphael Boll

Pages: 577 - 605

We prove the Bianchi permutability (existence of superposition principle) of Bäcklund transformations for asymmetric quad-equations. Such equations and their Bäcklund transformations form 3D consistent systems of a priori different equations. We perform this proof by using 4D consistent systems of quad-equations,...

**Research Article**

## The Heun equation and the Calogero-Moser-Sutherland system V: generalized Darboux transformations

Kouichi Takemura

Pages: 584 - 611

We obtain isomonodromic transformations for Heun's equation by generalizing the Darboux transformation, and we find pairs and triplets of Heun's equation which have the same monodromy structure. By composing generalized Darboux transformations, we establish a new construction of the commuting operator...

**Research Article**

## An exact solution for geophysical internal waves with underlying current in modified equatorial *β*-plane approximation*

Dong Su, Hongjun Gao

Pages: 579 - 603

In this paper, a modification of the standard geophysical equatorial β-plane model equations, incorporating a gravitational-correction term in the tangent plane approximation, is derived. We present an exact solution to meet the modified governing equations, whose form is explicit in the Lagrangian framework...

**Research Article**

## Group of Transformations with Respect to the Counterpart of Rapidity and Related Field Equations

Adán R. Rodríguez-Domínguez, Alejandro Martínez-González

Pages: 580 - 594

The Lorentz-group of transformations usually consists of linear transformations of the coordinates, keeping as invariant the norm of the four-vector in (Minkowski) space-time. Besides those linear transformations, one may construct different forms of nonlinear transformations of the coordinates keeping...

**Research Article**

## Asymptotic Approximations in Quantum Calculus

Ahmed Fitouhi, Kamel Brahim, Néji Bettaibi

Pages: 586 - 606

This paper aims to study the asymptotic approximation of some functions defined by the q-Jackson integrals, for a fix q ]0, 1[. For this purpose, we shall attempt to extend the classical methods by giving their q-analogues. In particular, a q-analogue of the Watson's lemma is discussed and new asymptotic...

**Research Article**

## Nonlocal symmetries and group invariant solutions for the coupled variable-coefficient Newell-Whitehead system

Yarong Xia, Ruoxia Yao, Xiangpeng Xin

Pages: 581 - 591

Starting from the Lax pairs, the nonlocal symmetries of the coupled variable-coefficient Newell-Whitehead system are obtained. By introducing an appropriate auxiliary dependent variable, the nonlocal symmetries are localized to Lie point symmetries and the coupled variable-coefficient Newell-Whitehead...

**Research Article**

## Laplace Transformation of Lie Class *ω* = 1 Overdetermined Systems

Boris Kruglikov

Pages: 583 - 611

In this paper, we investigate overdetermined systems of scalar PDEs on the plane with one common characteristic, whose general solution depends on one function of one variable. We describe linearization of such systems and their integration via Laplace transformation, relating this to Lie's integration...

**Research Article**

## On nonlocal symmetries of integrable three-field evolutionary systems

A.G. Meshkov

Pages: 589 - 611

Nonlocal symmetries for exactly integrable three-field evolutionary systems have been com- puted. Differentiation the nonlocal symmetries with respect to x gives a few hyperbolic systems for each evolution system. Zero curvature representations for all nonlocal systems and for some of the hyperbolic...

**Research Article**

## Liouville Integrability of Conservative Peakons for a Modified CH equation

Xiangke Chang, Jacek Szmigielski

Pages: 584 - 595

The modified Camassa-Holm (also called FORQ) equation is one of numerous cousins of the Camassa-Holm equation possessing non-smoth solitons (peakons) as special solutions. The peakon sector of solutions is not uniquely defined: in one peakon sector (dissipativea) the Sobolev H1 norm is not preserved,...

**Research Article**

## On the study of unitary representations of the twisted Heisenberg-Virasoro algebra via highest weight modules over affine Lie algebras*

Namhee Kwon

Pages: 584 - 592

In this paper, we first construct an analogue of the Sugawara operators for the twisted Heisenberg-Virasoro algebra. By using these operators, we show that every integrable highest weight module over an affine Lie algebra can be viewed as a unitary representation of the twisted Heisenberg-Virasoro algebra....

**Research Article**

## Morphisms Cohomology and Deformations of Hom-algebras

Anja Arfa, Nizar Ben Fraj, Abdenacer Makhlouf

Pages: 589 - 603

The purpose of this paper is to define cohomology complexes and study deformation theory of Hom-associative algebra morphisms and Hom-Lie algebra morphisms. We discuss infinitesimal deformations, equivalent deformations and obstructions. Moreover, we provide various examples.

**Research Article**

## Menelaus Relation, Hirota–Miwa Equation and Fay's Trisecant Formula are Associativity Equations

B. G. Konopelchenko

Pages: 591 - 603

It is shown that the celebrated Menelaus relation, Hirota–Miwa bilinear equation for KP hierarchy and Fay's trisecant formula similar to the WDVV equation are associativity conditions for structure constants of certain three-dimensional quasi-algebra.

**Research Article**

## Asymptotics behavior for the integrable nonlinear Schrödinger equation with quartic terms: Cauchy problem

Lin Huang

Pages: 592 - 615

We consider the Cauchy problem of integrable nonlinear Schrödinger equation with quartic terms on the line. The first part of the paper considers the Riemann-Hilbert formula via the unified method(also known as the Fokas method). The second part of the paper establishes asymptotic formulas for the solution...

**Research Article**

## Simple and collective twisted symmetries

G. Gaeta

Pages: 593 - 627

After the introduction of λ -symmetries by Muriel and Romero, several other types of so called “twisted symmetries” have been considered in the literature (their name refers to the fact they are defined through a deformation of the familiar prolongation operation); they are as useful as standard symmetries...

**Research Article**

## Generalised Inverse Scattering for a Linear PDE Associate to KdV

P.C. Sabatier

Pages: 599 - 613

Inverse Scattering methods for solving integrable nonlinear p.d.e. found their limits as soon as one tried to solve with them new boundary value problems. However, some of these problems, e.g. the quarter-plane problem, can be solved (e.g. by Fokas linear methods), for related linear p.d.e., (e.g. LKdV)....

**Research Article**

## Holomorphic last multipliers on complex manifolds

Mircea Crasmareanu, Cristian Ida, Paul Popescu

Pages: 596 - 619

The goal of this paper is to study the theory of last multipliers in the framework of complex manifolds with a fixed holomorphic volume form. The motivation of our study is based on the equivalence between a holomorphic ODE system and an associated real ODE system and we are interested how we can relate...

**Erratum**

## Erratum on "Some Symmetry Classifications of Hyperbolic Vector Evolution Equations": J Nonlinear Math. Phys. 12 suppl. 1 (2005), 1331.

Stephen C. Anco, Thomas Wolf

Pages: 607 - 608

**Research Article**

## Variational Operators, Symplectic Operators, and the Cohomology of Scalar Evolution Equations

M.E. Fels, E. Yaşar

Pages: 604 - 649

For a scalar evolution equation ut = K(t, x, u, ux, ..., u2m+1) with m ≥ 1, the cohomology space H1,2(ℛ∞) is shown to be isomorphic to the space of variational operators and an explicit isomorphism is given. The space of symplectic operators for ut = K for which the equation is Hamiltonian is also shown...

**Research Article**

## Lie point symmetries and ODEs passing the Painlevé test

D. Levi, D. Sekera, P. Winternitz

Pages: 604 - 617

The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlevé property are explored for ODEs of order n = 2, . . . , 5. Among the 6 ODEs identifying the Painlevé transcendents only PIII, PV and PVI have nontrivial symmetry algebras and that only for very...

**Research Article**

## Provenance of Type II hidden symmetries from nonlinear partial differential equations

Barbara Abraham-Shrauner, Keshlan S. Govinder

Pages: 612 - 622

The provenance of Type II hidden point symmetries of differential equations reduced from nonlinear partial differential equations is analyzed. The hidden symmetries are extra symmetries in addition to the inherited symmetries of the differential equations when the number of independent and dependent...

**Research Article**

## On a new technique to manufacture isochronous Hamiltonian systems: classical and quantal treatments

F. Calogero, F. Leyvraz

Pages: 612 - 636

We discuss a new technique to -modify real Hamiltonians so that they become isochronous while remaining real. Although the Ï‰-modified Hamiltonians thereby obtained often yield, in the classical context, singular motions, we exhibit and inves- tigate simple examples when this does not (quite) happen....

**Research Article**

## Four-Wave Semidiscrete Nonlinear Integrable System with *𝒫𝒯*-Symmetry

Oleksiy O. Vakhnenko

Pages: 606 - 622

The new type of third-order spectral operator suitable to generate new multifield semidiscrete nonlinear systems with two coupling parameters in the framework of zero-curvature equation is proposed. The evolution operator corresponding to the first integrable system in an infinite hierarchy is explicitly...

**Research Article**

## Darboux Integrability of a Simplified Friedman–Robertson–Walker Hamiltonian System

Jaume Llibre, Clàudia Valls

Pages: 607 - 617

We characterize the Darboux first integrals of a simplified Friedman–Robertson–Walker Hamiltonian system depending on one parameter.

**Research Article**

*SU*(1, 1) and *SU*(2) Perelomov number coherent states: algebraic approach for general systems

D. Ojeda-Guillén, M. Salazar-Ramírez, R. D. Mota, V. D. Granados

Pages: 607 - 619

We study some properties of the SU(1, 1) Perelomov number coherent states. The Schrödinger's uncertainty relationship is evaluated for a position and momentum-like operators (constructed from the Lie algebra generators) in these number coherent states. It is shown that this relationship is minimized...

**Research Article**

## Universal Solitonic Hierarchy

Alexei Shabat

Pages: 614 - 624

We describe recent results on the construction of hierarchies of nonlinear evolution equations associated to generalized second order spectral problems. The first results in this subject had been presented by Francesco Calogero.

**Research Article**

## Equivalence Classes of the Second Order Odes with the Constant Cartan Invariant

Vera V. Kartak

Pages: 613 - 640

Second order ordinary differential equations that possesses the constant invariant are investigated. Four basic types of these equations were found. For every type the complete list of nonequivalent equations is issued. As the examples the equivalence problem for the Painleve II equation, Painleve III...

**Research Article**

## On the discretization of Darboux Integrable Systems

Kostyantyn Zheltukhin, Natalya Zheltukhina

Pages: 616 - 632

We study the discretization of Darboux integrable systems. The discretization is done using x-, y-integrals of the considered continuous systems. New examples of semi-discrete Darboux integrable systems are obtained.

**Corrigendum**

## Corrigendum: “Classification of 3D Consistent Quad-Equations”

Raphael Boll

Pages: 618 - 620

**Research Article**

## Generating functions for characters and weight multiplicities of irreducible 𝓈𝓁(4)-modules

José Ferńandez Núñez, Wifredo García Fuertes, Askold M. Perelomov

Pages: 618 - 632

Generating functions for the characters of the irreducible representations of simple Lie algebras are rational functions where both the numerator and denominator can be expressed as polynomials in the characters corresponding to the fundamental weights. They encode much information on the representation...

**Research Article**

## RG Solutions for s at large Nc in d = 3 + 1 QCD

Yu.A. Simonov

Pages: 625 - 632

Solutions of RG equations for () and (Q) are found in the class of meromorphic functions satisfying asymptotic conditions at large Q (resp. small ), and analyticity properties in the Q2 plane. The resulting R(Q) is finite in the Euclidean Q2 region and agrees well at Q 1 GeV with the MS s(Q).

**Research Article**

## The modified Korteweg-de Vries equation on the quarter plane with *t*-periodic data

Guenbo Hwang

Pages: 620 - 634

We study the modified Korteweg-de Vries equation posed on the quarter plane with asymptotically t-periodic Dirichlet boundary datum u(0,t) in the sense that u(0,t) tends to a periodic function g̃0 (t) with period τ as t → ∞. We consider the perturbative expansion of the solution in a small ε > 0....

**Research Article**

## Discretization of Liouville type nonautonomous equations preserving integrals

Ismagil Habibullin, Natalya Zheltukhina

Pages: 620 - 642

The problem of constructing semi-discrete integrable analogues of the Liouville type integrable PDE is discussed. We call the semi-discrete equation a discretization of the Liouville type PDE if these two equations have a common integral. For the Liouville type integrable equations from the well-known...

**Research Article**

## Time Discretization of F. Calogero's "Goldfish" System

Yuri B. Suris

Pages: 633 - 647

Time-discretized versions of F. Calogero's rational and hyperbolic "goldfish" systems are presented, and their exact solutions are given.

**Research Article**

## Hybrid Ermakov-Painlevé IV Systems

Colin Rogers

Pages: 628 - 642

Ermakov-Painlevé IV coupled systems are introduced and associated Ermakov-type invariants isolated. These invariants are used to obtain systematic reduction of the system in terms of the canonical Painlevé IV equation. The procedure is applied to a Ermakov-Painlevé IV symmetry reduction of a coupled...

**Research Article**

## Finite genus solutions to the lattice Schwarzian Korteweg-de Vries equation

Xiaoxue Xu, Cewen Cao, Guangyao Zhang

Pages: 633 - 646

Based on integrable Hamiltonian systems related to the derivative Schwarzian Korteweg-de Vries (SKdV) equation, a novel discrete Lax pair for the lattice SKdV (lSKdV) equation is given by two copies of a Darboux transformation which can be used to derive an integrable symplectic correspondence. Resorting...

**Research Article**

## Riemann-Hilbert approach and *N*-soliton formula for a higher-order Chen-Lee-Liu equation

Juan Hu, Jian Xu, Guo-Fu Yu

Pages: 633 - 649

We consider a higher-order Chen-Lee-Liu (CLL) equation with third order dispersion and quintic nonlinearity terms. In the framework of the Riemann-Hilbert method, we obtain the compact N-soliton formula expressed by determinants. Based on the determinant solution, some properties for single soliton and...

**Research Article**

## Solitons of Wave Equation

Antoni Sym

Pages: 648 - 659

Modulated progressive wave solutions (solitons) to (3 + 1)dimensional wave equation are discussed within a general geometrical framework. The role of geodesic coordinates defined by hypersurfaces of Riemannian spaces is pointed out in this context. In particular in E3 orthogonal geodesic coordinates...

**Research Article**

## Symmetry reductions and exact solutions of Lax integrable 3-dimensional systems

H. Baran, I.S. Krasil'shchik, O.I. Morozov, P. Vojčák

Pages: 643 - 671

We present a complete description of 2-dimensional equations that arise as symmetry reductions of four 3- dimensional Lax-integrable equations: (1) the universal hierarchy equation uyy = uzuxy− uyuxz; (2) the 3D rdDym equation uty = uxuxy− uyuxx; (3) the equation uty = utuxy− uyutx, which we call modified...

**Research Article**

## Decomposition of 2-Soliton Solutions for the Good Boussinesq Equations

Vesselin Vatchev

Pages: 647 - 663

We consider decompositions of two-soliton solutions for the good Boussinesq equation obtained by the Hirota method and the Wronskian technique. The explicit forms of the components are used to study the dynamics of 2-soliton solutions. An interpretation in the context of eigenvalue problems arising from...

**Research Article**

## Systems of Hamilton-Jacobi equations

Julio Cambronero, Javier Pérez Álvarez

Pages: 650 - 658

In this article we develop a generalization of the Hamilton-Jacobi theory, by considering in the cotangent bundle an involutive system of dynamical equations.

**Review Article**

## Homogeneous Isotropic Turbulence: Geometric and Isometry Properties of the 2-point Velocity Correlation Tensor

Vladimir N. Grebenev, Martin Oberlack

Pages: 650 - 672

The emphasis of this review is both the geometric realization of the 2-point velocity correlation tensor field Bij(x,x′,t) and isometries of the correlation space K3 equipped with a (pseudo-) Riemannian metrics ds2(t) generated by the tensor field. The special form of this tensor field for homogeneous...

**Research Article**

## Quasi-Exactly Solvable Hamiltonians related to Root Spaces

Alexander V. Turbiner

Pages: 660 - 675

sl(2)-Quasi-Exactly-Solvable (QES) generalization of the rational An, BCn, G2, F4, E6,7,8 Olshanetsky-Perelomov Hamiltonians including many-body Calogero Hamiltonian is found. This generalization has a form of anharmonic perturbations and it appears naurally when the original rational Hamiltonian is...

**Research Article**

## Integrability conditions of a weak saddle in generalized Liénard-like complex polynomial differential systems

Jaume Giné, Claudia Valls

Pages: 664 - 678

We consider the complex differential system
x˙=x+yf(x), y˙=−y+xf(y),
where f is the analytic function f(z)=∑j=1∞ajzj with aj ∈ ℂ. This system has a weak saddle at the origin and is a generalization of complex Liénard systems. In this work we study its local analytic integrability.

**Research Article**

## Correlation Function of Asymmetric Simple Exclusion Process with Open Boundaries

Masaru Uchiyama, Miki Wadati

Pages: 676 - 688

We investigate the correlation functions of the one-dimensional Asymmetric Simple Exclusion Process (ASEP) with open boundaries. The conditions for the boundaries are made most general. The correlation function is expressed in a multifold integral whose behavior we study in detail. We present a phase...

**Research Article**

## Symmetry classification of scalar Ito equations with multiplicative noise

Giuseppe Gaeta, Francesco Spadaro

Pages: 679 - 687

We provide a symmetry classification of scalar stochastic equations with multiplicative noise. These equations can be integrated by means of the Kozlov procedure, by passing to symmetry adapted variables.

**Research Article**

## On the Equilibrium Configuration of the BC-type Ruijsenaars-Schneider System

Jan F. Van Diejen

Pages: 689 - 696

It is shown that the ground-state equilibrium configurations of the trigonometric Btype Ruijsenaars-Schneider systems are given by the zeros of Askey-Wilson polynomals.

**Research Article**

## Gambier lattices and other linearisable systems

Basil Grammaticos, Alfred Ramani

Pages: 688 - 696

We propose two different appraoches to extending the Gambier mapping to a two-dimensional lattice equation. A first approach relies on a hypothesis of separate evolutions in each of the two directions. We show that known equations like the Startsev-Garifullin-Yamilov equation, the Hietarinta equation,...

**Research Article**

## Trigonal Toda Lattice Equation

Shigeki Matsutani

Pages: 697 - 704

In this article, we give the trigonal Toda lattice equation,
−12d3dt3qℓ(t)=eqℓ+1(t)+eqℓ+ζ3(t)++eqℓ+ζ32(t)−3eqℓ(t),
for a lattice point ℓ ∈ [ζ3] as a directed 6-regular graph where ζ3=e2π−1/3, and its elliptic solution for the curve y(y − s) = x
3, (s ≠ 0).