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.
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...
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
(x)= w, where...
Pages: 539 - 550
The Hirota–Miwa equation is studied from the view point of derived category.
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...
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...
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,
. We then present a global existence result for strong solutions to the 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...
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.
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...
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...
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.
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...
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...
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...
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.
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.
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...
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...
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...
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),...
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...
Pages: 556 - 570
Einstein-like examples of four-dimensional Lorentzian Lie groups are listed and geometric properties of each class have been investigated.
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.
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...
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...
Pages: 565 - 598
In a previous paper (Regular and Chaotic Dynamics 7 (2002), 351391, Ref. ), 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. . Crucial...
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 , 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...
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...
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...
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.
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...
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....
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...
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...
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...
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...
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,...
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...
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...
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...
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...
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...
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...
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...
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,...
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....
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.
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.
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...
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...
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)....
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...
Stephen C. Anco, Thomas Wolf
Pages: 607 - 608
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...
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...
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....
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...
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...
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...
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.
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.
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...
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.
Pages: 618 - 620
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...
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).
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....
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...
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.
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...
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...
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...
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...
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...
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...
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.
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...
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...
Jaume Giné, Claudia Valls
Pages: 664 - 678
We consider the complex differential system
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.
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...
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.
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.
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,...
Pages: 697 - 704
In this article, we give the trigonal Toda lattice equation,
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).