Journal of Nonlinear Mathematical Physics

We show that every transitive Lie algebroid over a connected symplectic manifold comes from an intrinsic Lie algebroid of a symplectic leaf of a certain Poisson structure. The reconstruction of the corresponding Poisson structures from the Lie algebroid is given in terms of coupling tensors.

The maximal superintegrability of the intrinsic harmonic oscillator potential on N-dimensional spaces with constant curvature is revisited from the point of view of sl(2)-Poisson coalgebra symmetry. It is shown how this algebraic approach leads to a straightforward definition of a new large family of...

We present an explicit solution for the geophysical equatorial deep water waves in the f-plane approximation.

We consider symmetries and perturbed symmetries of canonical Hamiltonian equations of motion. Specifically we consider the case in which the Hamiltonian equations exhibit a Λ-symmetry under some Lie point vector field. After a brief survey of the relationships between standard symmetries and the existence...

In this paper we revisit Noether's theorem on the constants of motion for Lagrangian mechanical systems in the ODE case, with some new perspectives on both the theoretical and the applied side. We make full use of invariance up to a divergence, or, as we call it here, Bessel-Hagen (BH) invariance....

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

Nonlinear systems of differential equations for E and H which are compatible with the Galilei relativity principle are proposed. It is proved that the Schrödinger equation together with the nonlinear equation of hydrodynamic type for E and H are invariant with respect to the Galilei algebra. New Poincare-invariant...

In this paper, by using q-Volkenborn integral[10], the first author[25] constructed new generating functions of the new twisted (h, q)-Bernoulli polynomials and numbers. We define higher-order twisted (h, q)-Bernoulli polynomials and numbers. Using these numbers and polynomials, we obtain new approach...

Many methods of deriving Lie point symmetries for Itˆo stochastic ordinary differential equations (SODEs) have surfaced. In the Itˆo calculus context both the formal and intuitive understanding of how to construct these symmetries has led to seemingly disparate results. The impact of Lie point symmetries...

The nonclassical symmetries method is applied to a class of reaction-diffusion equations with nonlinear source, i.e. ut=uxx+cux+R(u, x). Several cases are obtained by using suitable solutions of the heir-equations as described in [M.C. Nucci, Nonclassical symmetries as special solutions of heir-equations,...

The Somos 4 sequences are a family of sequences satisfying a fourth order bilinear recurrence relation. In recent work, one of us has proved that the general term in such sequences can be expressed in terms of the Weierstrass sigma function for an associated elliptic curve. Here we derive the analogous...

It is shown that the group of geometrical symmetries of the Universal equation of D-dimensional space coincides with SL(D + 1, R).

We present a version of the conditional symmetry method in order to obtain multiple wave solutions expressed in terms of Riemann invariants. We construct an abelian distribution of vector fields which are symmetries of the original system of PDEs subjected to certain first order differential constraints....

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

In this work we derive potential symmetries for ordinary differential equations. By using these potential symmetries we find that the order of the ODE can be reduced even if this equation does not admit point symmetries. Moreover, in the case for which the ODE admits a group of point symmetries, we find...

This paper presents a new third-order trajectory solution in Lagrangian form for the water particles in a wave-current interaction flow based on an Euler–Lagrange transformation. The explicit parametric solution highlights the trajectory of a water particle and the wave kinematics above the mean water...

Motivated by the work of Kupershmidt (J. Nonlin. Math. Phys. 6 (1998), 222 –245) we discuss the occurrence of left symmetry in a generalized Virasoro algebra. The multiplication rule is defined, which is necessary and sufficient for this algebra to be quasi-associative. Its link to geometry and nonlinear...

Different kinds of reduction for ordinary differential equations, such as λ –symmetry and σ –symmetry reductions, are recovered as particular cases of Frobenius reduction theorem for distribution of vector fields. This general approach provides some hints to tackle the reconstruction problem and to solve...

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

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

A non-isospectral linear problem for an integrable 2+1 generalization of the non linear Schrödinger equation, which includes dispersive terms of third and fourth order, is presented. The classical symmetries of the Lax pair and the related reductions are carefully studied. We obtain several reductions...

In this paper we consider a semi-classical variation of the weight related to the q-Laguerre polynomials and study their recurrence coefficients. In particular, we obtain a second degree second order discrete equation which in particular cases can be reduced to either the qPV or the qPIII equation.

The paper investigates solutions and Lax pairs through bilinear Bäcklund transformations for some supersymmetric equations. We derive variety of solutions from the known bilinear Bäcklund transformations. Besides, using the gauge invariance of (super) Hirota bilinear derivatives we may get deformed bilinear...

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

We obtain a basis of joint or proper differential invariants for the scalar linear hperbolic partial differential equation in two independent variables by the infinitesimal method. The joint invariants of the hyperbolic equation consist of combinations of the coefficients of the equation and their derivatives...

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

Let Epn be the n-dimensional pseudo-Euclidean space of index p and M(n, p) the group of all transformations of Epn generated by pseudo-orthogonal transformations and parallel translations. We describe the system of generators of the differential field of all M(n, p)-invariant differential...

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

We study "tunnelling" in a one-dimensional, nonlocal evolution equation by assigning a penalty functional to orbits which deviate from solutions of the evolution equation. We discuss the variational problem of computing the minimal penalty for orbits which connect two stable, stationary solutions.

Algebraic integrability of the elliptic Calogero­Moser quantum problem related to the deformed root systems A2(2) is proved. Explicit formulae for integrals are found.

We present a consistent derivation of the pressure transfer function for small amplitude waves within the framework of linear wave theory and discuss some nonlinear aspects.

We study a generalization of the familiar Poincaré map, first implicitely introduced by N N Nekhoroshev in his study of persistence of invariant tori in hamiltonian systems, and discuss some of its properties and applications. In particular, we apply it to study persistence and bifurcation of invariant...

Symmetry classification of two-body central potentials in a two-particle Schrödinger equation in terms of contact transformations of the equation has been investigated. Explicit calculation has shown that they are of the same four different classes as for the point transformations. Thus in this problem...

A brief review is presented of the two recent perturbation algorithms. Their common idea lies in a not quite usual treatment of linear Schrödinger equations via nonlinear mathematical means. The first approach (let us call it a quasi-exact perturbation theory, QEPT) tries to get the very zero-order approximants...

We prove a regularity result for steady periodic travelling capillary waves of small amplitude at the free surface of water in a flow with constant vorticity over a flat bed.

The classical Lie method is applied to a nonisospectral problem associated with a system of partial differential equations in 2 + 1 dimensions (Maccari A, J. Math. Phys. 12 (1998) 6547–6551.). Identification of the classical Lie symmetries provides a set of reductions that give rise to different nontrivial...

The paper discusses fractional integrals and derivatives appearing in the so-called (q, h)-calculus which is reduced for h = 0 to quantum calculus and for q = h = 1 to difference calculus. We introduce delta as well as nabla version of these notions and present their basic properties. Furthermore, we...

This paper presents a linear transformation for low order nonlinear autonomous diferential equations. The procedure consists of a trajectory-based local linearization, which approximates the nonlinear system in the neighborhood of its equilibria. The approximation is possible even in the non-hyperbolic...

A reduction process to construct hidden hierarchies corresponding to the Gelfand­ Dickey ones is outlined in a specific example, not yet treated in the literature.

In this paper we describe the Hamiltonian dynamics, in some invariant manifolds of the mo- tion of a gyrostat in Newtonian interaction with a spherical rigid body. Considering a first integrable approximation of this roto-translatory problem, by means of Liouville-Arnold the- orem and some specifics...

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...

I propose an orthogonalization procedure preserving the grading of the initial graded set of linearly independent vectors. In particular, this procedure is applicable for orthonormalization of any countable set of polynomials in several (finitely many) ideterminates.

We investigate closed ideals in the Grassmann algebra serving as bases of Lie-invariant geometric objects studied before by E.Cartan. Especially, the E.Cartan theory is enlarged for Lax integrable nonlinear dynamical systems to be treated in the frame work of the Wahlquist Estabrook prolongation structures...

In this paper, we mainly investigate two kinds of gauge transformations for the constrained modified KP hierarchy in Kupershmidt-Kiso version. The corresponding gauge transformations are required to keep not only the Lax equation but also the Lax operator. For this, by selecting the special generating...

The Geng hierarchy is derived with the aid of Lenard recursion sequences. Based on the Lax matrix, a hyperelliptic curve 𝒦n + 1 of arithmetic genus n+1 is introduced, from which meromorphic function ϕ is defined. The finite genus solutions for Geng hierarchy are achieved according to asymptotic properties...

Let n be an integer such that n 3 and Cm denote a cyclic group of order m . It is proved that there exist exactly 17 non-isomorphic groups of order 22n+1 which can be represented as a semidirect product (C2n × C2n ) C2. These groups are given by generators and defining relations.

An explicit parameterization in terms of elliptic integrals (functions) for the Mylar balloon is found which then is used to calculate various geometric quantities as well as to study all kinds of geodesics on this surface.

We explicitly describe Heisenberg families of elements in an arbitrary grading subspaces of the quantized universal enveloping algebra Uq(𝒢^) of an affine Kac–Moody algebra 𝒢^ in the Drinfeld formulation.

In this paper we investigate relations between different transformations of the slutions of the sixth Painlevé equation. We obtain nonlinear superposition formulas linking solutions by means of the Bäcklund transformation. Algebraic solutions are also studied with the help of the Bäcklund transformation.

We investigate some classical evolution model in the discrete 2+1 space-time. A map, giving an one-step time evolution, may be derived as the compatibility condition for some systems of linear equations for a set of auxiliary linear variables. Dynamical variables for the evolution model are the coefficients...

The Yakushevich model provides a very simple pictures of DNA torsion dynamics, yet yields remarkably correct predictions on certain physical characteristics of the dynamics. In the standard Yakushevich model, the interaction between bases of a pair is modelled by a harmonic potential, which becomes anharmonic...

The higher order Painlevé system of type D2n+2(1) was proposed by Y. Sasano as an extension of PVI for the affine Weyl group symmetry with the aid of algebraic geometry for Okamoto initial value space. In this article, we give it as the monodromy preserving deformation of a Fuchsian system.

The first main aim of this article is to derive an explicit solution formula for the scalar two-dimensional Toda lattice depending on three independent operator parameters, ameliorating work in [31]. This is achieved by studying a noncommutative version of the 2d-Toda lattice, generalizing its soliton...

We construct non-localized, real global solutions of the Kadomtsev-Petviashvili-I eqution which vanish for x - and study their large time asymptotic behavior. We prove that such solutions eject (for t ) a train of curved asymptotic solitons which move behind the basic wave packet.

In this article, we present an explicit linearization of dynamical systems of RuijsenaarSchneider (RS) type and of the perturbations introduced by F Calogero [2] of these systems with all orbits periodic of the same period. The existence of this linearization and its algebraic nature relies on the dynamical...

In this note we describe the underlying principles — and pitfalls — of the process of non-dimensionalising and scaling the equations that model the classical problem in water waves. In particular, we introduce the two fundamental parameters (associated with amplitude and with wave length) and show how...

In this paper we obtain the dispersion relations for small-amplitude steady periodic water waves, which propagate over a flat bed with a specified mean depth, and which exhibit discontinuous vorticity. We take as a model an isolated layer of constant nonzero vorticity adjacent to the flat bed, with irrotational...

We consider surfaces arising from integrable partial differential equations and from their deformations. Symmetries of the equation, gauge transformation of the corrsponding Lax pair and spectral parameter transformations are the deformations which lead infinitely many integrable surfaces. We also study...

A depending on a complex parameter k superanalog SL of Calogero operator is costructed; it is related with the root system of the Lie superalgebra gl(n|m). For m = 0 we obtain the usual Calogero operator; for m = 1 we obtain, up to a change of indterminates and parameter k the operator constructed by...

The classical generation theorem of conservation laws from known ones for a system of differential equations which uses the action of a canonical Lie­Bäcklund generator is extended to include any Lie­Bäcklund generator. Also, it is shown that the Lie algebra of Lie­Bäcklund symmetries of a conserved...

We construct a number of ansatzes that reduce one-dimensional nonlinear heat equations to systems of ordinary differential equations. Integrating these, we obtain new exact solution of nonlinear heat equations with various nonlinearities.

Type II hidden symmetries of partial differential equations () are extra symme- tries in addition to the inherited symmetries of the differential equations which arise when the number of independent and dependent variables is reduced by a Lie point symmetry. (Type I hidden symmetries arise in the increase...

In this paper we prove the non–existence of Darboux first integrals for the Painlevé II equations x.=y−z2−x2,y.=α+12+2xy,ż=1 for all values of α ∊ ℂ \ {αn: n = 2,4,…}. These αn are real and larger than −1/2.

We apply a list of criteria which leads to a class of fifth-order symmetry-integrable evolution equations. The recursion operators for this class are given explicitly. Multipotentialisations are then applied to the equations in this class in order to extend this class of integrable equations.

We apply the method of Jacobi Last Multiplier to the fifty second-order ordinary differential equations of Painlevé type as given in Ince in order to obtain a Lagrangian and consequently solve the inverse problem of Calculus of Variations for those equations. The easiness and straightforwardness of Jacobi’s...

Spatial modulated coupled nonlinear Schrödinger systems with symmetry reduction to integrable Ermakov and Ermakov-Painlevé subsystems are investigated.

A hierarchy of new nonlinear evolution equations is proposed, which are composed of the positive and negative coupled KdV flows. Based on the theory of algebraic curve, the corresponding flows are straightened under the Abel-Jacobi coordinates. The meromorphic function ϕ, the Baker-Akhiezer vector ψ¯ ,...

The derivative nonlinear Schrödinger equation is shown to be locally well-posed in a class of functions analytic on a strip around the real axis. The main feature of the result is that the width of the strip does not shrink in time. To overcome the derivative loss, Kato-type smoothing results and space-time...

We exhibit a class of Dirac operators that possess Huygens' property, i.e., the support of their fundamental solutions is precisely the light cone. This class is obtained by considering the rational solutions of the modified Korteweg-de Vries hierarchy.

A new solvable N-body model of goldfish type is identified. Its Newtonian equations of motion read as follows: z¨n=-6z˙nzn-4zn3+32(z˙n+2zn2)∑k=1N(z˙ kzk+2zk)+2∑𝓁=1,𝓁≠nN[(z˙n+2zn2)(z˙𝓁+2z𝓁2)zn-z𝓁],      n=1,…,N, where...

We describe a few families of orthogonal matrix polynomials of size N × N satisfying first order differential equations. This problem differs from the recent efforts reported for instance in [7] (Orthogonal matrix polynomials satisfying second order differential equations, Internat. Math. Research Notices,...

Conditional symmetry We investigate conditional symmetry in three directions. The first direction is a research of the Q-conditional symmetry. The second direction is studying conditional symmetry when an algebra of invariance is known and an additional condition is unknown. The third direction is the...

Starting from any representation of the Lie algebra ℊ on the finite dimensional vector space V we can construct the representation on the space Aut(V ). These representations are of the type of ad. That is one of the reasons, why it is important to study the adjoint representation of the Lie algebra...

In this paper we give a brief review of the recent results obtained by the author and his co-authors for description of three-dimensional vortical incompressible flows in the hydrodynamic type systems. For such flows we introduce a new mixed LagrangiaEulerian description - the so called vortex line representation...

Integrable dispersionless Kadomtsev-Petviashvili (KP) hierarchy of B type is consiered. Addition formula for the -function and conformally invariant equations for the dispersionless BKP (dBKP) hierarchy are derived. Symmetry constraints for the dBKP hierarchy are studied.

In this work, we explain in what sense the generic level set of the constants of motion for the periodic nonlinear Schrödinger equation is an infinite dimensional torus on which each generalized nonlinear Schrödinger flow is reduced to straight line almost periodic motion, and describe how neighboring...

A special asymptotic solution of the Painlevé-2 equation with small parameter is stdied. This solution has a critical point t corresponding to a bifurcation phenomenon. When t t the solution oscillates very fast. We investigate the transitional layer in detail and obtain a smooth asymptotic solution,...

By use of nonlinearization method about spectral problem, a classical completely integrable system associated with the Harry-Dym (HD) hierarchy is obtained. Furthermore, the involutive solution of each equation in the HD hierarchy is presented, in particular, the involutive solution of the well-known...

In this paper, we give a proof of the existence of stationary dark soliton solutions of the cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity, together with an analytical example of a dark soliton.

It is shown that the exact solubility of the massless Thirring model in the canonical quantization scheme originates from the intrinsic hidden linearizability of its Heisenberg equations in the method of dynamical mappings. The corresponding role of inequivalent representations of free massless Dirac...

This paper is an informal collection of observations on how established rewriting techniques can be applied to or need to be adapted for use in non-associative algebras, operads, and PROPs.

Using the theory of self-adjoint extensions, we study the interaction model formally given by the Hamiltonian H + V (r), where H is the Aharonov-Bohm Hamiltonian and V (r) is the -type interaction potential on the cylinder of radius R . We give the mathematical definition of the model, the self-adjointness...

In this paper we study symmetry reductions of a class of nonlinear fourth order partial differential equations utt = u + u2 xx + uuxxxx + µuxxtt + uxuxxx + u2 xx, (1) where , , , and µ are arbitrary constants. This equation may be thought of as a fourth order analogue of a generalization of the Camassa-Holm...

A necessary condition for the existence of conserved densities and fluxes of a differential-difference equation which depend on q shifts, for q sufficiently large, is presented. This condition depends on the eigenvalues of the leading terms in the differential-difference equation. It also gives, explicitly,...

In this paper, we firstly investigate the successive applications of three elementary gauge transformation operators Ti with i = 1,2,3 for the mKP hierarchy in Kupershmidt-Kiso version, and find that the gauge transformation operators Ti can not commute with each other. Then two types of gauge transformation...

Painlevé equations belong to the class y +a1 y 3 +3a2 y 2 +3a3 y +a4 = 0, where ai = ai(x, y). This class of equations is invariant under the general point transformation x = (X, Y ), y = (X, Y ) and it is therefore very difficult to find out whether two equations in this class are related. We describe...

We investigate Lie symmetries of Einstein's vacuum equations in N dimensions, with a cosmological term. For this purpose, we first write down the second prolongation of the symmetry generating vector fields, and compute its action on Einstein's equations. Instead of setting to zero the coefficients of...

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...

In this paper we discuss the Moyal deformed 2D Euler flows and its Lax pairs. This in turn yields the semi-discrete version of 2D Euler equation.

We review the construction of Lax pairs with spectral parameter for twisted and utwisted elliptic Calogero-Moser systems defined by a general simple Lie algebra G, and the corresponding solution of N = 2 SUSY G Yang-Mills theories with a hypermultplet in the adjoint representation of G.

We consider the variational symmetry from the viewpoint of the non-integrability criterion towards dynamical systems. That variational symmetry can reduce complexity in determining non-integrability of general dynamical systems is illustrated here by a new non-integrability result about Hamiltonian systems...

Noether’s theorem is used to determine first integrals admitted by a generalised Lane-Emden equation of the second kind modelling a thermal explosion. These first integrals exist for rectangular and cylindrical geometry. For rectangular geometry the first integrals show the symmetry of the temperature...

We use an algorithm based on the notion of transvectants from classical invariant theory in determining the form of Stanley decomposition of the ring of invariants for the coupled Takens–Bogdanov systems when the Stanley decompositions of the Jordan blocks of the linear part are known at each stage....

In this study, we represent an application of the geometrical characterization of μ-prolongations of vector fields to the nonlinear partial differential Gardner equation with variable coefficients. First, μ-symmetries and the corresponding μ-symmetry classification are investigated and then μ-reduction...

We extend the notion of deformation to inverse operations of restrictions of completely integrable systems to regular or singular locus, and call the extended notion prolongation. We show that a prolongability determines uniquely a Fuchsian ordinary differential equation of rank three with three regular...

I discuss the connection of the three different questions: The existence of the Gibbs steady state distributions for the stochastic differential equations, the notion and the existence of the conservation laws for such equations, and the convergence of the smooth random perturbations of dynamical systems...

The Green function for Klein-Gordon-Dirac equation is obtained. The case with the dominating Klein-Gordon term is considered. There seems to be a formal analogy between our problem and a certain problem for a 4-dimensional particle moving in the external field. The explicit relations between the wave...