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The generalized vector is defined on an $n$ dimensional manifold. Interior product, Lie derivative acting on generalized $p$-forms, $-1\le p\le n$ are introduced. Generalized commutator of two generalized vectors are defined. Adding a correction term to Cartan's formula the generalized Lie derivative's action on a generalized vector field is defined. We explore various identities of the generalized Lie derivative with respect to generalized vector fields, and discuss an application., Comment: 10 pages

This paper describes a wide class of coupled KdV equa- tions. The first set of equations directly follow from the geodesic flows on the Bott-Virasoro group with a complex field. But the set of 2- component systems of nonlinear evolution equations, which includes dispersive water waves, Ito's equation, many other known and unknown equations, follow from the geodesic flows of the right invariant L 2 met- ric on the semidirect product group \ Diff(S 1 ) ⋉ C 1 (S 1 ), where Diff(S 1 ) is the group of orientation preserving diffeomorphisms on a circle. We compute the Lie-Poisson brackets of the Antonowicz-Fordy system, and the mode expansion of these beackets yield the twisted Heisenberg- Virasoro algebra. We also give an outline to study geodesic flows of a H 1 metric on \ Diff(S1) ⋉ C1(S1).

We examine the analytical structure of the nonlinear Lienard oscillator and show that it is a bi-Hamiltonian system depending upon the choice of the coupling parameters. While one has been recently studied in the context of a quantized momentum- dependent mass system, the other Hamiltonian also reflects a similar feature in the mass function and also depicts an isotonic character. We solve for such a Hamiltonian and give the complete solution in terms of a confluent hypergeometric function., 10 pages,Final Version

In this paper, we analyze the modification of integrable models in the $\kappa$-deformed space-time. We show that two dimensional isotropic oscillator problem, Kepler problem and MICZ-Kepler problem in $\kappa$-deformed space-time admit integrals of motion as in the commutative space. We also show that the duality equivalence between $\kappa$-deformed Kepler problem and $\kappa$-deformed two-dimensional isotropic oscillator explicitly, by deriving Bohlin-Sundman transformation which maps these two systems. These results are valid to all orders the the deformation parameter., Comment: 21 pages, references added, results are valid to all orders in the deformation parameter

In this paper we study generalized classes of volume preserving multidimensional integrable systems via Nambu-Poisson mechanics. These integrable systems belong to the same class of dispersionless KP type equation. Hence they bear a close resemblance to the self dual Einstein equation. Recently Takasaki-Takebe provided the twistor construction of dispersionless KP and dToda type equations by using the Gindikin's pencil of two forms. In this paper we generalize this twistor construction to our systems., 15 pages, Latex

In this paper we discuss a universal integrable model, given by a sum of two Wess-Zumino-Witten-Novikov (WZWN) actions, corresponding to two different orbits of the coadjoint action of a loop group on its dual, and the Polyakov-Weigmann cocycle describing their interactions. This is an effective action for free fermions on a torus with nontrivial boundary conditions. It is universal in the sense that all other known integrable models can be derived as reductions of this model. Hence our motivation is to present an unified description of different integrable models. We present a proof of this universal action from the action of the trivial dynamical system on the cotangent bundles of the loop group. We also present some examples of reductions.

We present a brief overview of classical isochronous planar differential systems focusing mainly on the second equation of the Liénard type ẍ + f(x)ẋ2 + g(x) = 0. In view of the close relation between Jacobi's last multiplier and the Lagrangian of such a second-order ordinary differential equation, it is possible to assign a suitable potential function to this equation. Using this along with Chalykh and Veselov's result regarding the existence of only two rational potentials which can give rise to isochronous motions for planar systems, we attempt to clarify some of the previous notions and results concerning the issue of isochronous motions for this class of differential equations. In particular, we provide a justification for the Urabe criterion besides giving a derivation of the Bolotin–MacKay potential. The method as formulated here is illustrated with several well-known examples like the quadratic Loud system and the Cherkas system and does not require any computation relying only on the standard techniques familiar to most physicists.

We explore the Jacobi last multiplier (JLM) as a means for deriving the Lagrangian of a fourth-order differential equation. In particular, we consider the classical Pais–Uhlenbeck problem and write down the accompanying Hamiltonian. We then compare such an expression with our alternative derivation of the Hamiltonian that makes use of the Ostrogradski's method and show how a mapping from the one to the other is achievable by variable transformations.

A variant of the generalized Riccati system of equations, , is considered. It is shown that for ? = 2n + 3 the system admits a bilagrangian description and the dynamics has a node at the origin, whereas for ? much smaller than a critical value the dynamics is periodic, the origin being a centre. It is found that the solution changes from being periodic to aperiodic at a critical point, , which is independent of the initial conditions. This behaviour is explained by finding a scaling argument via which the phase trajectories corresponding to different initial conditions collapse onto a single universal orbit. Numerical evidence for the transition is shown. Further, using a perturbative renormalization group argument, it is conjectured that the oscillator, , exhibits isochronous oscillations. The correctness of the conjecture is established numerically.

The periodic flag manifold (in the Sato Grassmannian context) description of the modified Korteweg--de Vries hierarchy is used to analyse the translational and scaling self--similar solutions of this hierarchy. These solutions are characterized by the string equations appearing in the double scaling limit of the symmetric unitary matrix model with boundary terms. The moduli space is a double covering of the moduli space in the Sato Grassmannian for the corresponding self--similar solutions of the Korteweg--de Vries hierarchy, i.e. of stable 2D quantum gravity. The potential modified Korteweg--de Vries hierarchy, which can be described in terms of a line bundle over the periodic flag manifold, and its self--similar solutions corresponds to the symmetric unitary matrix model. Now, the moduli space is in one--to--one correspondence with a subset of codimension one of the moduli space in the Sato Grassmannian corresponding to self--similar solutions of the Korteweg--de Vries hierarchy., 21 pages in LaTeX-AMSTeX

We consider a large class of polynomial planar differential equations proposed by Cherkas (1976 Differensial'nye Uravneniya 12 201?6), and show that these systems admit a Lagrangian description via the Jacobi last multiplier (JLM). It is shown how the potential term can be mapped either to a linear harmonic oscillator potential or into an isotonic potential for specific values of the coefficients of the polynomials. This enables the identification of the specific cases of isochronous motion without making use of the computational procedure suggested by Hill et al (2007 Nonlinear Anal.: Theor. Methods Appl. 67 52?69), based on the Pleshkan algorithm. Finally, we obtain a Lagrangian description and perform a similar analysis for a cubic system to illustrate the applicability of this procedure based on Jacobi's last multiplier.

The Darboux integrability method is particularly useful to determine first integrals of nonplanar autonomous systems of ordinary differential equations, whose associated vector fields are polynomials. In particular, we obtain first integrals for a variant of the generalized Raychaudhuri equation, which has appeared in string inspired modern cosmology.

Karasu-Kalkani et al (2008 J. Math. Phys. 49 073516) recently derived a new sixth-order wave equation KdV6, which was shown by Kupershmidt (2008 Phys. Lett. 372A 2634) to have an infinite commuting hierarchy with a common infinite set of conserved densities. Incidentally, this equation was written for the first time by Calogero and is included in the book by Calogero and Degasperis (1982 Lecture Notes in Computer Science vol 144 (Amsterdam: North-Holland) p 516). In this paper, we give a geometric insight into the KdV6 equation. Using Kirillov's theory of coadjoint representation of the Virasoro algebra, we show how to obtain a large class of KdV6-type equations equivalent to the original equation. Using a semidirect product extension of the Virasoro algebra, , we propose the nonholonomic deformation of the Ito equation. We also show that the Adler–Kostant–Symes scheme provides a geometrical method for constructing nonholonomic deformed integrable systems. Applying the Adler–Kostant–Symes scheme to loop algebra, we construct a new nonholonomic deformation of the coupled KdV equation.

The properties of higher-order Riccati equations are investigated. The second-order equation is a Lagrangian system and can be studied by using the symplectic formalism. The second-, third- and fourth-order cases are studied by proving the existence of Darboux functions. The corresponding cofactors are obtained and some related properties are discussed. The existence of generators of t-dependent constants of motion is also proved and then the expressions of the associated time-dependent first integrals are explicitly obtained. The connection of these time-dependent first integrals with the so-called master symmetries, characterizing some particular Hamiltonian systems, is also discussed. Finally the general n-th-order case is analyzed.

We report the connection of symplectic rectification in the construction of isochronous Hamiltonian systems.

We consider the role of the adjoint equation in determining explicit integrating factors and first integrals of nonlinear ODEs. In Chandrasekar et al (2006 J. Math. Phys. 47 023508), the authors have used an extended version of the Prelle–Singer method for a class of nonlinear ODEs of the oscillator type. In particular, we show that their method actually involves finding a solution of the adjoint symmetry equation. Next, we consider a coupled second-order nonlinear ODE system and derive the corresponding coupled adjoint equations. We illustrate how the coupled adjoint equations can be solved to arrive at a first integral.

The theory of Hamiltonian and quasi-Hamiltonian systems with respect to Nambu–Poisson structures is studied. It is proved that if a dynamical system is endowed with certain properties related to the theory of symmetries then it can be considered as a quasi-Hamiltonian (or Hamiltonian) system with respect to an appropriate Nambu–Poisson structure. Several examples of this construction are presented. These examples are related to integrability and also to superintegrability.

In this paper we compute first integrals of nonlinear ordinary differential equations using the extended Prelle-Singer method, as formulated by Chandrasekar et al in J. Math. Phys. 47 (2), 023508, ...

In analogy with the load and the metage in hydrodynamics, this paper defines magnetohydrodynamic load and magnetohydrodynamic metage in the case of magnetofluids. They can be used to write the magnetic field in MHD in Clebsch’s form. It has been shown herein how these two concepts can be utilized to derive the magnetic analog of the Ertel’s theorem and also, how in the presence of nontrivial topology of the magnetic field in the magnetofluid one may associate the linking number of the magnetic field lines with the invariant MHD loads. The paper illustrates that the symmetry translation of the MHD metage in the corresponding label space generates the conservation of cross helicity.

In this paper we propose an Euler-Poincare formalism of the Degasperis and Procesi (DP) equation. This is a second member of a one-parameter family of partial differential equations, known as b-field equations. This one-parameter family of pdes includes the integrable Camassa-Holm equation as a first member. We show that our Euler-Poincare formalism exactly coincides with the Degasperis-Holm-Hone (DHH) Hamiltonian framework. We obtain the DHH Hamiltonian structues of the DP equation from our method. Recently this new equation has been generalized by Holm and Staley by adding viscosity term. We also discuss Euler-Poincare formalism of the Holm-Staley equation. In the second half of the paper we consider a generalization of the Degasperis and Procesi (DP) equation with two dependent variables. we study the Euler-Poincare framework of the 2-component Degasperis-Procesi equation. We also mention about the b-family equation.

The standard embedding of the Lie algebra V ect(S 1 ) of smooth vector fields on the circle V ect(S 1 ) into the Lie algebra D(S 1 ) of pseudodierential symbols on S 1 identifies vector field f(x) @ @x 2 V ect(S 1 ) and its dual as (f(x) @ @x ) = f(x) (u(x)dx 2 ) = u(x) 2 . The space of symbols can be viewed as the space of functions on T S 1 . The natural lift of the action of Diff(S 1 ) yields Diff(S 1 )-module. In this paper we demonstate this construction to yield several examples of dispersionless integrable systems. Using Ovsienko and Roger method for nontrivial deformation of the standard embedding of V ect(S 1 ) into D(S 1 ) we obtain the celebrated HunterSaxton equation. Finally, we study the Moyal quantization of all such systems to construct noncommutative systems. Dedicated to Professor Dieter Mayer on his 60th birthday

In this paper we further investigate some applications of Nambu mechanics in hydro- dynamical systems. Using the Euler equations for a rotating rigid body Nevir and Blender (J. Phys. A 26 (1993), L1189-L1193) had demonstrated the connection be- tween Nambu mechanics and noncanonical Hamiltonian mechanics. Nambu mechanics is extended to incompressible ideal hydrodynamical fields using energy and helicity in three dimensional (enstrophy in two dimensional). In this paper we discuss the Lax representation of systems of Nevir-Blender type. We also formulate the three dimen- sional Euler equations of incompressible fluid in terms of Nambu-Poisson geometry. We discuss their Lax representation. We also briefly discuss the Lax representation of ideal incompressible magnetohydrodynamics equations.

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.

Monodromy deformation approach to nonlinear partial differential equation is discussed in a pedestrian's way. The whole methodology is discussed on the basis of Massive Thirring Model. In the first section of our paper we discuss the basic terminologies amociated with the deformation problem. In the next part the problem is defined on the basis of the Lax pairs for the Thirring model, and it is explicitly demonstrated that how one can determine the asymptotic expansions near a regular and irregular singularity, and hence the Stokes multipliers. Thirdly we show how to determine the “third” equation in according to Its. In the determination of the asymptotic expansion we have discussed the role played by both the WKB approximation and the series solution. In the fourth section we briefly consider the problem when the nonlinear field variables are taken to be fermionic.

Monodromy data for a nonlinear ordinary differential equation obtained as the reduction of the three-wave equation are determined by a new technique due to Gurarii and Mateev. The regions of growth and decay of the solutions are determined and the corresponding Stokes parameters are obtained. Earlier it was observed that this equation is not a member of the six Painleve equations but is concerned to the Painleve VI equation via a complicated transformation.

Using Grozman’s formalism of invariant differential operators we demonstrate the derivation of N=2 Camassa-Holm equation from the action of Vect(S 1|2) on the space of pseudo-differential symbols. We also use generalized logarithmic 2-cocycles to derive N=2 super KdV equations. We show this method is equally effective to derive Camassa-Holm family of equations and these system of equations can also be interpreted as geodesic flows on the Bott-Virasoro group with respect to right invariant H 1-metric. In the second half of the paper we focus on the derivations of the fermionic extension of a new peakon type systems. This new one-parameter family of N=1 super peakon type equations, known as N=1 super b-field equations, are derived from the action of Vect(S 1|1) on tensor densities of arbitrary weights. Finally, using the formal Moyal deformed action of Vect(S 1|1) on the space of Pseudo-differential symbols to derive the noncommutative analogues of N=1 super b-field equations.

We use a formula derived almost seventy years ago by Madhav Rao connecting the Jacobi Last Multiplier of a second-order ordinary differential equation and its Lagrangian and determine the Lagrangians of the Painleve equations. Indeed this method yields the Lagrangians of many of the equations of the Painleve–Gambier classification. Using the standard Legendre transformation we deduce the corresponding Hamiltonian functions. While such Hamiltonians are generally of non-standard form, they are found to be constants of motion. On the other hand for second-order equations of the Lienard class we employ a novel transformation to deduce their corresponding Lagrangians. We illustrate some particular cases and determine the conserved quantity (first integral) resulting from the associated Noetherian symmetry. Finally we consider a few systems of second-order ordinary differential equations and deduce their Lagrangians by exploiting again the relation between the Jacobi Last Multiplier and the Lagrangian.

We show that all the Antonowicz–Fordy type coupled KdV equations have the same symmetry group and similar bi-Hamiltonian structures. It turns out that their configuration space is Diff ( S 1 ) ⋉ C ∞ ( S 1 ) ˆ , where Diff ( S 1 ) ˆ is the Bott–Virasoro group of orientation preserving diffeomorphisms of the circle, and all these systems can be interpreted as equations of a geodesic flow with respect to L 2 metric on the semidirect product space Diff ( S 1 ) ⋉ C ∞ ( S 1 ) ˆ .

A metriplectic (or Leibniz) structure on a smooth manifold is a pair of skew-symmetric Poisson tensor P and symmetric metric tensor G. The dynamical system defined by the metriplectic structure can be expressed in terms of Leibniz bracket. This structure is used to model the geometry of the dissipative systems. The dynamics of purely dissipative systems are defined by the geometry induced on a phase space via a metric tensor. The notion of Leibniz brackets is extendable to infinite-dimensional spaces. We study metriplectic structure compatible with the Euler–Poincare framework of the Burgers and Whitham–Burgers equations. This means metriplectic structure can be constructed via Euler–Poincare formalism. We also study the Euler–Poincare frame work of the Holm–Staley equation, and this exhibits different type of metriplectic structure. Finally we study the 2D Navier–Stokes using metriplectic techniques.

Penile involvement was seen in 19.5% of 261 patients, aged 11 to 30 years, with tinea cruris. It was more common in patients under the age of 20 years (p less than 0.05). The increased incidence is probably related to the use of langota, a semiocclusive undergarment that may favor the growth of dermatophytes.