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Transition probabilities for a class of two level systems described by explicitly time dependent Hamiltonians are considered. Provided only that the approach to the infinite time limit is non-trivial falling at least as fast as 1/t for large t, the transition probability takes a particularly simple form depending only on the value of Hamiltonian parameters in this limit [ABSTRACT FROM AUTHOR]

We study a class of the isomonodromic deformation equations, which we call matrix Painlevé systems. They can be written in Hamiltonian form. The Hamiltonian is written as the trace of the Hamiltonian of the sixth Painlevé equation with canonical variables replaced by matrices. We also derive the matrix version of the fifth, fourth, third, and second Painlevé equation from the sixth matrix Painlevé system by the degeneration. [ABSTRACT FROM AUTHOR]

Without loss of generality the ABC systems reduce to two cases: either A = 0 and B, C >= 0, or A = 1 and 0 < B, C <= 1. In the first case it is known that the ABC system is completely integrable, here we provide its explicit first integrals. In the second case Ziglin ['Dichotomy of the separatrices and the nonexistence of first integrals in systems of differential equations of Hamiltonian type with two degrees of freedom,' Izv. Akad. Nauk SSSR, Ser. Mat. 51, 1088 (1987)] proved that the ABC system with 0 < B < 1 and C > 0 sufficiently small has no real meromorphic first integrals. We improve Ziglin's result showing that there are no C1 first integrals under convenient assumptions. [ABSTRACT FROM AUTHOR]

Based on a kind of new Lie superalgebras, we proposed a scheme for constructing nonlinear super integrable couplings. Super trace identity over the corresponding loop superalgebras is used to furnish the super Hamiltonian structures for the resulting nonlinear super integrable couplings. We illustrate the application of the scheme by means of an enlarged super Ablowitz-Kaup-Newell-Segur (AKNS) matrix spectral problem and present a nonlinear super integrable Hamiltonian couplings for the super AKNS hierarchy. [ABSTRACT FROM AUTHOR]

The classical Hill's problem is a simplified version of the restricted three-body problem where the distance of the two massive bodies (say, primary for the largest one and secondary for the smallest one) is made infinity through the use of Hill's variables. The Levi-Civita regularization takes the Hamiltonian of the Hill lunar problem into the form of two uncoupled harmonic oscillators perturbed by the Coriolis force and the Sun action, polynomials of degree 4 and 6, respectively. In this paper, we study periodic orbits of the planar Hill problem using the averaging theory. Moreover, we provide information about the C1 integrability or non-integrability of the regularized Hill lunar problem. [ABSTRACT FROM AUTHOR]

An elliptic vortex-type ansatz introduced into a 2+1-dimensional system governing rotating homentropic magneto-gasdynamics with a parabolic gas law is shown to lead to a finite-dimensional nonlinear dynamical system which admits exact analytical solution in terms of an elliptic function and integral representation. The dynamical system is demonstrated to be Hamiltonian and equivalent to the stationary reduction of the integrable nonlinear Schrödinger equation coupled with a Steen-Ermakov-Pinney equation. A novel magneto-gasdynamic analogue of the pulsrodon of shallow water f-plane theory is isolated thereby. Confined and time-periodic magneto-gasdynamic flows are constructed explicitly. [ABSTRACT FROM AUTHOR]

We investigate the classical exchange algebra of the monodromy matrix for a Green-Schwarz sigma model on supercoset target space with Z4m grading by using a first-order Hamiltonian formulation and by adding to the Lax connection terms proportional to constraints. This enables us to show that the conserved charges of the theory are in involution in the Poisson bracket sense. Our calculation is based on a general world-sheet metric. Taking a particular case of m = 1 (and a particular choice of supergroup), our results coincide with those of the Green-Schwarz superstring theory in AdS5 × S5 background obtained by Magro [J. High Energy Phys. 0901, 021 (2009)]. [ABSTRACT FROM AUTHOR]

We consider Schrödinger equations for the quantum Painlevé equations. We present hypergeometric solutions of the Schrödinger equations for the quantum Painlevé equations, as particular solutions. We also give a representation theoretic correspondence between Hamiltonians of the Schrödinger equations for the quantum Painlevé equations and those of the Knizhnik-Zamolodchikov (KZ) equation or the confluent KZ equations. [ABSTRACT FROM AUTHOR]

We consider a family of Hamiltonians describing a quantum particle interacting, via zero-range forces, with a localized spin in the plane. A sub-class of Hamiltonians shows eigenvalues and/or resonances at the threshold of the continuous spectrum and generates particle dynamics not affecting the localized spin. We examine the effect of a small particle-spin interaction on the spectral structure of such unperturbed Hamiltonians. [ABSTRACT FROM AUTHOR]

Wave attraction is a general phenomenon that was first established in the context of the attraction of the polarization between two counter-propagating waves in optical fibers. This phenomenon has been observed experimentally, and its properties were studied through numerical simulations. The relevant models are Hamiltonian hyperbolic systems of partial differential equations, with time-dependent boundary conditions on a finite interval. The underlying mechanism can be traced back to the existence of singular tori in the corresponding stationary equations. In this article, we analyze in detail the simplest example in this family of models. We show that most of the phenomena of the wave attraction process are already present in a linear model with resonant interaction. We establish the existence and regularity of the solutions and analyze the relaxation towards a stationary solution that features the wave attraction properties. [ABSTRACT FROM AUTHOR]

A generalized fractional trace identity and Hamiltonian structure of the fractional soliton equation hierarchy are presented by using the modified Riemann-Liouville derivative and exterior derivatives of fractional orders. Example of the fractional Kaup Newell (KN) soliton equation hierarchy and Hamiltonian system are considered, which is a new integrable hierarchy and possesses Hamiltonian structure. [ABSTRACT FROM AUTHOR]

We give an explicit and concise formula for higher degree polynomial first integrals of a family of Calogero-type Hamiltonian systems in dimension three. These first integrals, together with the already known ones, prove the maximal superintegrability of the systems. [ABSTRACT FROM AUTHOR]

In this paper we provide sufficient conditions for the existence of a nondegenerate Hamiltonian Hopf bifurcation at the equilibria corresponding to the rotation around the vertical axis of a symmetric gyrostat with a fixed point under the effect of an axially symmetric potential. [ABSTRACT FROM AUTHOR]

A Euclidean representation is given for a canonically quantized relativistic Maxwell-Chern-Simons field on a lattice, which approximates a complex measure on a space of distributions. Using a path-space formula for the nonself-adjoint Hamiltonian, the relation between Euclidean Osterwalder-Schrader positivity, the Krein metric, and Gauss' law is examined. [ABSTRACT FROM AUTHOR]

The averaging theory of first order is applied to study a generalized Yang-Mills system with two parameters. Two main results are proved. First, we provide sufficient conditions on the two parameters of the generalized system to guarantee the existence of continuous families of isolated periodic orbits parameterized by the energy, and these families are given up to first order in a small parameter. Second, we prove that for the nonintegrable classical Yang-Mills Hamiltonian systems, in the sense of Liouville-Arnold, which have the isolated periodic orbits found with averaging theory, cannot exist in any second first integral of class C1. This is important because most of the results about integrability deals with analytic or meromorphic integrals of motion. [ABSTRACT FROM AUTHOR]

Noncommutative oscillators are first-quantized through an abelian Drinfel'd twist deformation of a Hopf algebra and its action on a module. Several important and subtle issues making the quantization possible are solved. The spectrum of the single-particle Hamiltonians is computed. The multiparticle Hamiltonians are fixed, unambiguously, by the Hopf algebra coproduct. The symmetry under particle exchange is guaranteed. In d = 2 dimensions the rotational invariance is preserved, while in d = 3 the so(3) rotational invariance is broken down to an so(2) invariance. [ABSTRACT FROM AUTHOR]

The existence of a ground state of the Nelson Hamiltonian with perturbations of the form ∑j=14cjφj with c4 > 0 is considered. The self-adjointness of the Hamiltonian and the existence of a ground state are proven for arbitrary values of coupling constants. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying symmetries of certain kinds of k-cosymplectic Hamiltonian systems in first-order classical field theories. Thus, we introduce a particular class of symmetries and study the problem of associating conservation laws to them by means of a suitable generalization of Noether's theorem. [ABSTRACT FROM AUTHOR]

In this paper, the Hamiltonian of quantum electrodynamics with spatial cutoffs is investigated. A scaled total Hamiltonian is introduced and its asymptotic behavior is investigated. In the main theorem, it is shown that the scaled total Hamiltonian converges to a self-adjoint operator in the strong resolvent sense, and the effective potential of the Dirac field is derived. [ABSTRACT FROM AUTHOR]

As well known, cumulant expansion is an alternative way to moment expansion to fully characterize probability distributions provided all the moments exist. If this is not the case, the so-called escort mean values (or q-moments) have been proposed to characterize probability densities with divergent moments [C. Tsallis et al., J. Math. Phys. 50, 043303 (2009)]. We introduce here a new mathematical object, namely, the q-cumulants, which, in analogy to the cumulants, provide an alternative characterization to that of the q-moments for the probability densities. To illustrate the technical details of the procedure, we apply this new scheme to further study a recently proposed family of scale-invariant discrete probabilistic models [A. Rodríguez et al., J. Stat. Mech.: Theory Exp. 2008, P09006; R. Hanel et al., Eur. Phys. J. B 72, 263 (2009)] having q-Gaussians as limiting probability distributions. [ABSTRACT FROM AUTHOR]

We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, devising an expansion for the splitting matrix associated with a homoclinic point. This expansion consists of contributions that are manifestly exponentially small in the limit of vanishing hyperbolicity by a shift-of-contour argument. An exponentially small upper bound on the splitting is implied. The focus of this paper is on the method. [ABSTRACT FROM AUTHOR]

Hamiltonian function describing a system composed of n gravitating shells in general relativity is derived from general considerations and its dynamics is presented. The results appear to be promising for the description of colliding system of astrophysical and cosmological interest. [ABSTRACT FROM AUTHOR]

In this paper, we build on our earlier proposal for the construction of a positive inner product for pseudo-Hermitian Hamiltonians and present examples to clarify the procedure. We focus on two detailed calculations where the method is used, namely, a simple (generalized 2×2 matrix) pseudo-Hermitian Hamiltonian, which can be diagonalized, and a second system where the Hamiltonian cannot be diagonalized, but can be described as a perturbation of the harmonic oscillator. When the quantum mechanical system cannot be diagonalized exactly, our construction can be carried out perturbatively and we develop the general formalism for such a perturbative calculation systematically (for real eigenvalues). [ABSTRACT FROM AUTHOR]

We show how to apply to Hamiltonian differential systems recent results for studying the periodic orbits of a differential system using the averaging theory. We have chosen two classical integrable Hamiltonian systems, one with the Hooke potential and the other with the Kepler potential, and we study the periodic orbits which bifurcate from the periodic orbits of these integrable systems, first perturbing the Hooke Hamiltonian with a nonautonomous potential, and second perturbing the Kepler problem with an autonomous potential. [ABSTRACT FROM AUTHOR]

The dynamics of N point vortices in a fluid is described by the Helmholtz–Kirchhoff (HK) equations which lead to a completely integrable Hamiltonian system for N=2 or 3 but chaotic dynamics for N>3. Here we consider a generalization of the HK equations to describe the dynamics of magnetic vortices within a collective-coordinate approximation. In particular, we analyze in detail the dynamics of a system of three magnetic vortices by a suitable generalization of the solution for three point vortices in an ordinary fluid obtained by Gröbli more than a century ago. The significance of our results for the dynamics of ferromagnetic elements is briefly discussed. [ABSTRACT FROM AUTHOR]

Approximate analytical energy formulas for N-body semirelativistic Hamiltonians with one- and two-body interactions are obtained within the framework of the auxiliary field method. This method has already been proven to be a powerful technique in the case of two-body problems. A general procedure is given and applied to various Hamiltonians of interest, in atomic and hadronic physics in particular. A test of formulas is performed for baryons described as a three-quark system. [ABSTRACT FROM AUTHOR]

This paper is concerned with the superlinear periodic elliptic systems of Hamiltonian type in the whole space. The existence of a ground state solution as well as an infinite number of geometrically distinct solutions is obtained. [ABSTRACT FROM AUTHOR]

We consider quantum spin systems defined on finite sets V equipped with a metric. In typical examples, V is a large, but finite subset of Zd. For finite range Hamiltonians with uniformly bounded interaction terms and a unique, gapped ground state, we demonstrate a locality property of the corresponding ground state projector. In such systems, this ground state projector can be approximated by the product of observables with quantifiable supports. In fact, given any subset X⊂V the ground state projector can be approximated by the product of two projections, one supported on X and one supported on Xc, and a bounded observable supported on a boundary region in such a way that as the boundary region increases, the approximation becomes better. This result generalizes to multidimensional models, a result of Hastings that was an important part of his proof of an area law in one dimension [“An area law for one dimensional quantum systems,” J. Stat. Mech.: Theory Exp. 2007, 08024]. [ABSTRACT FROM AUTHOR]

We give a mathematically rigorous derivation of Ehrenfest’s equations for the evolution of position and momentum expectation values under general and natural assumptions which include atomic and molecular Hamiltonians with Coulomb interactions. [ABSTRACT FROM AUTHOR]

In this paper we construct a new 8×8 M matrix from the 4×4 M matrix, where M/M is the image of the braid group representation. The 8×8 M matrix and the 4×4 M matrix both satisfy extraspecial 2-group algebra relations. By Yang–Baxteration approach, we derive a unitary R(θ,φ) matrix from the M matrix with parameters φ and θ. Three-qubit entangled states can be generated by using the R(θ,φ) matrix. A Hamiltonian for three qubits is constructed from the unitary R(θ,φ) matrix. We then study the entanglement and Berry phase of the Yang–Baxter system. [ABSTRACT FROM AUTHOR]

A construction of the bi-Hamiltonian structures for integrable systems on regular time scales is presented. The trace functional on an algebra of δ-pseudodifferential operators, valid on an arbitrary regular time scale, is introduced. The linear Poisson tensors and the related Hamiltonians are derived. The quadratic Poisson tensors are given by the use of the recursion operators of the Lax hierarchies. The theory is illustrated by Δ-differential counterparts of Ablowitz–Kaup–Newell–Segur and Kaup–Broer hierarchies. [ABSTRACT FROM AUTHOR]

In this paper we study the relation between long cycles and Bose–Einstein condensation in the infinite-range Bose–Hubbard model. We obtain an expression for the cycle density involving the partition function for a Bose–Hubbard Hamiltonian with a single-site correction. Inspired by the approximating Hamiltonian method we conjecture a simplified expression for the short cycle density as a ratio of single-site partition functions. In the absence of condensation we prove that this simplification is exact and use it to show that in this case the long cycle density vanishes. In the presence of condensation we can justify this simplification when a gauge-symmetry breaking term is introduced in the Hamiltonian. Assuming our conjecture is correct, we compare numerically the long cycle density with the condensate and find that although they coexist, in general, they are not equal. [ABSTRACT FROM AUTHOR]

A complete solution to the problem of orthogonal separation of variables of the Hamilton–Jacobi equation in three-dimensional Minkowski space is obtained. The solution is based on the underlying ideas of Cartan geometry and ultimately developed into a general new algorithm that can be employed in the study of Hamiltonian systems defined by natural Hamiltonians within the framework of Hamilton–Jacobi theory. To demonstrate its effectiveness, we investigate from this viewpoint the Morosi–Tondo integrable system derived as a stationary reduction of the seventh-order Korteweg–de Vries flow to show explicitly that the system in question is an orthogonally separable Hamiltonian system. The latter result is a new characterization of the Morosi–Tondo system. [ABSTRACT FROM AUTHOR]

We construct quantum integrable systems associated with non-skew-symmetric gl(2)-valued classical r-matrices. We find a new explicit multiparametric family of such the non-skew-symmetric classical r-matrices. We consider two classes of examples of the corresponding integrable systems, namely generalized Gaudin systems with and without an external magnetic field. In the case of arbitrary r-matrices diagonal in a standard gl(2)-basis, we calculate the spectrum of the corresponding quantum integrable systems using the algebraic Bethe ansatz. We apply these results to a construction of integrable fermionic models and obtain a wide class of integrable Bardeen–Cooper–Schrieffer (BCS)-type fermionic Hamiltonians containing the pairing and electrostatic interaction terms. We also consider special cases when the corresponding integrable Hamiltonians contain only pairing interaction term and are exact analogs of the “reduced BCS Hamiltonian” of Richardson. [ABSTRACT FROM AUTHOR]

We construct a genuine Radon measure with values in B(ℓ2(Zd)) on the set of paths in Zd representing Feynman’s integral for the discrete Laplacian on ℓ2(Zd), and we prove the Feynman integral formula for the solutions of the Schrödinger equation with Hamiltonian H=-12Δ+V, where Δ is the discrete Laplacian and V is an arbitrary bounded potential. [ABSTRACT FROM AUTHOR]

Energy of waves (or eigenmodes) in an ideal fluid and plasma is formulated in the noncanonical Hamiltonian context. By imposing the kinematical constraint on perturbations, the linearized Hamiltonian equation provides a formal definition of wave energy not only for eigenmodes corresponding to point spectra but also for singular ones corresponding to a continuous spectrum. The latter becomes dominant when mean fields have inhomogeneity originating from shear or gradient of the fields. The energy of each wave is represented by the eigenfrequency multiplied by the wave action, which is nothing but the action variable and, moreover, is associated with a derivative of a suitably defined dispersion relation. The sign of the action variable is crucial to the occurrence of Hopf bifurcation in Hamiltonian systems of finite degrees of freedom [M. G. Krein, Dokl. Akad. Nauk SSSR, Ser. A 73, 445 (1950)]. Krein’s idea is extended to the case of coalescence between point and continuous spectra. [ABSTRACT FROM AUTHOR]

We present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of regular and nonregular time-dependent mechanical systems, which is based on the approach of Skinner and Rusk [“Generalized Hamiltonian dynamics I. Formulation on T*QxTQ,” J. Math. Phys. 24, 2589 (1983)]. The dynamical equations of motion and their compatibility and consistency are carefully studied, making clear that all the characteristics of the Lagrangian and the Hamiltonian formalisms are recovered in this formulation. As an example, a semidiscretization of the nonlinear wave equation is studied, proving the applicability of the proposed formalism. [ABSTRACT FROM AUTHOR]

Starting from a discrete spectral problem, we derive a hierarchy of nonlinear discrete equations which includes the Ablowitz–Ladik hierarchy and a new hierarchy as special cases. Especially, we investigate in detail the integrability and resolvability of the new hierarchy. It is shown that the new hierarchy is integrable in Liouville’s sense and possesses multi-Hamiltonian structure. A Darboux transformation is established for a typical discrete system in the new hierarchy with the help of the gauge transformation of its Lax pair. As applications of the Darboux transformation, new exact solutions of the discrete system are explicitly given. [ABSTRACT FROM AUTHOR]

For the rational, elliptic, and trigonometric r-matrices, we exhibit the links between three “levels” of Poisson spaces: (a) some finite-dimensional spaces of matrix-valued meromorphic functions on the complex line, (b) spaces of spectral curves and sheaves supported on them, and (c) symmetric products of a surface. We have, at each level, a linear space of compatible Poisson structures, and the maps relating the levels are Poisson. This leads in a natural way to the Nijenhuis coordinates for these spaces. At level (b), there are Hamiltonian systems on these spaces which are integrable for each Poisson structure in the family and which are such that the Lagrangian leaves are the intersections of the symplectic leaves over the Poisson structures in the family. Specific examples include many of the well-known integrable systems. [ABSTRACT FROM AUTHOR]

We study statistical models, specifically transfer matrices corresponding to a multiparameter hierarchy of braid matrices of (2n)2×(2n)2 dimensions with 2n2 free parameters (n=1,2,3,...). The simplest, 4×4 case, is treated in detail. Powerful recursion relations are constructed by explicitly giving the dependence on the spectral parameter θ of the eigenvalues of the transfer matrix at each level of coproduct sequence. A brief study of higher dimensional cases (n>=2) is presented, pointing out features of particular interest. Spin chain Hamiltonians are also briefly presented for the hierarchy. In a long final section, basic results are recapitulated with systematic analysis of their contents. Our eight vertex 4×4 case is compared to standard six vertex and eight vertex models. [ABSTRACT FROM AUTHOR]

Hamiltonian n-dimensional Lotka–Volterra systems are introduced that have n-1 conserved quantities. The explicit integrability in quadratures is demonstrated. [ABSTRACT FROM AUTHOR]

We cannot perform the projective measurement of a momentum on a half line since it is not an observable. Nevertheless, we would like to obtain some physical information of the momentum on a half line. We define an optimality for measurement as minimizing the variance between an inferred outcome of the measured system before a measuring process and a measurement outcome of the probe system after the measuring process, restricting our attention to the covariant measurement studied by Holevo [Rep. Math. Phys. 13, 379 (1978)]. Extending the domain of the momentum operator on a half line by introducing a two dimensional Hilbert space to be tensored, we make it self-adjoint and explicitly construct a model Hamiltonian for the measured and probe systems. By taking the partial trace over the newly introduced Hilbert space, the optimal covariant positive operator valued measure of a momentum on a half line is reproduced. We physically describe the measuring process to optimally evaluate the momentum of a particle on a half line. [ABSTRACT FROM AUTHOR]

We discuss the possibility of defining an algebraic dynamics within the settings of O*-algebras. Compared to our previous results on this subject, the main improvement here is that we are not assuming the existence of some Hamiltonian for the full physical system. We will show that, under suitable conditions, the dynamics can still be defined via some limiting procedure starting from a given regularized sequence. [ABSTRACT FROM AUTHOR]

Form-invariant Poisson brackets of hydrodynamic type with several spatial variables x1,...,xp are introduced for any commuting vector fields U1,...,Um on a manifold Mn. The incompressibility equation for certain parameters of the Poisson brackets is derived. Poisson brackets connected with an arbitrary dynamical system on Mn are constructed. A gauge transform for the Poisson brackets with one spatial variable x is defined and its action onto the tensor invariants is studied. A general identity between the coefficients bkij(u) and gij(u) is found. [ABSTRACT FROM AUTHOR]

We suggest a systematic method of extension of quasiexactly solvable (QES) systems. We construct finite-dimensional subspaces on the basis of special functions (hypergeometric, Airy, Bessel ones) invariant with respect to the action of differential operators of the second order with polynomial coefficients. As an example of physical applications, we show that the known two-photon Rabi Hamiltonian becomes quasiexactly solvable at certain values of parameters when it can be expressed in terms of corresponding QES operators related to the hypergeometric function. [ABSTRACT FROM AUTHOR]

The long time dynamics of solitary wave solutions of the nonlinear Schrödinger equation in time-dependent external potentials is rigorously studied. To set the stage, the well-posedness of the Cauchy problem for a generalized nonautonomous nonlinear Schrödinger equation with time-dependent nonlinearities and potential is established. Afterward, the dynamics of NLS solitary waves in time-dependent potentials is studied. It is shown that in the space-adiabatic regime where the external potential varies slowly in space compared to the size of the soliton, the dynamics of the center of the soliton is described by Hamilton’s equations, plus terms due to radiation damping. Finally, two physical applications are discussed: the first is adiabatic transportation of solitons and the second is the Mathieu instability of trapped solitons due to time-periodic perturbations. [ABSTRACT FROM AUTHOR]

We consider an invariant quantum Hamiltonian H=-ΔLB+V in the L2 space based on a Riemannian manifold M with a countable discrete symmetry group Γ. Typically, M is the universal covering space of a multiply connected Riemannian manifold M and Γ is the fundamental group of M. On the one hand, following the basic step of the Bloch analysis, one decomposes the L2 space over M into a direct integral of Hilbert spaces formed by equivariant functions on M. The Hamiltonian H decomposes correspondingly, with each component HΛ being defined by a quasiperiodic boundary condition. The quasiperiodic boundary conditions are in turn determined by irreducible unitary representations Λ of Γ. On the other hand, fixing a quasiperiodic boundary condition (i.e., a unitary representation Λ of Γ) one can express the corresponding propagator in terms of the propagator associated with the Hamiltonian H. We discuss these procedures in detail and show that in a sense they are mutually inverse. [ABSTRACT FROM AUTHOR]

The Faddeev–Popov gauge fixing in Yang–Mills theory is interpreted as equivariant localization. It is shown that the Faddeev–Popov procedure amounts to a construction of a symplectic manifold with a Hamiltonian group action. The BRST cohomology is shown to be equivalent to the equivariant cohomology based on this symplectic manifold with Hamiltonian group action. The ghost operator is interpreted as a (pre)symplectic form and the gauge condition as the moment map corresponding to the Hamiltonian group action. This results in the identification of the gauge fixing action as a closed equivariant form, the sum of an equivariant symplectic form, and a certain closed equivariant 4-form, which ensures convergence. An almost complex structure compatible with the symplectic form is constructed. The equivariant localization principle is used to localize the path integrals onto the gauge slice. The Gribov problem is also discussed in the context of equivariant localization principle. As a simple illustration of the methods developed in the paper, the partition function of N=2 supersymmetric quantum mechanics is calculated by equivariant localization. [ABSTRACT FROM AUTHOR]

A supertrace identity on Lie superalgebras is established. It provides a tool for constructing super-Hamiltonian structures of zero curvature equations associated with Lie superalgebras. Applications in the case of the Lie superalgebra B(0,1) present super-Hamiltonian structures of a super-AKNS soliton hierarchy and a super-Dirac soliton hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we consider the weakly dissipative Camassa–Holm equation. First, we try to improve other authors’ results and get some new criterion on blow up, then discuss the global existence of the solution. Finally, we intend to establish sufficient conditions on the propagation speed for the weakly dissipative Camassa–Holm equation. [ABSTRACT FROM AUTHOR]