soliton dynamics in computational anatomy computational anatomy ca has introduced the idea of anatomical structures being transformed by geodesic deformations on groups of diffeomorphisms among these geometric structures landmarks and image outlines in ca are shown to be singular solutions of a partial differential equation that is called the geodesic epdiff equation a recently discovered momentum map for singular solutions of epdiff yields their canonical hamiltonian formulation which in turn provides a complete parameterization of the landmarks by their canonical positions and momenta the momentum map provides an isomorphism between landmarks and outlines for images and singular soliton solutions of the epdiff equation this isomorphism suggests a new dynamical paradigm for ca as well as new data representation

geometrical phase in the cyclic evolution of non hermitian systems the authors derive by a biorthonormal state approach the analogy of berry s phase factor for open non conservative systems for both adiabatic and non adiabatic evolution in the latter case a non unitary evolution operator method is exploited an application is given to the optical supermode propagation in the free electron laser

the need for a flat higher gauge structure to describe a berry phase associated with some resonance phenomena in the presence of a resonance crossing producing splitting of the base manifold for example a circle crossing in a plane we show that the rigorous geometrical structure within which the berry phase arises may be a 2 bundle a structure related to gerbes and to category theory rather than a fiber bundle the bloch wave operator plays an important role in the associated theory

semidirect products and reduction in mechanics this paper shows how to reduce a hamiltonian system on the cotangent bundle of a lie group to a hamiltonian system in the dual of the lie algebra of a semidirect product the procedure simplifies unifies and extends work of greene guillemin holm holmes kupershmidt marsden morrison ratiu sternberg and others the heavy top compressible fluids magnetohydrodynamics elasticity the maxwell vlasov equations and multifluid plasmas are presented as examples starting with lagrangian variables our method explains in a direct way why semidirect products occur so frequently in examples it also provides a framework for the systematic introduction of clebsch or canonical variables

on the cauchy problem for the two component camassa holm system in this paper we establish the local well posedness for the two component camassa holm system in a range of the besov spaces we also derive a wave breaking mechanism for strong solutions in addition we determine the exact blow up rate of such solutions to the system

cyclic states berry phases and the schrodinger operator the evolution of a quantum mechanical system under a nonadiabatic external perturbation at a time interval 0 t is considered it is shown that all the cyclic states of the system mod psi t ei phi mod psi 0 are determined by the eigenvalues in quasi energies and associated eigenvectors mod phi in t of the schrodinger operator sq t h t i delta delta t acting on some hilbert space the set of linearly independent cyclic quasi energy states possess some properties similar to the properties of the stationary states of a closed system the berry phases of the states associated with eigenvectors of the discrete spectrum of sq t which are single valued functions of omega 2 pi t are supplied by the partial derivatives of the corresponding eigenvalues quasi energies with respect to omega the approach developed is illustrated by several applications to time dependent systems the system under an adiabatic perturbation the forced harmonic oscillator and the two level system even in the case of a system with a time independent hamiltonian there exist nonstationary quasi energy states that have nontrivial berry phases beta 2 pi not an integer the criterion of existence of such states is formulated in terms of the energies of the system and the corresponding expression for the berry phases is obtained some examples of nonstationary cyclic states of closed systems including the coherent states of a force free oscillator and the wannier states of electrons in the parabolic band are considered

existence and approximate lp and continuous solutions of nonlinear integral equations of the hammerstein and volterra types in this paper we consider a compact interval a b a positive real number p 1 and give different existence results for l p a b and c a b solutions of some nonlinear integral equations of the hammerstein and volterra types the main ingredients of our existence results are the shaefer s and schauder s fixed point theorems combined with a general version of gronwall s inequality moreover we give a numerical method for the approximation of the solutions of the volterra hammerstein integral equations finally to illustrate the results of this work we provide the reader with some numerical examples

linear dynamical systems coherent state manifolds flows and matrix riccati equation the classical motion and the quantum evolution determined by linear hamiltonians on coherent state manifolds of compact and noncompact hermitian symmetric space structure are considered a matrix riccati equation with opposite signs of the quadratic terms in the compact and noncompact cases determines the classical motion and the quantum evolution the geometric meaning of equations of motion as flow on a complex grassmann manifold and his noncompact dual is emphasized possibilities of generalizing the results to flag manifolds are pointed out

a topological investigation of the quantum adiabatic phase using algebraic topology the appearance of the quantum adiabatic phase over various parameter manifolds is investigated the relation with nontrivial gauge bundles both abelian and non abelian is studied and it is shown that the phase appears as a result of homotopically non trivial mappings induced by the hamiltonian in the space of wave functions the cohomological picture is developed and some topological considerations concerning field theory anomalies in the hamiltonian picture are presented a proof of the nielsen ninomiya theorem is given inspired from the notion of the adiabatic phase

generalized coherent states as eigenstates of linearized quadratic casimir operators it is shown that the generalized coherent states for several groups and a supergroup are eigenstates of the linearized quadratic casimir operator of the associated algebra or superalgebra these operators may be useful in the study of geometrical phases

the piecewise polynomial collocation method for nonlinear weakly singular volterra equations second kind volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration using an adaptation of the analysis originally developed for nonlinear weakly singular fredholm integral equations we present a complete discussion of the optimal global and local order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear volterra integral equations with algebraic or logarithmic singularities in their kernels

hamilton s turn and geometric phase for two level systems hamilton in the course of his studies on quaternions introduced an elegant geometric representation for the composition of su 2 elements in terms of turns on the unit sphere s2 the authors use these turns to study two level systems with particular reference to geometric phase the special roles played by piecewise geodesic circuits in the state space and evolution under constant hamiltonian are recognized

angle variable holonomy in adiabatic excursion of an integrable hamiltonian if an integrable classical hamiltonian h describing bound motion depends on parameters which are changed very slowly then the adiabatic theorem states that the action variables i of the motion are conserved here the fate of the angle variables is analysed because of the unavoidable arbitrariness in their definition angle variables belonging to distinct initial and final hamiltonians cannot generally be compared however they can be compared if the hamiltonian is taken on a closed excursion in parameter space so that initial and final hamiltonians are the same the result shows that the angle variable change arising from such an excursion is not merely the time integral of the instantaneous frequency omega dh di but differs from it by a definite extra angle which depends only on the circuit in parameter space not on the duration of the process the 2 form which describes this angle variable holonomy is calculated

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

generalized coherent states and berry s phase the quantum dynamics of time dependent hamiltonians of the kind h t u t h 0 u t is studied in the framework of generalized coherent states it is shown that the quantum and the classical dynamics are isomorphic and that the phase of t is simply the classical action in adiabatic approximation it is easy to extract berry s phase from this a discussion of generalized coherent states cs s necessary to deal with excited states is given in particular for su 2 su 1 1 and e 2 their structure is completely characterized showing that the deep connection with geometric quantization extends to these generalized cs s this technique may be employed to estimate higher order correction to the adiabatic approximation

integrable systems on semidirect product lie groups we study integrable systems on the semidirect product of a lie group and its lie algebra as the representation space of the adjoint action regarding the tangent bundle of a lie group as phase space endowed with this semidirect product lie group structure we construct a class of symplectic submanifolds equipped with a dirac bracket on which integrable systems in the adler kostant symes sense are naturally built through collective dynamics in doing so we address other issues such as factorization poisson lie structures and dressing actions we show that the procedure becomes recursive for some particular hamilton functions giving rise to a tower of nested integrable systems

quantum phase corrections from adiabatic iteration the phase change gamma acquired by a quantum state psi t rangle driven by a hamiltonian h 0 t which is taken slowly and smoothly round a cycle is given by a sequence of approximants gamma k obtained by a sequence of unitary transformations the phase sequence is not a perturbation series in the adiabatic parameter epsilon because each gamma k except gamma 0 contains epsilon to infinite order for spin frac 1 2 systems the iteration can be described in terms of the geometry of parallel transport round loops c k on the hamiltonian sphere non adiabatic effects transitions must cause the sequence of gamma k to diverge for spin systems with analytic h 0 t this happens in a universal way the loops c k are sinusoidal spirals which shrink as epsilon k until k sim epsilon 1 and then grow as k the smallest loop has a size exp 1 epsilon comparable with the non adiabaticity

classical adiabatic angles and quantal adiabatic phase a semiclassical connection is established between quantal and classical properties of a system whose hamiltonian is slowly cycled by varying its parameters round a circuit the quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n nl the classical property is a shift delta theta l i in the lth angle variable for motion round a phase space torus with actions i il the connection is delta theta l delta gamma delta nl two applications are worked out in detail the generalised harmonic oscillator with quadratic hamiltonian whose parameters are the coefficients of q2 qp and p2 and the rotated rotator consisting of a particle sliding freely round a non circular hoop slowly turned round once in its own plane

geometric phase estimation using time frequency analysis a method for geometric phase estimation from the scalar vector product of the initial and final state vectors of a system is proposed the method is based on the time frequency distribution namely the short time fourier transform stft of a complex signal by adjusting the width of the analysis window of the stft an estimated geometric phase can be obtained which closely matches a true geometric phase the computational algorithm for geometric phase decomposition based on the gabor expansion is presented numerical results are verified by several examples of geometric phase decompositions for su 2 evolutions

geodesic flows on semidirect product lie groups geometry of singular measure valued solutions the epdiff equation or dispersionless camassa holm equation in 1d is a well known example of geodesic motion on the diff group of smooth invertible maps diffeomorphisms its recent two component extension governs geodesic motion on the semidirect product mathrm diff circleds mathcal f where mathcal f denotes the space of scalar functions this paper generalizes the second construction to consider geodesic motion on mathrm diff circleds mathfrak g where mathfrak g denotes the space of scalar functions that take values on a certain lie algebra for example mathfrak g mathcal f otimes mathfrak so 3 measure valued delta like solutions are shown to be momentum maps possessing a dual pair structure thereby extending previous results for the epdiff equation the collective hamiltonians are shown to fit into the kaluza klein theory of particles in a yang mills field and these formulations are shown to apply also at the continuum pde level in the continuum description the kaluza klein approach produces the kelvin circulation theorem

classical particles with internal structure ii second order internal spaces a systematic study of classical relativistic particles with internal structure initiated in a previous paper is continued and a study of second order internal spaces sos is presented within the framework of the lagrangian form of constrained dynamics such internal spaces q are those for which a phase space treatment must necessarily use the cotangent bundle t q the large variety of possible sos s ten discrete cases and two one parameter families is separated into those capable of a manifestly covariant description and those for which special methods based on the transitive action of sl 2 c on a coset space are needed the concept of the isotopy representation plays an important role in this context seven of the possible discrete sos s are shown to be describable in a manifestly covariant way two discrete and one one parameter family of sos s are shown to be unphysical in the sense that no lagrangians can be written in which the internal and the space time position variables are nontrivially coupled and the remaining single discrete and one one parameter family are shown to be physical though not describable in a manifestly covariant way general phase space methods uniformly applicable to all sos s are developed and as an illustrative example a lagrangian model for an sos in which the internal space is spanned by two orthonormal spacelike unit vectors is presented

runge kutta theory for volterra integral equations of the second kind the present paper develops the theory of general runge kutta methods for volterra integral equations of the second kind the order conditions are derived by using the theory of p series which for our problem reduces to the theory of v series these results are then applied to two special classes of runge kutta methods introduced by pouzet and by bel tyukov

geometric phase bundle classification and group representation the line bundles that arise in the holonomy interpretations of the geometric phase display curious similarities to those encountered in the statement of the borel weil bott theorem of the representation theory the remarkable relationship between the mathematical structure of the geometric phase and the classification theorem for complex line bundles provides the necessary tools for establishing the relevance of the borel weil bott theorem to berry s adiabatic phase this enables one to define a set of topological charges for arbitrary compact connected semisimple dynamical lie groups these charges signify the topological content of the phase they can be explicitly computed in this paper the problem of the determination of the parameter space of the hamiltonian is also addressed it is shown that in general the parameter space is either a flag manifold or one of its submanifolds a simple topological argument is presented to indicate the relation between the riemannian structure on the parameter spa

berry s phase and hannay s angle from quantum canonical transformations using an action angle coherent states formalism introduced by the authors in a preceding paper it is shown that berry s phase and hannay s angle can both be derived from the same quantum unitary transformation their relationship is easily established in this framework

non adiabatic berry phase for periodic hamiltonians berry s 1984 formulation of the topological phase for periodic hamiltonians is extended to the case of non adiabatic evolution by proper choice of initial states this restores a full parallel with the non adiabatic formalism of aharonov and anandan 1987 for periodic orbits in a projective hilbert space the berry phase for the novel example of a single electron in a rigidly rotating octahedral environment is calculated using both this formalism and that used previously by aharonov and anandan and by page 1987

vector coherent state theory as a theory of induced representations a general definition is given of vector coherent state vcs representation theory it is shown that the theory is more general than suggested by previous applications and that it incorporates the standard theories of induced representations as special cases the associated k matrix theory is also given a fuller treatment than hitherto and shown to provide a rather general algorithm both for projecting vcs representations from larger representations in which they are embedded and for determining the hermitian form with respect to which an isometric equivalent representation is in fact isometric

a note on multi dimensional camassa holm type systems on the torus we present a 2n component nonlinear evolutionary pde which includes the n dimensional versions of the camassa holm and the hunter saxton systems as well as their partially averaged variations our goal is to apply arnold s geometric formalism arnold 1966 ann inst fourier grenoble 16 319 61 ebin and marsden 1970 ann math 92 102 63 to this general equation in order to obtain results on well posedness conservation laws or stability of its solutions following the line of arguments of kohlmann 2011 j phys a math theor 44 465205 we present geometric aspects of a two dimensional periodic b equation on the diffeomorphism group of the torus in this context

order of convergence of one step methods for volterra integral equations of the second kind nice proofs of convergence and asymptotic expansions are known for one step methods for ordinary differential equations it is shown that these proofs can be generalized in a natural way to extended one step methods for volterra integral equations of the second kind furthermore the convergence of mixed one step methods is investigated for both types general volterra runge kutta methods are considered as examples

the hamiltonian structure of classical chromohydrodynamics noncanonical hamiltonian structures are presented both for yang mills vlasov plasmas and for ideal fluids interacting with yang mills fields the hamiltonian structure for the yang mills vlasov system passes over to that for the yang mills fluid in the cold plasma limit the resulting hamiltonian structure is shown to correspond to a lie algebra

invariance of the generalised berry phase under unitary transformations application to the time dependent generalised harmonic oscillator a generalisation of the berry phase which reduces to the usual berry phase in the adiabatic and cyclic case is derived in a gauge invariant way this generalised berry phase unlike the berry phase is invariant under unitary transformations a time dependent generalised harmonic oscillator with a cross term in the hamiltonian is taken as an example the hamiltonian of this system is not in general the energy an energy the time derivative of which is the power is obtained from the equation of motion when the system is quantised the generalised berry phase is zero and remains invariant under unitary transformations on the other hand if the energy is chosen to be the hamiltonian a non zero generalised berry phase is obtained but the power is given incorrectly in this case the total phase which includes the dynamical phase and the generalised berry phase is equal in the adiabatic limit to the correct total phase

on the solution of a nonlinear integral equation on the basis of a fixed point technique and cubic b spline scaling functions in this paper we apply the fixed point method to solve some nonlinear functional volterra integral equations which appear in many physical chemical and biological problems in each iteration of this method cubic semi orthogonal compactly supported b spline wavelets are used as basis functions to approximate the solution also the convergence of this numerical method is investigated and some examples are presented to show the accuracy and convergence of the method

reduction theory for symmetry breaking with applications to nematic systems we formulate euler poincare and lagrange poincare equations for systems with broken symmetry we specialize the general theory to present explicit equations of motion for nematic systems ranging from single nematic molecules to biaxial liquid crystals the geometric construction applies to order parameter spaces consisting of either unsigned unit vectors directors or symmetric matrices alignment tensors on the hamiltonian side we provide the corresponding poisson brackets in both lie poisson and hamilton poincare formulations the explicit form of the helicity invariant for uniaxial nematics is also presented together with a whole class of invariant quantities casimirs for two dimensional incompressible flows

on geometric phases and dynamical invariants a formalism is developed for calculating berry phases for nonadiabatic time periodic quantum systems when a dynamical invariant is known it is found that when the invariant is periodic and has a nondegenerate spectrum this method allows a convenient way to obtain generalized berry phases and the proper cyclic initial states the method is applied to the generalized harmonic oscillator and the two level system where the invariant operators are explicitly constructed formulae for the conventional berry s phases are readily obtained by taking the adiabatic limit of the exact results

thomas rotation and the mixed state geometric phase it is shown that uhlmann s parallel transport of purifications along a path of mixed states represented by 2 2 density matrices is just the path ordered product of thomas rotations these rotations are invariant under hyperbolic translations inside the bloch sphere that can be regarded as the poincare ball model of hyperbolic geometry a general expression for the mixed state geometric phase for an arbitrary geodesic triangle in terms of the bures fidelities is derived the formula gives back the solid angle result well known from studies of the pure state geometric phase it is also shown that this mixed state anholonomy can be reinterpreted as the pure state non abelian anholonomy of entangled states living in a suitable restriction of the quaternionic hopf bundle in this picture uhlmann s parallel transport is just the pancharatnam transport of quaternionic spinors

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

analysis of berry s phase by the evolution operator method berry 1984 discovered that an eigenstate undergoing an adiabatic evolution in the parameter space will acquire a topological phase aharonov and anandan 1987 on the other hand showed that an eigenstate transporting round a closed circuit in the projective hilbert space suffices to generate the topological phase the authors employ the evolution operator method to study the propagation of an eigenstate they show that berry s phase can be represented as a closed path integral in the hilbert space and is independent of the choice for the base mod n t the treatments of berry and aharonov and anandan are shown to correspond to two different choices of the base therefore their two approaches are unified they have acquired a more general viewpoint on the origin of berry s phase

existence and uniqueness of solutions of abel integral equations with power law non linearities abstract in this article a class of nonlinear abel type integral equations is studied these equations have at least the trivial solution but we are interested in a continuous positive solution we show that there exists a unique positive solution in an order interval of a cone in a banach space

the curvature of semidirect product groups associated with two component hunter saxton systems in this paper we study two component versions of the periodic hunter saxton equation and its variant considering both equations as a geodesic flow on the semidirect product of the circle diffeomorphism group with a space of scalar functions on we show that both equations are locally well posed the main result of this paper is that the sectional curvature associated with the 2hs is constant and positive and that 2 hs allows for a large subspace of positive sectional curvature the issues of this paper are related to some of the results for 2ch and 2dp presented in escher et al 2011 j geom phys 61 436 52

quantum canonical transformations and exact solution of the schrodinger equation time dependent unitary transformations are used to study the schrodinger equation for explicitly time dependent hamiltonians of the form h t r t j where r is an arbitrary real vector valued function of time and j is the angular momentum operator the solution of the schrodinger equation for the most general hamiltonian of this form is shown to be equivalent to the special case r 1 0 t this corresponds to the problem of a driven two level atom for the spin half representation of j it is also shown that by requiring the magnitude of r to depend on its direction in a particular way one can solve the schrodinger equation exactly in particular it is shown that for every hamiltonian of the form h t r t j there is another hamiltonian with the same eigenstates for which the schrodinger equation is exactly solved the application of the results to the exact solution of the parallel transport equation and exact holomony calculation for su 2 principal bundles yang mills gauge theory is also pointed

generalized berry phase for the most general time dependent damped harmonic oscillator the most general time dependence hamiltonian for a harmonic oscillator is both linear and quadratic in the coordinate and the canonical momentum it describes in general a harmonic oscillator with a time dependent mass a time dependent friction or antifriction constant and a time dependent spring constant acted upon by a time dependent force the energy whose time derivative is the power is in general different from the hamiltonian the generalized berry phase for a given energy eigenstate has a state dependent part which vanishes if there is no damping and an arbitrary state independent part if the hamiltonian is identified as the energy both the energy eigenvalues and the generalized berry phase are different in the adiabatic limit both approaches give the same total phase which is the sum of the dynamical and berry phases

classical particles with internal structure general formalism and application to first order internal spaces group theoretic methods are used to systematically classify all possible internal structures for an elementary classical relativistic particle in terms of coset spaces of sl 2 c with respect to its continuous subgroups the allowed internal spaces q are separated into first and second order ones depending on whether a canonical description can be given using q itself or if it needs the cotangent bundle t q three of the former are found one corresponding to the use of a majorana spinor as the internal variable the other two related to orbits in the lie algebra of so 3 1 under the adjoint action for the latter two a lagrangian description of an elementary object with the corresponding internal space is set up and the dynamics studied

numerical solution of nonlinear volterra integral equations of the second kind by using chebyshev polynomials orthogonal chebyshev polynomials are developed to approximate the solutions of linear and nonlinear volterra integral equations properties of these polynomials and some operational matrices are first presented these properties are then used to reduce the integral equations to a system of linear or nonlinear algebraic equations numerical examples illustrate the pertinent features of the method

chebyshev spectral solution of nonlinear volterra hammerstein integral equations abstract in this paper the chebyshev spectral cs method for the approximate solution of nonlinear volterra hammerstein integral equations f o k r g r y r d r 0 t is investigated the method is applied to approximate the solution not to the equation in its original form but rather to an equivalent equation z t g t y t t 1 1 the function z is approximated by the nth degree interpolating polynomial zn with coefficients determined by discretizing g t y t at the chebyshev gauss labatto nodes we then define the approximation to y to be of the form y n t f t 1 1 k t s z n s d s 1 1 and establish that under suitable conditions limn yn t y t uniformly in t finally a numerical experiment for a nonlinear volterra hammerstein integral equation is presented which confirms the convergence demonstrates the applicability and the accuracy of the chebyshev spectral cs method

geometric phases in adiabatic floquet theory abelian gerbes and cheon s anholonomy we study the geometric phase phenomenon in the context of the adiabatic floquet theory so called t t floquet theory a double integration appears in the geometric phase formula because of the presence of two time variables within the theory we show that the geometric phases are then identified with horizontal lifts of surfaces in an abelian gerbe with connection rather than with horizontal lifts of curves in an abelian principal bundle this higher degree in the geometric phase gauge theory is related to the appearance of changes in the floquet blocks at the transitions between two local charts of the parameter manifold we present the physical example of a kicked two level system where the changes are involved by a cheon s anholonomy in this context the analogy between the usual geometric phase theory and the classical field theory also provides an analogy with the classical string theory

a parallel fifth order algorithm for the numerical solution of volterra equations an iterative algorithm based upon the use of a nonpurely polynomial extension of the standard newton cotes quadrature rules is constructed to yield the numerical solution of volterra type integral equations with nonsingular kernel with an accuracy of fifth order in the distance between grid points this algorithm which is easily parallelizable for a small number of processors is illustrated and analysed by means of three test equations

singular solutions of a modified two component camassa holm equation the camassa holm ch equation is a well known integrable equation describing the velocity dynamics of shallow water waves this equation exhibits spontaneous emergence of singular solutions peakons from smooth initial conditions the ch equation has been recently extended to a two component integrable system ch2 which includes both velocity and density variables in the dynamics although possessing peakon solutions in the velocity the ch2 equation does not admit singular solutions in the density profile we modify the ch2 system to allow a dependence on the average density as well as the pointwise density the modified ch2 system mch2 does admit peakon solutions in the velocity and average density we analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data numerical results for the mch2 system are given and compared with the pure ch2 case these numerics show that the modification in the mch2 system to introduce the average density has little short time effect on the emergent dynamical properties however an analytical and numerical study of pairwise peakon interactions for the mch2 system shows a different asymptotic feature namely besides the expected soliton scattering behavior seen in overtaking and head on peakon collisions the mch2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes

on the removability of berry s phase the authors present a thorough analysis of the removability of the berry phase they show that for any system they can by properly restricting the parameter space turn the geometrical phase into an extra shift of the dynamical one however this extra shift in the dynamical phase retains its truly geometrical character the removability is thus only apparent and the geometrical phase can still be measured in interference experiments as indeed has already been done several times

wigner rotations bargmann invariants and geometric phases the concept of the wigner rotation familiar from the composition law of pure lorentz transformations is described in the general setting of lie group coset spaces and the properties of coset representatives examples of abelian and non abelian wigner rotations are given the lorentz group wigner rotation occurring in the coset space sl 2 r so 2 simeq so 2 1 so 2 is shown to be an analytic continuation of a wigner rotation present in the behaviour of particles with nonzero helicity under spatial rotations belonging to the coset space su 2 u 1 simeq so 3 so 2 the possibility of interpreting these two wigner rotations as geometric phases is shown in detail essential background material on geometric phases bargmann invariants and null phase curves all of which are needed for this purpose is provided

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

geometric framework for phase synchronization in coupled noisy nonlinear systems a geometric approach is introduced for understanding the phenomenon of phase synchronization in coupled nonlinear systems in the presence of additive noise we show that the emergence of cooperative behavior through a change of stability via a hopf bifurcation entails the spontaneous appearance of a gauge structure in the system arising from the evolution of the slow dynamics but induced by the fast variables the conditions for the oscillators to be synchronised in phase are obtained the role of weak noise appears to be to drive the system towards a more synchronized behavior our analysis provides a framework to explain recent experimental observations on noise induced phase synchronization in coupled nonlinear systems

taylor polynomial solutions of nonlinear volterra fredholm integral equations the kanwal and liu method for the solution of fredholm integral equations is applied to certain nonlinear volterra fredholm integral equations in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed

the removability of the hopf term in the 2 1d model and aharonov bohm effect in this paper the removability of the hopf term in the 2 1d model is studied a quantum unitary transformation is found and used to remove the hopf term from the hamiltonian for the model it is then shown that there is an intimate connection between the hopf term and the multivaluedness of the wavefunctional describing the model

weak geodesic flow on a semidirect product and global solutions to the periodic hunter saxton system abstract we give explicit solutions for the two component hunter saxton system on the unit circle moreover we show how global weak solutions can be naturally constructed using the geometric interpretation of this system as a re expression of the geodesic flow on the semidirect product of a suitable subgroup of the diffeomorphism group of the circle with the space of smooth functions on the circle these spatially and temporally periodic solutions turn out to be conservative

geometric phase decomposition for exactly solvable cases with interaction in the paper a problem of geometric phase decomposition for general evolutions in the hilbert space is addressed the decomposition of total phase into dynamical and geometrical parts employs a state representation in the noninteraction picture the noninteraction picture is introduced with the help of decomposition of the system evolution operator into two parts the one pertaining to free evolution and the other to interaction the procedure requires the problem to be an exactly solvable case of the schrodinger equation the most common class of such problems includes dipole hamiltonians for which the evolution operator can be decomposed into a combination of unitary operators geometric phase decomposition in the noninteraction picture can be applied to general noncyclic evolutions but for the cyclic states it reduces to the floquet decomposition defined this way geometric phase possesses characteristic features of geometric phase for the free system but extends some stationary properties to temporal dependences as a case the time dependent relationship between geometric phase and the nonstationarity of a quantum state is illustrated by an example of a spin in a rotating magnetic field it is shown that geometric phase reaches a maximum when the spin state becomes completely nonstationary

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

floquet theory and the non adiabatic berry phase an efficient calculational algorithm is provided for the operator decomposition approach to non adiabatic berry phases for systems with periodic hamiltonians

bloch sphere like construction of su 3 hamiltonians using unitary integration the bloch sphere is a familiar and useful geometrical picture of the time evolution of a single spin or a quantal two level system the analogous geometrical picture for three level systems is presented with several applications the relevant su 3 group and su 3 algebra are eight dimensional objects and are realized in our picture as two four dimensional manifolds that describe the time evolution operator the first called the base manifold is the counterpart of the s2 bloch sphere whereas the second called the fiber generalizes the single u 1 phase of a single spin now four dimensional it breaks down further into smaller objects depending on alternative representations that we discuss geometrical phases are also developed and presented for specific applications arbitrary time dependent couplings between three levels or between two spins qubits with su 3 hamiltonians can be conveniently handled through these geometrical objects

poisson brackets and clebsch representations for magnetohydrodynamics multifluid plasmas and elasticity abstract poisson brackets are constructed by the same mathematical procedure for three physical theories ideal magnetohydrodynamics multifluid plasmas and elasticity each of these brackets is given a simple lie algebraic interpretation moreover each bracket is induced to physical space by use of a canonical poisson bracket in the space of clebsch potentials which are constructed for each physical theory by the standard procedure of constrained lagrangians

on coadjoint orbits of rotational perfect fluids in this paper the structure of vortex coadjoint orbits pertaining to perfect fluids having smooth vorticities in r3 within the framework set up by j marsden and a weinstein physica d 7 305 323 1983 in terms of an associated hamiltonian kahler manifold the clebsch manifold described in terms of the so called clebsch variables is investigated the topological quantization of mikhailov and kuznetsov is related to geometric quantization natural candidates for the coherent states on the clebsch manifold are also exhibited

berry s phase in the floquet representation by means of the floquet theory the investigation of berry s phase for the quantum system with time dependent periodic hamiltonian is transformed into the study of a time independent eigenvalue equation an expression for the infinite dimensional matrix is obtained describing the berry s phase in terms of the floquet state

berry phases and unitary transformations berry phases for spin are defined for any closed loop made by a vector changing direction in three dimensional space a sequence of rotations moves the vector along the loop each rotation is around the axis perpendicular to the moving vector the berry phases come from the eigenvalues of the unitary operator for the product of these rotations the angle of the product rotation is shown to be the solid angle enclosed by the loop the proof uses the ordinary language of quantum mechanics the product is calculated from the commutation relations for spin a general framework is set up to define berry phases for other transformations and states like those for rotations and spin the integral formula is derived alternatives for dynamics are shown to provide different applications and interpretations of the same mathematics an example is used to show how one hamiltonian may be simpler than others adiabatic evolution is obtained in the limit as a coupling constant goes to zero so the adiabatic changes are m

on blow up of solutions to the two component π camassa holm system abstract we investigate cauchy problem of the two component camassa holm system which arises from geodesic flow on a semidirect product lie group of the circle the precise blow up scenarios of strong solutions are derived for the system then several criteria to guarantee blow up of strong solutions are presented finally the exact blow up rate is determined

non abelian geometric phase floquet theory and periodic dynamical invariants for a periodic hamiltonian periodic dynamical invariants may be used to obtain non degenerate cyclic states this observation is generalized to the degenerate cyclic states and the relation between the periodic dynamical invariants and the floquet decompositions of the time evolution operator is elucidated in particular a necessary condition for the occurrence of cyclic non adiabatic non abelian geometrical phase is derived degenerate cyclic states are obtained for a magnetic dipole interacting with a precessing magnetic field