the complex geometry of weak piecewise smooth solutions of integrable nonlinear pde s of shallow water and dym type an extension of the algebraic geometric method for nonlinear integrable pde s is shown to lead to new piecewise smooth weak solutions of a class of n component systems of nonlinear evolution equations this class includes among others equations from the dym and shallow water equation hierarchies the main goal of the paper is to give explicit theta functional expressions for piecewise smooth weak solutions of these nonlinear pde s which are associated to nonlinear subvarieties of hyperelliptic jacobians

finite element approximation of surfactant spreading on a thin film we consider a fully practical finite element approximation of the following system of nonlinear degenerate parabolic equations begin alignat 2 textstyle frac partial u partial t textstyle frac 1 2 nabla u 2 nabla sigma v textstyle frac 1 3 nabla u 3 nabla w 0 quad mbox in omega t qquad hspace 2cm nonumber w c delta u a u 3 delta u nu nonumber textstyle frac partial v partial t nabla u v nabla sigma v rho delta v textstyle frac 1 2 nabla u 2 v nabla w 0 quad mbox in omega t nonumber end alignat the above models a surfactant driven thin film flow in the presence of both attractive a 0 and repulsive delta 0 with nu 3 van der waals forces where u is the height of the film v is the concentration of the insoluble surfactant monolayer and sigma v 1 v is the typical surface tension here rho geq 0 and c 0 are the inverses of t

a linear finite volume method for nonlinear cross diffusion systems in this paper we propose and analyze a linear finite volume scheme for general nonlinear cross diffusion systems the scheme consists of discretization of linear elliptic equations and pointwise explicit algebraic corrections at each time step therefore the scheme can be implemented very easily moreover it is unconditionally stable we establish error estimates in the l 2 l2 norm

fractional hamiltonian monodromy from a gauss manin monodromy fractional hamiltonian monodromy is a generalization of the notion of hamiltonian monodromy recently introduced by nekhoroshev sadovskii and zhilinskii c r acad sci paris ser 1 335 985 2002 nekhoroshev sadovskii and zhilinskii ann henri poincare 7 1099 2006 for energy momentum maps whose image has a particular type of nonisolated singularities in this paper we analyze the notion of fractional hamiltonian monodromy in terms of the gauss manin monodromy of a riemann surface constructed from the energy momentum map and associated with a loop in complex space which bypasses the line of singularities we also prove some propositions on fractional hamiltonian monodromy for 1 n and m n resonant systems

hamiltonian monodromy via picard lefschetz theory in this paper we investigate the hamiltonian monodromy of the fibration in liouville tori of certain integrable systems via real algebraic geometry using picard lefschetz theory in a relative prym variety we determine the hamiltonian monodromy of the geodesic flow on so 4 using a relative generalized jacobian we prove that the hamiltonian monodromy of the spherical pendulum can also be obtained by the picard lefschetz formula

wave solutions of evolution equations and hamiltonian flows on nonlinear subvarieties of generalized jacobians the algebraic geometric approach is extended to study evolution equations associated with the energy dependent schr dinger operators having potentials with poles in the spectral parameter in connection with hamiltonian flows on nonlinear subvarieties of jacobi varieties the general approach is demonstrated by using new parametrizations for constructing quasi periodic solutions of the shallow water and dym type equations in terms of theta functions a qualitative description of real valued solutions is provided

motion of four dimensional rigid body around a fixed point an elementary approach i the goal of this letter is to give an elementary approach to the solution of euler frahm equations for the manakov four dimensional case for this we use the kotter approach and some results from the original papers by schottky weber and caspary we hope that such an approach will be useful for the solution of the problem of an n dimensional top

error estimates for discrete time approximations of nonlinear cross diffusion systems this paper is concerned with nonlinear and linear discrete time algorithms for cross diffusion systems the nonlinear scheme corresponds to backward differences in time and the linear algorithm is a very easy to implement scheme proposed by the author esaim math model numer anal 45 2011 pp 1141 1161 the main purpose of this paper is to derive convergence rates of the discrete time schemes we obtain the same orders for both the nonlinear and the linear scheme moreover these orders are optimal we also establish uniqueness and regularity results of weak solutions of the cross diffusion systems

on a cross diffusion population model deduced from mutation and splitting of a single species we deduce a particular case of the population cross diffusion model introduced by shigesada et al 1979 1 by using the ideas of mutation and splitting from a single species as described by sanchez palencia for ode s systems sanchez palencia 2011 21 the resulting equations of the pde system only differ in the cross diffusion terms the corresponding diffusion matrix being self diffusion dominated which implies that the well known population segregation patterns of the shigesada et al model do not appear in this case we prove existence and uniqueness of solutions of the pde system and use a finite element approximation to discuss numerically stability properties of solutions with respect to the parameters in comparison with related models

strong coupling schemes for interaction of thin walled structures and incompressible flows the contribution addresses the specific physical sensitivity inherent in the interaction of slender structures like membranes or shells and instationary incompressible flows and its consequences for numerical solution schemes different partitioned strong coupling approaches are presented and compared with respect to their efficiency in addition to the classical block gauss seidel iteration and a newton krylov approach a two level coupling scheme is introduced it is based on a coarse grid predictor followed by the usual iterative fine scale solution the performance of the three coupling schemes is investigated applying a two dimensional model problem of a membrane roof three dimensional numerical examples demonstrate the mechanical and numerical sensitivity for these especially delicate fluid structure interaction problems copyright 2010 john wiley sons ltd

the kowalewski and henon heiles motions as manakov geodesic flows on so 4 a two dimensional family of lax pairs the invariant surfaces for the kowalewski top the henon heiles system and the manakov geodesic flow onso 4 complete into abelian surfacesa by adjoining in each case a divisord of arithmetic genus 9 these divisors belong to the same linear system ona and they each define a polarization 2 4 therefore there are rational maps transforming the kowalewski top and the henon heiles system into manakov s geodesic flow onso 4 this paper deals with the precise geometric relationship between these three problems it is based on the splitting of the 8 dimensional space of sections ofd theta functions into an even and an odd part and also on a normal form for the six quadrics describinga as embedded in 7 as a byproduct we get a 2 dimensional family of lax pairs for both the kowalewski top and the henon heiles system

algebraic geometry and stability for integrable systems abstract in 1970s a method was developed for integration of nonlinear equations by means of algebraic geometry starting from a lax representation with spectral parameter the algebro geometric method allows to solve the system explicitly in terms of theta functions of riemann surfaces however the explicit formulas obtained in this way fail to answer qualitative questions such as whether a given singular solution is stable or not in the present paper the problem of stability for equilibrium points is considered and it is shown that this problem can also be approached by means of algebraic geometry

the full geometry of kowalewski s top and 1 2 abelian surfaces revue sur les varietes abeliennes les automorphismes d une courbe elliptique surfaces abeliennes t 2 de polarisation 1 2 jacobiens hyperelliptiques comme recouvrements doubles de t 2 les tores t 2 comme doubles recouvrements des jacobiens hyperelli courbe de kowalewski et surfaces abeliennes de polarisation 2 4

singular fiber of the mumford system and rational solutions to the kdv hierarchy we study the singular isolevel manifold mg 0 of the genus g mumford system associated to the spectral curve y2 x2g 1 we show that mg 0 is stratified by g 1 open subvarieties of additive algebraic groups of dimension 0 1 g and we give an explicit description of mg 0 in terms of the compactification of the generalized jacobian as a consequence we obtain an effective algorithm to compute rational solutions to the genus g mumford system which is closely related to rational solutions of the kdv hierarchy 2009 wiley periodicals inc

modeling spatial adaptation of populations by a time non local convection cross diffusion evolution problem abstract in 19 sighesada et al presented a system of partial differential equations for modeling spatial segregation of interacting species apart from competitive lotka volterra reaction and population pressure cross diffusion terms a convective term modeling the populations attraction to more favorable environmental regions is included in this article we introduce a modification of their convective term to take account for the notion of spatial adaptation of populations after describing the model we briefly discuss its well possedness and propose a numerical discretization in terms of a mass preserving time semi implicit finite differences scheme finally we provide the results of two biologically inspired numerical experiments showing qualitative differences between the original model of 19 and the model proposed in this article

hyperelliptic quasi periodic and soliton solutions of the nonlinear schrodinger equation on considere la construction de solutions par des methodes de geometrie algebriques et en particulier par les fonctions theta hyperelliptiques

a class of integrable flows on the space of symmetric matrices for a given skew symmetric real n n matrix n the bracket x y n xny ynx defines a lie algebra structure on the space sym n n of symmetric n n real matrices and hence a corresponding lie poisson structure the purpose of this paper is to investigate the geometry integrability and linearizability of the hamiltonian system dot x x 2 n or equivalently in lax form the equation dot x x xn nx on this space along with a detailed study of the poisson geometry itself if n has distinct eigenvalues it is proved that this system is integrable on a generic symplectic leaf of the lie poisson structure of sym n n this is established by finding another compatible poisson structure

the algebraic complete integrability of geodesic flow on so n we study for which left invariant diagonal metrics onso n the euler arnold equations dot x x lambda x x x ij in so n lambda ij x ij lambda ij lambda ji can be linearized on an abelian variety i e are solvable by quadratures we show that merely by requiring that the solutions of the differential equations be single valued functions of complex timet suffices to prove that under a non degeneracy assumption on the metric the only such metrics are those which satisfy manakov s conditions ij bi bj ai aj 1 the case of degenerate metrics is also analyzed forn 4 this provides a new and simpler proof of a result of adler and van moerbeke 3

fluid structure interaction with large structural displacements the calculation of viscous flows inside deformable structures faces major difficulties when enforcing the kinematic compatibility at the fluid structure interface and updating the geometry of the domain this paper proposes to overcome these problems by considering the fluid and the structure as a unique continuous medium to be studied in a fixed reference configuration the resulting problem is then split into a fluid and a structural part through an additive decomposition of the space of kinematically admissible test functions this approach treats the structure in a fully lagrangian way and uses an associated arbitrary lagrangian eulerian ale formulation for the fluid such a strategy has three advantages which are detailed in the present paper it can be discretised in time by implicit stable energy conserving time integration schemes and solved by simple iterative uncoupled algorithms the fluid and the structural problems can be approximated by independent finite element spaces the most recent geometrically exact nonlinear shell models can be used for the structures the full strategy has been implemented into an industrial cfd code results obtained in the simulation of industrial hydraulic shock absorbers are presented at the end of the paper

kowalewski s asymptotic method kac moody lie algebras and regularization we use an effective criterion based on the asymptotic analysis of a class of hamiltonian equations to determine whether they are linearizable on an abelian variety i e solvable by quadrature the criterion is applied to a system with hamiltonian h 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 sum limits i 1 l p i 2 sum limits i 1 l 1 exp left sum limits j 1 l n ij x j right parametrized by a real matrixn n ij of full rank it will be solvable by quadrature if and only if for alli j 2 n nt ij n n t jj 1 is a nonpositive integer i e the interactions correspond to the toda systems for the kac moody lie algebras the criterion is also applied to a system of gross neveu

a numerical method for the accurate solution of the fokker planck landau equation in the nonhomogeneous case abstract a new approach for the accurate numerical solution of the fokker planck landau fpl equation in the nonhomogeneous case is presented the method couples through a time splitting algorithm a finite volume scheme for the transport with a fast spectral solver for the efficient solution of the collision operator recently introduced the scheme allows the use of different grids in the velocity space for the transport and the collision phases the use of a suitable explicit runge kutta solver for the time integration of the collision phase permits avoid once of excessive small time steps induced by the stiffness of the diffusive collision operator numerical results for both space homogeneous and space nonhomogeneous situations show the efficiency and the accuracy of the present method

coarse grid correction scheme for implicit multiblock euler calculations a coarse grid correction scheme is used to bring global influence to implicit multiblock calculations compared to using only explicit coupling between the blocks a considerable reduction in the number of time steps needed to reach steady state has been observed for transonic and subsonic flows the additional cpu time per time step associated with the method is small resulting in a highly efficient parallel implicit scheme

isospectral hamiltonian flows in finite and infinite dimensions the approach to isospectral hamiltonian flow introduced in part i is further developed to include integration of flows with singular spectral curves the flow on finite dimensional ad invariant poisson submanifolds of the dual widetilde gl r of the positive part of the loop algebra widetilde gl r is obtained through a generalization of the standard method of linearization on the jacobi variety of the invariant spectral curves these curves are embedded in the total space of a line bundlet p1 c allowing an explicit analysis of singularities arising from the structure of the image of a moment map tilde j r m n r times m n r to widetilde gl r from the space of rank r deformations of a fixedn n matrixa it is shown how the linear flow of line bundles e t to tilde s over a suitably desingularized curve tilde s may be used to determine both the flow of matricial polynomialsl and the hamiltonian flow in the spacemn r mn r in terms of functions the resulting flows are proved to be completely integrable the reductions to subalgebras developed in part i are shown to correspond to invariance of the spectral curves and line bundles e t to tilde s under certain linear or anti linear involutions the integration of two examples from part i is given to illustrate the method the rosochatius system and the cnls coupled non linear schrodinger equation

the euler equations on lie algebra so 4 an elementary approach to integrability condition we give a new derivation of the manakov and the product case integrability conditions of the euler equations on lie algebra so 4 fourth first integral functionally independent of the three already known integrals is obtained explicitly for all values of the parameters by an algorithmic approach

convergent discretizations for the nernst planck poisson system we propose and compare two classes of convergent finite element based approximations of the nonstationary nernst planck poisson equations whose constructions are motivated from energy versus entropy decay properties for the limiting system solutions of both schemes converge to weak solutions of the limiting problem for discretization parameters tending to zero our main focus is to study qualitative properties for the different approaches at finite discretization scales like conservation of mass non negativity discrete maximum principle decay of discrete energies and entropies to study long time asymptotics

positivity preserving numerical schemes for lubrication type equations lubrication equations are fourth order degenerate diffusion equations of the form h t nabla cdot f h nabla delta h 0 describing thin films or liquid layers driven by surface tension recent studies of singularities in which h to 0 at a point describing rupture of the fluid layer show that such equations exhibit complex dynamics which can be difficult to simulate accurately in particular one must ensure that the numerical approximation of the interface does not show a false premature rupture generic finite difference schemes have the potential to manifest such instabilities especially when underresolved we present new numerical methods in one and two space dimensions that preserve positivity of the solution regardless of the spatial resolution whenever the pde has such a property we also show that the schemes can preserve positivity even when the pde itself is only known to be nonnegativity preserving we prove that positivity preserving finite difference schemes have unique positive solutions at all times we prove stability and convergence of both positivity preserving and generic methods in one and two space dimensions to positive solutions of the pde showing that the generic methods also preserve positivity and have global solutions for sufficiently fine meshes we generalize the positivity preserving property to a finite element framework and show via concrete examples how this leads to the design of other positivity preserving schemes

separation of variables and backlund transformations for the symmetric lagrange top we construct the one and two point integrable maps backlund transformations for the symmetric lagrange top we show that the lagrange top has the same algebraic poisson structure that belongs to the sl 2 gaudin magnet the two point map leads to a real time discretization of the continuous flow therefore it provides an integrable numerical scheme for integrating the physical flow we illustrate the construction by a few pictures of the discrete flow calculated in matlab

semi global symplectic invariants of the spherical pendulum abstract we compute the semi global symplectic invariants near the focus focus point of the spherical pendulum a modified birkhoff normal form procedure is presented to compute the expansion of the hamiltonian near the focus focus point in eliasson variables combining this with explicit formulas for the action we find the semi global symplectic invariants near the focus focus point introduced by v ng c 2003 32 we show that the birkhoff normal form at the focus focus point is the inverse of a complete elliptic integral over a vanishing cycle we close with some remarks about the pendulum for which the invariants can be related to theta functions in a beautiful way

finite volume element approximation of an inhomogeneous brusselator model with cross diffusion this paper is concerned with the study of pattern formation for an inhomogeneous brusselator model with cross diffusion modeling an autocatalytic chemical reaction taking place in a three dimensional domain for the spatial discretization of the problem we develop a novel finite volume element fve method associated to a piecewise linear finite element approximation of the cross diffusion system we study the main properties of the unique equilibrium of the related dynamical system a rigorous linear stability analysis around the spatially homogeneous steady state is provided and we address in detail the formation of turing patterns driven by the cross diffusion effect in addition we focus on the spatial accuracy of the fve method and a series of numerical simulations confirm the expected behavior of the solutions in particular we show that depending on the spatial dimension the magnitude of the cross diffusion influences the selection of spatial patterns

quasi periodic solution of the 2 1 dimensional boussinesq burgers soliton equation abstract a 2 1 dimensional bossinesq burgers soliton equation is proposed which has a affinitive connection with the boussinesq burgers soliton hierarchy through a natural nonlinearization of the boussinesq burgers s eigenvalue problems a finite dimensional hamiltonian system is obtained and is proved to be completely integrable in liouville sense the abel jacobi coordinates are constructed to straighten out the hamiltonian flows from which the quasi periodic solutions of the 2 1 dimensional boussinesq burgers equation are derived by resorting to the riemann theta functions

prym varieties and the geodesic flow onso n on etudie la relation entre l espace ou le flot apparait et la variete abelienne sur laquelle le flot se linearise

partitioned procedures for the transient solution of coupled aroelastic problems part i model problem theory and two dimensional application abstract in order to predict the dynamic response of a flexible structure in a fluid flow the equations of motion of the structure and the fluid must be solved simultaneously in this paper we present several partitioned procedures for time integrating this focus coupled problem and discuss their merits in terms of accuracy stability heterogeneous computing i o transfers subcycling and parallel processing all theoretical results are derived for a one dimensional piston model problem with a compressible flow because the complete three dimensional aeroelastic problem is difficult to analyze mathematically however the insight gained from the analysis of the coupled piston problem and the conclusions drawn from its numerical investigation are confirmed with the numerical simulation of the two dimensional transient aeroelastic response of a flexible panel in a transonic non linear euler flow regime

quasi periodic solution of a new 2 1 dimensional coupled soliton equation a new 2 1 dimensional integrable soliton equation is proposed which has a close connection with the levi soliton hierarchy through the nonlinearization of the levi eigenvalue problems we obtain a finite dimensional integrable system the abel jacobi coordinates are constructed to straighten out the hamiltonian flows by which the solutions of both the 1 1 and 2 1 levi equations are obtained through linear superpositions an inversion procedure gives the quasi periodic solution in the original coordinates in terms of the riemann theta functions

on the construction of monopoles we show that any self dual su 2 monopole may be constructed either by ward s twistor method or nahm s use of the adhm construction the common factor in both approaches is an algebraic curve whose jacobian is used to linearize the non linear ordinary differential equations which arise in nahm s method we derive the non singularity condition for the monopole in terms of this curve and apply the result to prove the regularity of axially symmetric solutions

the arnol d formula for algebraically completely integrable systems soit f v n r m une application algebrique reelle et d r m l ensemble de ses valeurs critiques on etudie la monodromie reelle des applications f definies sur une variete symplectique v 2m et dont les fibres generiques sont lagrangiennes pour la forme symplectique de tels applications sont des applications impulsion de systemes hamiltoniens integrables

nonnegativity preserving convergent schemes for the thin film equation we present numerical schemes for fourth order degenerate parabolic equations that arise e g in lubrication theory for time evolution of thin films of viscous fluids we prove convergence and nonnegativity results in arbitrary space dimensions a proper choice of the discrete mobility enables us to establish discrete counterparts of the essential integral estimates known from the continuous setting hence the numerical cost in each time step reduces to the solution of a linear system involving a sparse matrix furthermore by introducing a time step control that makes use of an explicit formula for the normal velocity of the free boundary we keep the numerical cost for tracing the free boundary low

error amplification analysis of subiteration preconditioned gmres for fluid structure interaction the customary subiteration method for solving fluid structure interaction problems exhibits several deficiencies viz only conditional stability potential convergence difficulties due to non normality induced divergence and the inability to reuse information from previously solved similar problems to overcome these deficiencies a novel solution method is considered in which subiteration is used as a preconditioner to gmres this paper treats the linear algebra aspects of the subiteration method and of the subiteration preconditioned gmres method on the basis of properties of the error amplification matrix for the aggregated fluid structure system an analysis of the error amplification matrix of subiteration establishes that subiteration condenses errors into a low dimensional subspace which can be associated with the interface degrees of freedom therefore the gmres acceleration of subiteration can be confined to the interface degrees of freedom the error amplification analysis provides a clear explanation of the relation between the local gmres acceleration i e on the interface degrees of freedom and the global error amplification properties i e for the aggregated system moreover we show that the subiteration iterates span a krylov space corresponding to a preconditioned aggregated system we then address the implications of the non normality of the subiteration preconditioner for the convergence of gmres the subiteration preconditioned gmres method enables the optional reuse of krylov vectors in subsequent invocations of gmres which can substantially enhance the efficiency of the method to assess the potential and the limitations of the reuse option we analyse the error amplification matrix of the gmres method with reuse furthermore we establish that the gmres acceleration on the interface degrees of freedom generates an approximation to the schur complement for the aggregated system the gmres acceleration and the reuse of krylov vectors are then assessed in terms of the approximation properties for the schur complement and in terms of the properties of the corresponding error amplification matrices numerical experiments on a model fluid structure interaction problem illustrate the developed theory in particular we analyse the convergence of the respective methods in terms of spectral radii matrix norms and sharp convergence upper bounds

a monolithical fluid structure interaction algorithm applied to the piston problem abstract an investigation of time marching computational fluid structure interaction algorithms is presented the analysis is applied to the piston problem attention is focussed on the time integration properties of the coupling algorithms the staggered scheme is first investigated where fluid and structure are alternately integrated by separate solvers in a predictor corrector fashion this algorithm suffers from a time lag between the integration of the fluid and structure the influence of the time lag is investigated by the comparison of different predictions for the structure a novel monolithical algorithm is then introduced in order to annihilate the time lag this algorithm integrates fluid structure and interaction as a single system by an implicit algorithm linear acoustic as well as nonlinear euler equations for gas dynamics are investigated the numerical results of the staggered scheme reveal a non physical deviation of the mean position of the piston at higher cfl numbers the deviation of the mean position is not present in the calculation with the monolithical scheme stability analysis shows the unconditional stability of the monolithical scheme for the acoustic equations whereas the staggered scheme has a limited domain of stability this domain can be enlarged by an improvement of the structural prediction analysis of the damping shows an energy production in the staggered scheme while the monolithical scheme has no energy production term the analyses lead to the formulation of the interaction consistency law which prescribes the relation for the time discretisation between fluid and structure solvers and their boundary conditions

two level algorithms for partitioned fluid structure interaction computations abstract in this paper we use the multigrid algorithm commonly used to improve the efficiency of the flow solver to improve the efficiency of partitioned fluid structure interaction iterations coupling not only the structure with the fine flow mesh but also with the coarse flow mesh often present due to the multigrid scheme leads to a significant efficiency improvement as solution of the flow equations typically takes much longer than the structure solve and as multigrid is not standard in structure solvers we do not coarsen the structure or the interface as a result the two level method can be easily implemented into existing solvers two types of two level algorithms were implemented 1 coarse grid correction of the partitioning error and 2 coarse grid prediction or full multigrid to generate a better initial guess the resulting schemes are combined with a fourth order runge kutta implicit time integration scheme for the linear one dimensional piston problem with compressible flow the superior stability accuracy and efficiency of the two level algorithms is shown the parameters of the piston problem were chosen such that both a weak and a strong interaction case were obtained even the strong interaction case with a flexible structure could be solved with our new two level partitioned scheme with just one iteration on the fine grid this is a major accomplishment as most weakly coupled methods fail in this case of the two algorithms the coarse grid prediction or full multigrid method was found to perform best the resulting efficiency gain for our one dimensional problem is around a factor of ten for the coarse to intermediate time steps at which the high order time integration methods should be run for two and three dimensional problems the efficiency gain is expected to be even larger

finite element approximation of a nonlinear cross diffusion population model we consider a fully discrete finite element approximation of the nonlinear cross diffusion population model find ui the population of the ith species i 1 and 2 such that where j i and gi u1 u2 i ii ui ij uj ui in the above the given data is as follows v is an environmental potential ci ai are diffusion coefficients bi are transport coefficients i are the intrinsic growth rates and ii are intra specific whereas ij i j are interspecific competition coefficients in addition to showing well posedness of our approximation we prove convergence in space dimensions d 3 finally some numerical experiments in one space dimension are presented

the toda lattice dynkin diagrams singularities and abelian varieties the periodic l 1 particle toda lattices coresponding to extended dynkin diagrams have l 1 polynomial constants of the motion namely as many as there are dots in the dynkin diagram for most values of the constants their intersection defines as l dimensional affine invariant manifold which completes into a complex algebraic torus abelian variety by glueing on a divisor entirely specified by the extended dynkin diagram therefore the global geometry of the complex invariant tori such as polarization dimension of certain linear systems divisor equivalences etc is entirely given by the extended dynkin diagram in particular an explicit geometric description for the case of three particle toda lattices is given which describes the divisors their singularities and how the various divisors intersect

generalized jacobians of spectral curves and completely integrable systems consider an ordinary differential equation which has a lax pair representation dot a x a x b x where a x is a matrix polynomial with a fixed regular leading coefficient and the matrix b x depends only on a x such an equation can be considered as a completely integrable complex hamiltonian system we show that the generic complex invariant manifold of this lax pair is an affine part of a non compact commutative algebraic group the generalized jacobian of the spectral curve x y p x y 0 with its points at infinity identified moreover for suitable b x the hamiltonian vector field defined by the lax pair on the generalized jacobian is translation invariant

classical bethe ansatz and normal forms in an integrable version of the dicke model abstract the dicke jaynes cummings gaudin model describes a collection of n spins coupled to a harmonic oscillator it is known to be integrable so one can define a moment map which associates to each point in phase space the list of values of the n 1 conserved hamiltonians we identify all the critical points of this map and we compute the corresponding quadratic normal forms using the lax matrix representation of the model the normal coordinates are constructed by a procedure which appears as a classical version of the bethe ansatz used to solve the quantum model we show that only elliptic or focus focus singularities are present in this model which provides an interesting example of a symplectic toric action with singularities to explore these we study in detail the degeneracies of the spectral curves for the n 1 and n 2 cases this gives a complete picture for the image of the momentum map imm and the associated bifurcation diagram for n 2 we found in particular some lines of rank 1 which lie for one part on the boundary of the imm where they behave like an edge separating two faces and which go for another part inside the imm

semi discretization in time and numerical convergence of solutions of a nonlinear cross diffusion population model a positivity preserving numerical scheme for a strongly coupled cross diffusion model for two competing species is presented based on a semi discretization in time the variables are the population densities of the species existence of strictly positive weak solutions to the semidiscrete problem is proved moreover it is shown that the semidiscrete solutions converge to a non negative solution of the continuous system in one space dimension the proofs are based on a symmetrization of the problem via an exponential transformation of variables and the use of an entropy functional

algebraic integrability of the confluent neumann system in this paper we study the neumann system which describes the harmonic oscillator of arbitrary dimension constrained to the sphere in particular we will consider the confluent case where two eigenvalues of the potential coincide which implies that the system has s1 symmetry we will prove complete algebraic integrability of the confluent neumann system and show that its flow can be linearized on the generalized jacobian torus of some singular algebraic curve the symplectic reduction of s1 action will be described and we will show that the general rosochatius system is a symplectic quotient of the confluent neumann system where all the eigenvalues of the potential are double

new integrable problem of classical mechanics complete integrability in liouville s sense is proven for rotation of an arbitrary rigid body with a fixed point in a newtonian field with an arbitrary homogeneous quadratic potential a consequences is the complete integrability of rotation of a rigid body with fixed center of mass in the field of arbitrary sufficiently remote objects in the second approximation explicit formulae are obtained expressing angular velocities of the rigid body in terms of functions for riemannian surfaces integrable cases are found for rotation of a rigid body in nonlinear newtonian potential fields