higher order first integrals in classical mechanics we present a practical algorithm for computing first integrals of motion which are polynomial in the momenta for natural hamiltonian systems defined in a flat pseudo riemannian space of arbitrary dimension and signature we then apply our algorithm to explore the integrability of two physical systems first we study the holt potential in two dimensions and derive analogous potentials which admit an additional first integral quartic in the momenta second we analyze a class of cylindrically symmetric potentials in three dimensional euclidean space and recover known families of second order maximally superintegrable systems

group invariant classification of separable hamiltonian systems in the euclidean plane and the o 4 symmetric yang mills theories of yatsun we present a new and effective method of determining separable coordinate systems for natural hamiltonians with two degrees of freedom in flat riemannian space the method is based on intrinsic properties of the associated killing tensors and their invariants under the group of rigid motions e 2 applications to the hamiltonian systems derived by the late v a yatsun from o 4 symmetric yang mills theories are presented in addition an equivalence between separability of two dimensional hamiltonian systems and the existence of pfaffian quasi bi hamiltonian representations is specified

existence and uniqueness of solution in sobolev space for an unsteady crystal growth problem with zero surface tension abstract we study an initial value problem for a two dimensional dendritic crystal growth model with zero surface tension if the initial data is in sobolev space h 2 r it is proved that an unique local solution exists in proper sobolev space

analyticity and nonexistence of classical steady hele shaw fingers this paper concerns analyticity of a classical steadily translating nger in a hele shaw cell and nonexistence of solutions when the relative nger width is smaller than 1 it is proven that any classical solution to the nger problem if it exists for sufciently small but nonzero surface tension is close to some saffman taylor zero surface tension solution and satises some algebraic decay conditions at 1 must belong to the analytic function space a0 as dened in section 1 and chosen in a previous study 34 of the existence of nger solutions further it is proven that for any x ed 2 0 1 there can be no classical steadynger solution when surface tension is sufciently small which contradicts a previous conclusion based on numerical simulation c 2002 wiley periodicals inc

rigorous results in steady finger selection in viscous fingering this paper concerns the existence of a steadily translating finger solution in a hele shaw cell for small but non zero surface tension 2 though there are numerous numerical and formal asymptotic results for this problem we know of no mathematically rigorous results that address the selection problem we rigorously conclude that for relative finger width in the range with small analytic symmetric finger solutions exist in the asymptotic limit of surface tension if and only if the stokes constant for a relatively simple nonlinear differential equation is zero this stokes constant s depends on the parameter and earlier calculations by a number of authors have shown it to be zero for a discrete set of values of a the methodology consists of proving the existence and uniqueness of analytic solutions for a weak half strip problem for any in a compact subset of 0 1 the weak problem is shown to be equivalent to the original finger problem in the function space considered provided we invoke a symmetry condition next we consider the behavior of the solution in a neighborhood of an appropriate complex turning point for the restricted case for some this turning point accounts for exponentially small terms in as 0 that generally violate the symmetry condition we prove that the symmetry condition is satisfied for small when the parameter a is constrained appropriately

generalized lenard chains separation of variables and superintegrability we show that the notion of generalized lenard chains naturally allows formulation of the theory of multiseparable and superintegrable systems in the context of bi hamiltonian geometry we prove that the existence of generalized lenard chains generated by a hamiltonian function defined on a four dimensional n manifold guarantees the separation of variables as an application we construct such chains for the henon heiles systems and for the classical smorodinsky winternitz systems new bi hamiltonian structures for the kepler potential are found

superintegrable three body systems on the line we consider classical three body interactions on a euclidean line depending on the reciprocal distance of the particles and admitting four functionally independent quadratic in the momentum first integrals these systems are multiseparable superintegrable and equivalent up to rescalings to a one particle system in the three dimensional euclidean space common features of the dynamics are discussed we show how to determine quantum symmetry operators associated with the first integrals considered here but do not analyze the corresponding quantum dynamics the conformal multiseparability is discussed and examples of conformal first integrals are given the systems considered here in generality include the calogero wolfes and other three body interactions widely studied in mathematical physics

the quasi bi hamiltonian formulation of the lagrange top starting from the tri hamiltonian formulation of the lagrange top lt in a six dimensional phase space we discuss the possible reductions of the poisson tensors the vector field and its hamiltonian functions on a four dimensional space we show that the vector field of the lt possesses on the reduced phase space a quasi bi hamiltonian formulation which provides a set of separation variables for the corresponding hamilton jacobi equation

lyapunov schmidt reduction algorithm for three dimensional discrete vortices abstract we address the persistence and stability of three dimensional vortex configurations in the discrete nonlinear schrodinger equation and develop a symbolic package based on wolfram s mathematica for computations of the lyapunov schmidt reduction method the lyapunov schmidt reduction method is a theoretical tool which enables us to study continuations and terminations of the discrete vortices for small coupling between lattice nodes as well as the spectral stability of the persistent configurations the method was developed earlier in the context of the two dimensional lattice and applied to the onsite and offsite configurations called the vortex cross and the vortex cell by using semianalytical computations d e pelinovsky p g kevrekidis d frantzeskakis physica d 212 2005 20 53 p g kevrekidis d e pelinovsky proc r soc a 462 2006 2671 2694 the present treatment develops a full symbolic computational package which takes a desired waveform at the anticontinuum limit of uncoupled sites performs a required number of lyapunov schmidt reductions and outputs the predictions on whether the configuration persists for finite coupling in the three dimensional lattice and whether it is stable or unstable it also provides approximations for the eigenvalues of the linearized stability problem we report a number of applications of the algorithm to important multisite three dimensional configurations such as the simple cube the double cross and the diamond for each configuration we identify exactly one solution which is stable for small coupling between lattice nodes

exact solvability of superintegrable benenti systems we establish quantum and classical exact solvability for two large classes of maximally superintegrable benenti systems in n dimensions with arbitrarily large n namely we solve the hamilton jacobi and schrodinger equations for the systems in question the results obtained are illustrated for a model with the cubic potential

multiple solutions for discrete periodic nonlinear schrodinger equations in this paper we study the existence of multibump solutions for discrete nonlinear schrodinger equations with periodic potentials we first reduce the existence of multibump homoclinic solutions to the existence of an isolated homoclinic solution with a nontrivial critical group then we study the existence of homoclinics with nontrivial critical groups for both superlinear and asymptotically linear discrete periodic nonlinear schrodinger equations and we provide simple sufficient conditions for the existence of homoclinics with nontrivial critical groups in the positive definite case as an application we get without any symmetry assumptions infinitely many geometrically distinct homoclinic solutions with exponential decay at infinity

bi differential calculi bi hamiltonian systems and conformal killing tensors the theory of dimakis and m ller hoissen dimakis a and m ller hoissen f 2000 j phys a math gen 33 957 74 concerning bi differential calculi and completely integrable systems is related to bi hamiltonian systems of the poisson nijenhuis type in the special case where the ambient manifold is a cotangent bundle one is able to recover and elucidate the theory of ibort et al ibort a magri f and marmo g 2000 j geom phys 33 210 23 which is in turn a reworking in the bi hamitonian context of benenti s theory of hamilton jacobi separable systems in particular it is shown that benenti s conformal killing tensor which is central to his theory has an even more special form than has hitherto been realized and that when it is converted into a field of endomorphisms by raising an index with the ambient metric it necessarily has vanishing nijenhuis torsion

a unified treatment of quartic invariants at fixed and arbitrary energy two dimensional hamiltonian systems admitting second invariants which are quartic in the momenta are investigated using the jacobi geometrization of the dynamics this approach allows for a unified treatment of invariants at both arbitrary and fixed energy in the differential geometric picture the quartic invariant corresponds to the existence of a fourth rank killing tensor expressing the jacobi metric in terms of a kahler potential the integrability condition for the existence of the killing tensor at fixed energy is a nonlinear equation involving the kahler potential at arbitrary energy further conditions must be imposed which lead to an overdetermined system with isolated solutions we obtain several new integrable and superintegrable systems in addition to all previously known examples

discrete vector on site vortices we study discrete vortices in coupled discrete nonlinear schrodinger equations we focus on the vortex cross configuration that has been experimentally observed in photorefractive crystals stability of the single component vortex cross in the anti continuum limit of small coupling between lattice nodes is proved in the vector case we consider two coupled configurations of vortex crosses namely the charge one vortex in one component coupled in the other component to either the charge one vortex forming a double charge vortex or the charge negative one vortex forming a so called hidden charge vortex we show that both vortex configurations are stable in the anti continuum limit if the parameter for the inter component coupling is small and both of them are unstable when the coupling parameter is large in the marginal case of the discrete two dimensional manakov system the double charge vortex is stable while the hidden charge vortex is linearly unstable analytical predictions are corroborated with numerical observations that show good agreement near the anti continuum limit but gradually deviate for larger couplings between the lattice nodes

the super separability of the three body inverse square calogero system the geometrical theory of the variable separation for the hamilton jacobi equation is applied to the classical three body inverse square calogero system it is proved that this system is separable in infinitely many inequivalent ways related to five different kinds of separable webs in the euclidean three space and the corresponding systems of independent first integrals in involution are computed

from stackel systems to integrable hierarchies of pde s benenti class of separation relations we propose a general scheme of constructing of soliton hierarchies from finite dimensional stackel systems and related separation relations in particular we concentrate on the simplest class of separation relations called benenti class i e certain stackel systems with quadratic in momenta integrals of motion

symmetries separability and volume forms in order to clarify certain misconceptions in the literature we discuss the details of the way determining equations for general symmetries of a lagrangian system differ from determining equations for noether symmetries and establish the minimal set of extra conditions which have to be imposed in practical situations for the former to reduce to the latter we further derive properties by which in situations where the system is integrable through quadratic integrals in involution the components of the corresponding noether symmetries themselves can be used to compute the separation variables if they exist for the hamilton jacobi equation

killing tensors warped products and the orthogonal separation of the hamilton jacobi equation we study killing tensors in the context of warped products and apply the results to the problem of orthogonal separation of the hamilton jacobi equation this work is motivated primarily by the case of spaces of constant curvature where warped products are abundant we first characterize killing tensors which have a natural algebraic decomposition in warped products we then apply this result to show how one can obtain the killing stackel space ks space for separable coordinate systems decomposable in warped products this result in combination with benenti s theory for constructing the ks space of certain special separable coordinates can be used to obtain the ks space for all orthogonal separable coordinates found by kalnins and miller in riemannian spaces of constant curvature next we characterize when a natural hamiltonian is separable in coordinates decomposable in a warped product by showing that the conditions originally given by benenti can be reduced finally we use this characterization and concircular tensors a special type of torsionless conformal killing tensor to develop a general algorithm to determine when a natural hamiltonian is separable in a special class of separable coordinates which include all orthogonal separable coordinates in spaces of constant curvature

a class of liouville integrable hamiltonian systems with two degrees of freedom a class of two dimensional liouville integrable hamiltonian systems is studied separability of the corresponding hamilton jacobi equation for these systems is shown to be equivalent to other fundamental properties of hamiltonian systems such as the existence of the lax and bi hamiltonian representations of certain fixed types applications to physical models including the calogero moser model an integrable case of the henon heiles potential and the nonperiodic toda lattice are presented

analyticity of classical steady needle crystals abstract this paper is a continuation of our previous work rigorous results in selection of steady needle crystals j differential equations 197 2004 349 426 it concerns analyticity of a classical steadily translating needle crystal it is proved that any classical solution to the needle crystal problem with sufficiently small but nonzero surface tension if its slope deviation is close to some ivantsov zero surface tension solution and if its curvature satisfies some algebraic decay conditions at must belong to the analytic function space a 0 defined in 1 and chosen in the previous study mentioned above the analyticity result implies that there can be no classical steady needle crystal solution when anisotropy is zero

orthogonal separation of variables for the hamilton jacobi and wave equations in three dimensional minkowski space we give a complete solution of the eisenhart integrability conditions in three dimensional minkowski space obtaining 39 orthogonally separable webs and 58 inequivalent metrics in adapted coordinate systems which permit orthogonal separation of variables for the associated hamilton jacobi and wave equations the corresponding transformations from canonical separable coordinates to canonical pseudo cartesian coordinates are listed for each of the 58 cases and characteristic killing tensors are given for each of the 39 webs

intrinsic characterization of the variable separation in the hamilton jacobi equation the nonorthogonal separation of variables in the hamilton jacobi equation corresponding to a natural hamiltonian h 12gijpipj v with a metric tensor of any signature is intrinsically characterized by geometrical objects on the riemannian configuration manifold killing vectors killing tensors and killing webs comparisons with previous characterizations and some illustrative examples are given

interlaced solitons and vortices in coupled dnls lattices in the present work we propose a new set of coherent structures that arise in nonlinear dynamical lattices with more than one component namely interlaced solitons in the anti continuum limit of uncoupled sites these are waveforms whose one component has support where the other component does not we illustrate systematically how one can combine dynamically stable unary patterns to create stable ones for the binary case of two components for the one dimensional setting we provide a detailed theoretical analysis of the existence and stability of these waveforms while in higher dimensions where such analytical computations are far more involved we resort to corresponding numerical computations lastly we perform direct numerical simulations to showcase how these structures break up when they are exponentially or oscillatorily unstable to structures with a smaller number of participating sites

an extension of the classical theory of algebraic invariants to pseudo riemannian geometry and hamiltonian mechanics we develop a new approach to the study of killing tensors defined in pseudo riemannian spaces of constant curvature that is ideologically close to the classical theory of invariants the main idea which provides the foundation of the new approach is to treat a killing tensor as an algebraic object determined by a set of parameters of the corresponding vector space of killing tensors under the action of the isometry group the spaces of group invariants and conformal group invariants of valence two killing tensors defined in the minkowski plane are described the group invariants which are the generators of the space of invariants are applied to the problem of classification of orthogonally separable hamiltonian systems defined in the minkowski plane transformation formulas to separable coordinates expressed in terms of the parameters of the corresponding space of killing tensors are presented the results are applied to the problem of orthogonal separability of the drach superintegrable potentials

separation of variables on n dimensional riemannian manifolds i the n sphere sn and euclidean n space rn the following problem is solved what are all the different separable coordinate systems for the laplace beltrami eigenvalue equation on the n sphere sn and euclidean n space rn and how are they constructed this is achieved through a combination of differential geometric and group theoretic methods a graphical procedure for construction of these systems is developed that generalizes vilenkin s construction of polyspherical coordinates the significance of these results for exactly soluble dynamical systems on these manifolds is pointed out the results are also of importance for the analysis of the special functions appearing in the separable solutions of the laplace beltrami eigenvalue equation on these manifolds

geometrical classification of killing tensors on bidimensional flat manifolds valence two killing tensors in the euclidean and minkowski planes are classified under the action of the group which preserves the type of the corresponding killing web the classification is based on an analysis of the system of determining partial differential equations for the group invariants and is entirely algebraic the approach allows one to classify both characteristic and noncharacteristic killing tensors

stability of discrete solitons in nonlinear schrodinger lattices we consider the discrete solitons bifurcating from the anti continuum limit of the discrete nonlinear schrodinger nls lattice the discrete soliton in the anti continuum limit represents an arbitrary finite superposition of in phase or anti phase excited nodes separated by an arbitrary sequence of empty nodes by using stability analysis we prove that the discrete solitons are all unstable near the anti continuum limit except for the solitons which consist of alternating anti phase excited nodes we classify analytically and confirm numerically the number of unstable eigenvalues associated with each family of the discrete solitons

complex variables for separation of the hamilton jacobi equation on real pseudo riemannian manifolds in this paper the geometric theory of separation of variables for the time independent hamilton jacobi equation is extended to include the case of complex eigenvalues of a killing tensor on pseudo riemannian manifolds this task is performed without complexifying the manifold but just by considering complex valued functions on it the simple formalism introduced in the paper allows us to extend in a very natural way the classical results on separation of variables including levi civita criterion and stackel eisenhart theory to the complex case only orthogonal variables are considered

separability preserving dirac reductions of poisson pencils on riemannian manifolds dirac deformation of poisson operators of arbitrary rank is considered the question when dirac reduction does not destroy linear poisson pencils is studied a class of separability preserving dirac reductions in the corresponding quasi bi hamiltonian systems of benenti type is discussed two examples of such reductions are given

maximal superintegrability of benenti systems for a class of hamiltonian systems naturally arising in the modern theory of separation of variables we establish their maximal superintegrability by explicitly constructing the additional integrals of motion

when is a hamiltonian system separable the hamiltonian system given by h 1 2 p2 v q with v c rn is considered a method for integrating such a system is that of separating the variables in the hamilton jacobi equation it is known that if such a separation is possible then it can take place only when the equation is expressed in terms of generalized elliptic coordinates or in a degeneration of these a criterion is proposed for deciding if separation is possible and if it is in which degeneration of elliptic coordinates it takes place

rigorous results in selection of steady needle crystals abstract this paper concerns the existence of solutions to a steady needle crystal growth problem in a one sided model we rigorously prove that for small nonzero anisotropy analytic symmetric needle crystal solutions exist in the limit of surface tension e 2 if only if the stokes constant s for a relatively simple nonlinear differential equation is zero this stokes constant s depends on the parameter 2 9 7 e 8 7 and earlier numerical calculations by a number of investigators have shown this to be zero for a discrete set of values of it is also proven that for 0 there can be no symmetric needle crystal solution in the considered space the methodology consists of two steps first the original problem is reduced to a weak half strip problem for any in a compact set of 0 1 by relaxation of the symmetry condition the weak problem is shown to have a unique solution in the function space considered for any 0 m for some m 0 when a symmetry is invoked the weak problem is shown equivalent to the original needle crystal problem next by considering the behavior of the solution in neighborhood of an appropriate complex turning point for 0 m we extract an exponentially small term in e as e 0 that generally violates the symmetric condition we prove that the symmetry condition is satisfied for small e when the parameter is constrained appropriately

the complete whittaker theorem for two dimensional integrable systems and its application the authors prove for a particle moving in a plane under the influence of a conservative force that when the motion is constrained by a second invariant quadratic in the velocities then the potential allows separability of the hamilton jacobi equation in rectangular polar elliptical cylinder or parabolic cylinder coordinates this link shows the intimate connection between quadratic invariants and the two dimensional hamilton jacobi equation they give examples of the utility of parabolic cylinder coordinates in cases of recent study

multistable solitons in the cubic quintic discrete nonlinear schrodinger equation we analyze the existence and stability of localized solutions in the one dimensional discrete nonlinear schrodinger dnls equation with a combination of competing self focusing cubic and defocusing quintic onsite nonlinearities we produce a stability diagram for different families of soliton solutions that suggests the co existence of infinitely many branches of stable localized solutions bifurcations that occur with an increase in the coupling constant are studied in a numerical form a variational approximation is developed for accurate prediction of the most fundamental and next order solitons together with their bifurcations salient properties of the model which distinguish it from the well known cubic dnls equation are the existence of two different types of symmetric solitons and stable asymmetric soliton solutions that are found in narrow regions of the parameter space the asymmetric solutions appear from and disappear back into the symmetric ones via loops of forward and backward pitchfork bifurcations

intrinsic characterisation of orthogonal separation of one coordinate in the hamilton jacobi equation the authors extend an idea due to woodhouse 1975 to give a coordinate free characterisation of the orthogonal separation of one coordinate for the hamilton jacobi equation on a pseudo riemannian manifold in terms of an involutive family of killing tensors the coordinates can be computed from the killing tensors

existence and uniqueness of analytic solution for unsteady crystals with zero surface tension abstract we study the initial value problem for two dimensional dendritic crystal growth with zero surface tension if the initial data is analytic and close to ivantsov steady solution it is proved that unique analytic solution exists locally in time the analysis is based on a nirenberg theorem on abstract cauchy kovalevsky problem in properly chosen banach spaces

stability of localized structures in generalized dnls equations near the anti continuum limit in this work we consider the stability of localized structures in discrete nonlinear schrodinger lattices with generalized nonlinearities depending on the absolute value of the field we illustrate how the continuation of solutions in one as well as higher dimensions proceeds from the anti continuum limit and show how to generalize the results of pelinovsky et al 2005 physica d 212 1 for arbitrary nonlinearities as a case example of particular experimental relevance we showcase our main findings in the special setting of the lattice with the saturable photorefractive nonlinearity in one and two dimensions our analytical results are found to be in good agreement with direct numerical computations

periodic and decaying solutions in discrete nonlinear schrodinger with saturable nonlinearity we demonstrate the existence of solutions in the discrete nonlinear schrodinger equation dnls with saturable nonlinearity we consider two types of solutions to dnls periodic and vanishing at infinity calculus of variations and the nehari manifolds are employed to establish the existence of these solutions we present some extensions of our results combining the nehari manifold approach and the mountain pass argument

separation of variables in quasi potential systems of bi cofactor form we perform variable separation in the quasi potential systems of equations of the form q a 1 k a 1 k w herea and a are killing tensors by embedding these systems into a bi hamiltonian chain and by calculating the corresponding darboux nijenhuis coordinates on the symplectic leaves of one of the hamiltonian structures of the system we also present examples of the corresponding separation coordinates in two and three dimensions

variable separation for natural hamiltonians with scalar and vector potentials on riemannian manifolds the additive variable separation in the hamilton jacobi equation is studied for a natural hamiltonian with scalar and vector potentials on a riemannian manifold with positive definite metric the separation of this hamiltonian is related to the separation of a suitable geodesic hamiltonian over an extended riemannian manifold thus the geometrical theory of the geodesic separation is applied and the geometrical characterization of the separation is given in terms of killing webs killing tensors and killing vectors the results are applicable to the case of a nondegenarate separation on a manifold with indefinite metric where no null essential separable coordinates occur

bi hamiltonian representation of stackel systems it is shown that linear separation relations are fundamental objects for integration by quadratures of stackel separable liouville integrable systems the so called stackel systems these relations are further employed for the classification of stackel systems moreover we prove that any stackel separable liouville integrable system can be lifted to a bi hamiltonian system of gel fand zakharevich type in conjunction with other known result this implies that the existence of bi hamiltonian representation of liouville integrable systems is a necessary condition for stackel separability

separability and lax pairs for henon heiles system the hamiltonian system corresponding to the generalized henon heiles hamiltonian h 1 2 px2 py2 1 2ax2 1 2by2 x2y ey3 is known to be integrable in the following three cases a b e 1 3 e 2 b 16a e 16 3 in the first two the system has been integrated by making use of genus one and genus two theta functions we show that in the third case the system can also be integrated by making use of elliptic functions finally using the fairbanks theorem we find lax pairs for each of the three integrable systems under consideration

one parameter localized traveling waves in nonlinear schrodinger lattices we address the existence of traveling single humped localized solutions in the spatially discrete nonlinear schr nls equation a mathematical technique is developed for analysis of persistence of these solutions from a certain limit in which the dispersion relation of linear waves contains a triple zero the technique is based on using the implicit function theorem for solution of an appropriate differential advance delay equation in exponentially weighted spaces the resulting melnikov calculation relies on a number of assumptions on the spectrum of the linearization around the pulse which are checked numerically we apply the technique to the so called salerno model and the translationally invariant discrete nls equation with a cubic nonlinearity we show that the traveling solutions terminate in the salerno model whereas they generally persist in the translationally invariant nls lattice as a one parameter family of solutions these results are found to be in a close correspondence with numerical approximations of traveling solutions with zero radiation tails analysis of persistence also predicts the spectral stability of the one parameter family of traveling solutions under time evolution of the discrete nls equation c 2007 elsevier b v all rights reserved

superintegrability on curved spaces orbits and momentum hodographs revisiting a classical result by hamilton the equation of the orbits in the configuration space and of the hodographs in the momentum plane for the curved kepler and harmonic oscillator systems living in a configuration space of any constant curvature and either signature type are derived by purely algebraic means this result extends to the curved kepler or harmonic oscillator for the classical hamilton derivation of the orbits of the euclidean kepler problem through its hodographs in both cases the fundamental property allowing these derivations to work is the superintegrability of the curved kepler and harmonic oscillator no matter whether the constant curvature of the configuration space is zero or not or whether the configuration space metric is riemannian or lorentzian in the curved case the basic result does not refer to the velocity hodograph but to the momentum hodograph both coincide in a euclidean configuration space but only the latter is unambiguously defined in all curved spaces

existence and selection of steady bubbles in a hele shaw cell abstract this paper concerns the existence of a steadily translating bubble in a hele shaw cell for small but non zero surface tension 2 we rigorously conclude that for bubble velocity u relative to the fluid velocity at infinity in the interval 1 2 analytic symmetric solutions exist in the asymptotic limit of surface tension 2 0 if and only if the stokes constant for a relatively simple nonlinear differential equation is zero this stokes constant s depends on the parameter 0 1 corresponding to bubble size and a u 1 2 2 u 1 2 2 2 1 2 4 3 earlier calculations have shown s to be zero for a discrete set of values of a

generalized stackel transform and reciprocal transformations for finite dimensional integrable systems we present a multiparameter generalization of the stackel transform the latter is also known as the coupling constant metamorphosis and show that under certain conditions this generalized stackel transform preserves liouville integrability noncommutative integrability and superintegrability the corresponding transformation for the equations of motion proves to be nothing but a reciprocal transformation of a special form and we investigate the properties of this reciprocal transformation finally we show that the hamiltonians of the systems possessing separation curves of apparently very different form can be related through a suitably chosen generalized stackel transform

persistence and stability of discrete vortices in nonlinear schrodinger lattices abstract we study discrete vortices in the two dimensional nonlinear schrodinger lattice with small coupling between lattice nodes the discrete vortices in the anti continuum limit of zero coupling represent a finite set of excited nodes on a closed discrete contour with a non zero charge using the lyapunov schmidt reductions we analyze continuation and termination of the discrete vortices for small coupling between lattice nodes an example of a square discrete contour is considered that includes the vortex cell also known as the off site vortex we classify families of symmetric and asymmetric discrete vortices that bifurcate from the anti continuum limit we predict analytically and confirm numerically the number of unstable eigenvalues associated with each family of such discrete vortices