the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

the arithmetic of the digital computer a new approach a new approach to the arithmetic of the digital computer is surveyed the methodology for defining and implementing floating point arithmetic is described shortcomings of elementary floating point arithmetic are revealed through sample problems the development of automatic computation with emphasis on the user control of errors is reviewed the limitations of conventional rule of thumb procedures for error control in scientific computation are demonstrated by means of examples computer arithmetic is extended so that the arithmetic operations in the linear spaces and their interval correspondents which are most commonly used in computation can be performed with maximum accuracy on digital computers a new fundamental computer operation the scalar product is introduced to develop this advanced computer arithmetic a process of automatic error control called validation which delivers high accuracy with guarantees for scientific computations is described validation of computations for a large class of nu

painleve expansions for nonintegrable evolution equations abstract the wtc method is applied to a variety of nonintegrable nonlinear evolution equations and shown to provide a systematic procedure for obtaining special solutions some of the special techniques developed for integrable systems are also shown to be applicable to nonintegrable equations

phase field study of the tip operating state of a freely growing dendrite against convection using a novel parallel multigrid approach alloy dendrite growth during solidification with coupled thermal solute convection fields has been studied by phase field modeling and simulation the coupled transport equations were solved using a novel parallel multigrid numerical approach with high computational efficiency that has enabled the investigation of dendrite growth with realistic alloy values of lewis number 10 4 and prandtl number 10 2 the detailed dendrite tip shape and character were compared with widely recognized analytical approaches to show validity and shown to be highly dependent on undercooling solute concentration and lewis number in a relatively low flow velocity regime variations in the ratio of growth selection parameter with and without convection agreed well with theory

topology and bifurcations of the invariant level sets of a fokker planck hamiltonian through two coupled anisotropic quartic anharmonic oscillators integrable hamiltonians with velocity dependent potentials including those of fokker planck hamiltonians h px2 py2 kxpx kypy are constructed from integrable hamiltonians of type h px2 py2 v x y in order to carry out the analytical investigations we convert the problem into that of two coupled anisotropic quartic anharmonic oscillators using certain canonical transformations afterwards we give a complete description of the real phase space topology of the system we give also an explicit periodic solution for singular common level sets of the first integrals all generic bifurcations of liouville tori were determined analytically and numerically

backlund transformations for many body systems related to kdv we present backlund transformations bts with parameter for certain classical integrable n body systems namely the many body generalized henon heiles garnier and neumann systems our construction makes use of the fact that all these systems may be obtained as particular reductions stationary or restricted flows of the kdv hierarchy alternatively they may be considered as examples of the reduced sl 2 gaudin magnet the bts provide exact time discretizations of the original continuous systems preserving the lax matrix and hence all integrals of motion and satisfy the spectrality property with respect to the backlund parameter

similarity reductions from extended painleve expansions for nonintegrable evolution equations abstract the use of singular manifold expansions to find exact solutions to nonintegrable evolution equations is extended to include arbitrary resonance coefficients in such a way as to make the resulting infinite series exactly resummable the technique involves the use of a rescaling ansatz analogous to that used to analyze the psi series of nonintegrable ordinary differential equations the result is a similarity reduction of the equation in which the constrained singular manifold plays the role of a similarity variable the method is capable of yielding new solutions corresponding to either classical or nonclassical conditional lie symmetries

the weiss tabor carnevale painleve test and burgers hierarchy the weiss tabor carnevale wtc painleve test and its recent perturbative extension provide necessary conditions for a partial differential equation to have the painleve property it follows that burgers hierarchy must pass the wtc painleve test the aim here is to prove this explicitly in addition the backlund transformation for burgers equation obtained by wtc via truncation is extended to the entire hierarchy the recursion operator is found to be related to a simple first order system

on existence theorems of positive solutions to nonlinear singular differential equations abstract suppose i f t u 0 1 0 0 is continuous and is increasing on u ii there exists a function g 1 0 g b b and g b b 2 is integrable on 1 such that f t bu g b f t u t u 0 1 0 consider the singular boundary value problem u t f t u t 0 0 t 1 u 0 au u 1 0 then a necessary and sufficient condition for eq to have c 0 1 positive solutions is that 0 0 1 s 1 s f s 1 d s a necessary and sufficient condition for eq to have c 1 0 1 positive solutions is that 0 0 1 f s 1 s d s also the uniqueness iterative methods and computational methods of c 1 0 1 positive solutions have been studied our nonlinearity may be singular at t 0 and or t 1

the painleve property and backlund transformations for the sequence of boussinesq equations we investigate the sequence of boussinesq equations by the method of singular manifolds for the boussinesq equation which is known to possess the painleve property a backlund transformation is defined this backlund transformation which is formulated in terms of the schwarzian derivative obtains the system of modified boussinesq equations and the resulting miura type transformation the modified boussinesq equations are found to be invariant under a discrete group of symmetries acting on the dependent variables by linearizing the miura transformation and modified equations the lax pair is readily obtained furthermore by a result of fokas and anderson the recursion operators defining the sequence of higher order boussinesq equations may be constructed from the miura transformation this allows the recursive definition of backlund transformations for this sequence of equations the recursion operator is found to preserve the discrete symmetries of the modified boussinesq equations this leads to the conclusion that the sequences of boussinesq and modified boussinesq equations identically possess the painleve property are meromorphic we also find that by a simple reduction the sequences of caudrey dodd gibbon and kuperschmidt equations are contained within the boussinesq sequence rational solutions are iteratively constructed for the boussinesq equation and a criterion is proposed for the existence of rational solutions of general integrable systems

pdeptest a package for the painleve test of nonlinear partial differential equations the so called wtc kruskal algorithm is presented in order to study the painleve integrability of nonlinear partial differential equations which combines the wtc method and kruskal s simplification method based on the wtc kruskal and wtc kruskal algorithms we give an implementation in maple called pdeptest this package can handle nonlinear partial differential equations which can be transformed to polynomial ones and its effectiveness is illustrated by applying it to a variety of equations some new results are reported

isotropic finite volume discretization finite volume methods traditionally employ dimension by dimension extension of the one dimensional reconstruction and averaging procedures to achieve spatial discretization of the governing partial differential equations on a structured cartesian mesh in multiple dimensions this simple approach based on tensor product stencils introduces an undesirable grid orientation dependence in the computed solution the resulting anisotropic errors lead to a disparity in the calculations that is most prominent between directions parallel and diagonal to the grid lines in this work we develop isotropic finite volume discretization schemes which minimize such grid orientation effects in multidimensional calculations by eliminating the directional bias in the lowest order term in the truncation error explicit isotropic expressions that relate the cell face averaged line and surface integrals of a function and its derivatives to the given cell area and volume averages are derived in two and three dimensions respectively it is found that a family of isotropic approximations with a free parameter can be derived by combining isotropic schemes based on next nearest and next next nearest neighbors in three dimensions use of these isotropic expressions alone in a standard finite volume framework however is found to be insufficient in enforcing rotational invariance when the flux vector is nonlinear and or spatially non uniform the rotationally invariant terms which lead to a loss of isotropy in such cases are explicitly identified and recast in a differential form various forms of flux correction terms which allow for a full recovery of rotational invariance in the lowest order truncation error terms while preserving the formal order of accuracy and discrete conservation of the original finite volume method are developed numerical tests in two and three dimensions attest the superior directional attributes of the proposed isotropic finite volume method prominent anisotropic errors such as spurious asymmetric distortions on a circular reaction diffusion wave that feature in the conventional finite volume implementation are effectively suppressed through isotropic finite volume discretization furthermore for a given spatial resolution a striking improvement in the prediction of kinetic energy decay rate corresponding to a general two dimensional incompressible flow field is observed with the use of an isotropic finite volume method instead of the conventional discretization c 2014 elsevier inc all rights reserved

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

new variable separation approach application to nonlinear diffusion equations the concept of the derivative dependent functional separable solution ddfss as a generalization to the functional separable solution is proposed as an application it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach as a consequence a complete list of canonical forms for such equations which admit the ddfss is obtained and some exact solutions to the resulting equations are described

darboux transformation and hamiltonian structure for the jaulent miodek hierarchy abstract starting from the jaulent miodek spectral problem we derive the associated hierarchy of nonlinear evolution equations in this paper it is shown that this hierarchy is completely integrable in the liouville sense and possesses the hamiltonian structure moreover by virtue of symbolic computation two types of darboux transformations for the whole hierarchy are explicitly constructed which enables us to find the new soliton like shock and anti shock solutions for the jaulent miodek hierarchy figures are presented to discuss the properties of the new soliton like solutions

variable separation solutions to the coupled integrable dispersionless equations abstract in this paper the variable separation solutions of the coupled integrable dispersionless equations are derived these solutions include two arbitrary functions and they are more general than the results presented before furthermore some coherent structures of the physical quantity including decaying bell solitary single dromion soliton type breather peakon type breather and loop type breather are constructed

exact solutions of the multidimensional derivative nonlinear schrodinger equation for many body systems of criticality the authors investigate the problem of strongly interacting many body systems near criticality as recently described in terms of nonlinear dynamics by dixon and tuszynski 1989 they show that in the first order of approximation the equation of motion for the order parameter can be mapped on the derivative nonlinear schrodinger equation and thus can be solved exactly they solve the equations both in the absence and presence of current densities

the sine gordon equations complete and partial integrability the sine gordon equation in one space one time dimension is known to possess the painleve property and to be completely integrable it is shown how the method of singular manifold analysis obtains the backlund transform and the lax pair for this equation a connection with the sequence of higher order kdv equations is found the modified sine gordon equations are defined in terms of the singular manifold these equations are shown to be identically painleve also certain rational solutions are constructed iteratively the double sine gordon equation is shown not to possess the painleve property however if the singular manifold defines an affine minimal surface then the equation has integrable solutions this restriction is termed partial integrability the sine gordon equation in n 1 variables n space 1 time where n is greater than one is shown not to possess the painleve property the condition of partial integrability requires the singular manifold to be an einstein space with null scalar curvature the known integrable solutions satisfy this constraint in a trivial manner finally the coupled kdv or hirota satsuma equations possess the painleve property the associated modified equations are derived and from these the lax pair is found

integrability of kersten krasil shchik coupled kdv mkdv equations singularity analysis and lax pair the integrability of a coupled kdv mkdv system is tested by means of singularity analysis the true lax pair associated with this system is obtained by the use of prolongation technique

symmetry reductions and exact solutions of a class of nonlinear heat equations abstract classical and nonclassical symmetries of the nonlinear heat equation ut uxx f u are considered the method of differential grobner bases is used both to find the conditions on f u under which symmetries other than the trivial spatial and temporal translational symmetries exist and to solve the determining equations for the infinitesimals a catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of the nonlinear heat equation for cubic f u in terms of the roots of f u 0

non classical symmetries and the singular manifold method revisited the connection between the singular manifold method painleve expansions truncated at the constant term and symmetry reductions of two members of a family of cahn hilliard equations is considered the conjecture that similarity information for a nonlinear partial differential equation may always be fully recovered from the singular manifold method is violated for these equations and is thus shown to be invalid in general given that several earlier examples demonstrate the connection between the two techniques in some cases it now becomes necessary to establish when such a relationship exists a question related to a deeper understanding of painleve analysis this issue is also briefly discussed

on the one and two dimensional toda lattices and the painleve property the toda lattice and the two dimensional toda lattice 2 dtl are shown to possess a type of painleve property that is based on the use of separate singular manifolds for each dependent variable the isospectral problem for the 2 dtl found by both mikhailov and by fordy and gibbons can be simply and logically derived from this analysis some remarks are made about the connection between our work and independent work of kametaka and airhault on the relationship between the toda lattice and the second painleve transcendent

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

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

direction aware slope limiter for three dimensional cubic grids with adaptive mesh refinement abstract in the context of finite volume methods for hyperbolic systems of conservation laws slope limiters are an effective way to suppress creation of unphysical local extrema and or oscillations near discontinuities we investigate properties of these limiters as applied to piecewise linear reconstructions of conservative fluid quantities in three dimensional simulations in particular we are interested in linear reconstructions on cartesian adaptively refined meshes where a reconstructed fluid quantity at a face center depends on more than a single gradient component of the quantity we design a new slope limiter which combines the robustness of a minmod limiter with the accuracy of a van leer limiter the limiter is called direction aware limiter dal because the combination is based on a principal flow direction in particular dal is useful in situations where the barth jespersen limiter for general meshes fails to maintain global linear functions such as on cubic computational meshes with stencils including only face neighboring cells we verify the new slope limiter on a suite of standard hydrodynamic test problems on cartesian adaptively refined meshes we demonstrate reduced mesh imprinting for radially symmetric problems such as the sedov blast wave or the noh implosion test cases the results with dal show better preservation of radial symmetry compared to the other standard methods on cartesian meshes

derivation and implicit solution of the harry dym equation and its connections with the korteweg de vries equation the harry dym equation which is related to the classical string problem is derived in three different ways an implicit cusp solitary wave solution is constructed via a simple direct method the existing connections between the harry dym and the korteweg de vries equations are uniformised and simplified and transformations between their respective solutions are carried out explicitly whenever possible physical insights are provided

elliptic solutions and blow up in an integrable henon heiles system we consider an integrable case of the henon heiles system and use an isomorphism with the two gap kdv flow to construct families of real elliptic trajectories which are associated with two gap elliptic solitons of the kdv equation some of these solutions exhibit blow up in finite time

new similarity reductions and painleve analysis for the symmetric regularised long wave and modified benjamin bona mahoney equations the author discusses similarity reductions and painleve analysis for the symmetric regularised long wave and modified benjamin bona mahoney equations 1972 both of which arise in several physical applications including shallow water waves both equations are thought to non integrable i e not solvable by inverse scattering since numerical studies show that the interaction of solitary waves in inelastic in particular we determine some new similarity reductions of the symmetric regularised long wave equation these new similarity reductions are not obtainable using the classical lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method which involves no group theoretical techniques it is shown that every similarity reduction of both the symmetric regularised long wave and modified benjamin bona mahoney equations obtained using the classical lie group method reduces the associated partial differential equation to an ordinary differential equation of painleve type whereas the new similarity solution of the symmetric regularised long wave equation reduces it to an ordinary differential equation which is not of painleve type it is also shown that neither the symmetric regularised long wave equation nor the modified benjamin bona mahoney equation possesses the painleve property for partial differential equations as defined by weiss et al 1983

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

short note isotropic discrete laplacian operators from lattice hydrodynamics we show that discrete schemes developed for lattice hydrodynamics provide an elegant and physically transparent way of deriving laplacians with isotropic discretisation error isotropy is guaranteed whenever the laplacian weights follow from the discrete maxwell boltzmann equilibrium since these are by construction isotropic on the lattice we also point out that stencils using as few as 15 points in three dimensions generate isotropic laplacians these computationally efficient laplacians can be used in cell dynamical and hybrid lattice boltzmann simulations in favor of popular anisotropic laplacians which make use of larger stencils the method can be extended to provide discretisations of higher order and for other differential operators such the gradient divergence and curl

rational solutions for schwarzian integrable hierarchies we give an approach to finding rational solutions of completely intagrable hierarchies which makes use of the relationship between modifications and the schwarzian equations obtained via the singular manifold method this extends the recent work of kudryashov which allowed a simple derivation of the iteration used to construct sequences of such solutions we also give a closed form for the index polynomial of the schwarzian korteweg de vries hierarchy in addition we consider the representation of rational solutions using lower families of the hierarchy we give a simple representation under which the rational solutions remain solutions of every flow of the hierarchy this representation also allows the inclusion of arbitrary data corresponding to negative indices we use our method to derive an alternative form of the backlund transformation for the hierarchy of the second painleve equation as well as new solutions of a hierarchy of breaking soliton equations we also present here for the first time a schwarzian version of this breaking soliton hierarchy

separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source abstract a solution of a partial differential equation with two real variables t and x is functionally separable in these variables if q u x t for some single variable functions q and in this paper the generalized conditional symmetry approach is used to study the separation of variables of quasilinear diffusion equations with nonlinear source we obtain a complete list of canonical forms for such equations which admit the functionally separable solutions as a result we get broad families of exact solutions to some quasilinear diffusion equations with nonlinear source the behavior and blow up properties of some solutions are described

painleve test generalized symmetries backlund transformations and exact solutions to the third order burgers equations in this paper the painleve analysis is performed on the physical form of the third order burgers equation the painleve property and integrability c integrable of the equation is verified then the generalized symmetries of the equation are presented and the generalized symmetries of the other equation are given by the symmetry transformation method the backlund transformations of the equations are constructed based on the symmetries respectively furthermore the exact explicit solutions to the equations are investigated in terms of the symmetries backlund transformations and transformations of the equations

finite difference approximations of first derivatives for three dimensional grid singularities explicit finite difference approximations of first derivatives are developed for two dimensional three way and five way grid singularities the schemes for three way singularities also apply for six way singularities the development objectives are such that the finite difference schemes are non dissipative and compatible with regular difference approximations for convection dominated phenomena special emphasis is given to the directional dependency of numerical phase speeds and stability limits for explicit time integration for that purpose free coefficients are used to minimize the isotropy error of the numerical phase velocity it is shown that numerical phase speeds and stability limits are almost isotropic hence employment of the proposed schemes is nearly independent of the local orientation of the differencing stencil the time step limits are greater than those of regular finite differences indicating that the overall time step of a computation is not reduced by employing the proposed schemes the developed schemes for three way singularities are of second fourth and sixth order of accuracy whereas the five way difference approximations are second and fourth order accurate respectively the spectral characteristics the formal order of accuracy and the stability limits of the proposed schemes imply that they can favorably be used in combination with regular difference approximations of the same order of accuracy on block structured grids

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

the singularity analysis for nearly integrable systems homoclinic intersections and local multivaluedness abstract in this study a new perturbative scheme for nonintegrable ordinary differential equations is proposed these perturbative expansions are based on the singularity analysis of the unperturbed system and is performed in the neighborhood of its singularities under suitable conditions on the homoclinic structure of the unperturbed system the melnikov vector can be computed based on the knowledge of the laurent expansions of the solutions the existence of transverse homoclinic intersections is therefore explicitly related to the existence of critical points for the solutions in the complex plane of the independent variable

a cascaded phase field lattice boltzmann model for the simulation of incompressible immiscible fluids with high density contrast abstract in this work a conservative phase field model for the simulation of immiscible multiphase flows is developed using an incompressible velocity based cascaded lattice boltzmann method clbm extensions are made to the lattice boltzmann lb equations for interface tracking and incompressible hydrodynamics proposed by fakhari et al 1 by performing relaxation operations in central moment space this was motivated by the work of fei et al 2 3 where promising results from such a transformation were observed the relaxation of central moments is defined in a reference frame moving with the fluid while the existing multiple relaxation time 4 5 scheme performs collision in a fixed frame of reference moreover the derivations make use of continuous maxwellian distribution functions as a result the clbm enhances the galilean invariance and stability of the method when high lattice mach numbers are evident the cascaded scheme has been previously used in the literature to simulate multiphase flows based on the pseudo potential model where it allowed for high density and viscosity contrasts to be captured 6 7 here the clbm is implemented within the phase field framework and is verified through the analysis of a layered poiseuille flow the performance of the clbm is then investigated in terms of spurious currents galilean invariance and computational efficiency finally the work of fakhari et al 1 is extended by validating the model s ability to capture the relation between surface tension and the rise velocity of a planar taylor bubble in both stagnant and flowing fluids new counter current results indicate that the rise velocity model of ha ngoc and fabre 8 also applies in this regime

exact solutions for a class of integrable henon heiles type systems we study the exact solutions of a class of integrable henon heiles type systems according to the analysis of bountis et al phys rev a 25 1257 1982 these solutions are expressed in terms of two dimensional kleinian functions special periodic solutions are expressed in terms of the well known weierstrass function we extend some of our results to a generalized henon heiles type system with n 1 degrees of freedom

separation of variables of a generalized porous medium equation with nonlinear source this paper considers a general form of the porous medium equation with nonlinear source term ut d u u n x f u n 1 the functional separation of variables of this equation is studied by using the generalized conditional symmetry approach we obtain a complete list of canonical forms for such equations which admit the functional separable solutions as a consequence some exact solutions to the resulting equations are constructed and their behavior are also investigated 2002 elsevier science usa all rights reserved

precise evaluation of a polynomial at a point given in staggered correction format abstract the problem of the evaluation in floating point arithmetic of a polynomial with floating point coefficients at a point which is a finite sum of floating point numbers is studied the solution is obtained as an infinite convergent series of floating point numbers the algorithm requires a precise scalar product but this can always be implemented by software in a high level language without assembly language routines as we indicate a convergence result is proved under a very weak restriction on the size of the degree of the polynomial in terms of the unit roundoff u roughly speaking the degree should not be larger than the square root of 1 u 2u even in the particular case when the point at which to evaluate the polynomial reduces to one floating point number we find a new simplified algorithm among the whole family that the preceding convergence result allows this problem occurs among others in the convergence of the newton method to some real root of the given polynomial p if we simply use the horner scheme to evaluate the polynomial p in a neighbourhood of the root in some cases the evaluation will contain no correct digits and will prevent us from getting convergence even to machine accuracy the convergence of iterative methods among which the newton method with added perturbations was the central theme of my talk given at the iccam 92 the second part will appear in a forthcoming paper these added perturbations can represent for example forward or backward errors occurring in finite precision computations the problem discussed here appears in validating some hypotheses of these general convergence results see the forthcoming paper

analytic solutions of a microstructure pde and the kdv and kadomtsev petviashvili equations by invariant painleve analysis and generalized hirota techniques abstract truncated painleve expansions invariant painleve analysis and generalized hirota expansions are employed in combination to solve partially reduce to quadrature the integrable kdv and kp equations and a nonintegrable generalized microstructure gms equation although the multisolitons of the kdv and kp equations are very well known the solutions obtained here for all the three nlpdes are novel and non trivial the solutions obtained via invariant painleve analysis are all complicated rational functions with arguments which themselves are confluent hypergeometric kdv or trigonometric gms functions of various distinct non traveling kdv and traveling wave variables in some cases this is slightly reminiscent of doubly periodic elliptic function solutions when nonlinear ode systems are reduced to quadratures the solutions obtained by the use of recently generalized hirota type expansions in the truncated painleve expansions are closer in functional form to conventional hyperbolic secant solutions although with non trivial traveling wave arguments which are distinct for the three nlpdes considered here

symmetry reductions of partial differential equations related to singular manifold expansions it is shown that all the information yielded by the reduction methods of bluman and cole 1974 and clarkson and kruskal 1989 can be obtained using the singular manifold expansion the burgers equation modified korteweg de vries equation caudrey dodd gibbon equation and the fitzhugh nagumo equation are used as illustrative examples several new exact solutions are presented

a necessary and sufficient condition for the existence of positive solutions to singular sublinear three point boundary value problems abstract this paper investigates the existence of positive solutions for second order singular sublinear three point boundary value problems a necessary and sufficient condition for the existence of c 0 1 as well as c 1 0 1 positive solutions is given by constructing lower and upper solutions and with the maximal theorem also the uniqueness iterative methods convergence rate and computational methods of the c 1 0 1 positive solutions are studied our nonlinear term may be singular at t 0 and or t 1

a unified approach to painleve expansions abstract the painleve test for partial differential equations developed by weiss tabor and carnevale wtc is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations a simple modification of the wtc procedure used for partial differential equations enables us to determine the lax pairs hirota equations and auto backlund transformations for ordinary differential equations including a new lax pair for an integrable case of the henon heiles system a detailed study of the kdv hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained the role of the singular branches is examined in detail and important new insights obtained in particular we find that each singular branch is simply a re expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce a parallel analysis is carried out for the akns hierarchy

a bidirectional kaup kupershmidt equation and directionally dependent solitons exact multisoliton solutions are obtained for the bidirectional kaup kupershmidt equation that was reported in a recent paper j math phys 42 2567 2001 this fills the gap that was left in the prior study these solutions are found to possess the same remarkable and unusual property as the solitary wave solution namely their wave profiles are directionally dependent in particular solitons are presented which describe multiple head on collisions between single and double humped solitary waves explicit expressions are given for the first few multisoliton solutions and a procedure is described for constucting the general n soliton by iteration on the solitons of lower order

crum transformation and rational solutions of the non focusing nonlinear schrodinger equation a crum transformation for the linear problem of the non focusing nonlinear schrodinger nls equation is used to construct the sequence of rational solutions given in terms of tau functions expressed as double wronskian determinants the rational solutions of the adler kaup newell segur akns hierarchy have already been obtained in the work of sachs using the equivalence with the classical boussinesq hierarchy however we find that sachs description of the restrictions that must be placed on the akns tau functions in order to obtain nls rational solutions is not entirely correct we briefly comment on the constrained calogero moser systems associated with the nls rational solutions and indicate how the calogero moser equations can be obtained from a trilinear equation for the tau function which arises by reduction of the kadomtsev petviashvili kp hierarchy

lax representation for the systems of s kowalevskaya type we describe a number of dynamical systems that are generalizations of the s kowalevskaya system and admit the lax representation

modified singular manifold expansion application to the boussinesq and mikhailov shabat systems we present a modified treatment of the singular manifold method as an improved variant of the painleve analysis for partial differential equations with two branches in the painleve expansion we illustrate the method by fully applying it to the classical boussinesq system and to the mikhailov shabat system

instability of pole singularities for the chazy equation we prove that the negative resonances of the chazy equation in the sense of painleve analysis can be related directly to its group invariance properties these resonances indicate in this case the instability of pole singularities depending on the value of a parameter in the equation an unstable isolated pole may turn into the familiar natural boundary or split into several isolated singularities in the first case a convergent series representation involving exponentially small corrections can be given this reconciles several earlier approaches to the interpretation of negative resonances on the other hand we also prove that pole singularities with the maximum number of positive resonances are stable the proofs rely on general properties of nonlinear fuchsian equations

level set jet schemes for stiff advection equations the semijet method abstract many interfacial phenomena in physical and biological systems are dominated by high order geometric quantities such as curvature here a semi implicit method is combined with a level set jet scheme to handle stiff nonlinear advection problems the new method offers an improvement over the semi implicit gradient augmented level set method previously introduced by requiring only one smoothing step when updating the level set jet function while still preserving the underlying methods higher accuracy sample results demonstrate that accuracy is not sacrificed while strict time step restrictions can be avoided

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

isospectral hamiltonian flows in finite and infinite dimensions i generalized moser systems and moment maps into loop algebras a moment map jr jia gl r is constructed from the poisson manifold j a of rank r perturbations of a fixed n xn matrix a to the dual r f e positive part of the formal loop algebra i r g r c 1 the adler kostant symes theorem is used to give hamiltonians which generate commutative isospectral flows on g r the pull back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in j a the latter may be identified with flows on finite dimensional coadjoint orbits in g and linearized on the jacobi variety of an invariant spectral curve xr which generically is an r sheeted riemann surface reductions of j a are derived corresponding to subalgebras of gl r c and sl r c determined as the fixed point set of automorphism groupes generated by involutions i e jdl the classical algebras as well as reductions to twisted subalgebras of s r c the theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of pde s

integrability test for spin chains we examine a simple heuristic test of integrability for quantum chains this test is applied to a variety of systems including isotropic spin 1 models with nearest neighbour interaction potts type inhomogeneous spin 1 2 chains and a multiparameter family of spin 1 2 models generalizing the xyz chain with next to nearest neighbour interactions and bond alternation within the latter family we determine all the integrable models with an o 2 symmetry

geodesic flow on so 4 and the intersection of quadrics this note studies the relationship between the complete integrability of geodesic flow on so 4 with a left invariant metric and the geometry of the intersection of four quadrics in p 6

elliptic baker akhiezer functions and an application to an integrable dynamical system an ansatz for elliptic baker akhiezer functions is used to find new solutions for various spectral problems the sturm liouville problem with four and five gap lame potentials and two third order problems one of the third order problems is the classical halphen equation with a spectral parameter the eigenfunctions are used to construct elliptic solutions for integrable cases of a generalized henon heiles system

modified equations rational solutions and the painleve property for the kadomtsev petviashvili and hirota satsuma equations we propose a method for finding the lax pairs and rational solutions of integrable partial differential equations that is when an equation possesses the painleve property a backlund transformation is defined in terms of an expansion about the singular manifold this backlund transformation obtains 1 a type of modified equation that is formulated in terms of schwarzian derivatives and 2 a miura transformation from the modified to the original equation by linearizing the ricati type miura transformation the lax pair is found on the other hand consideration of the distinct backlund transformations of the modified equations provides a method for the iterative construction of rational solutions this also obtains the lax pairs for the modified equations in this paper we apply this method to the kadomtsev petviashvili equation and the hirota satsuma equations

universal dynamics of phase field models for dendritic growth we compare time dependent solutions of different phase field models for dendritic solidification in two dimensions including a thermodynamically consistent model and several ad hoc models the results are identical when the phase field equations are operating in their appropriate sharp interface limit the long time steady state results are all in agreement with solvability theory no computational advantage accrues from using a thermodynamically consistent phase field model

the painleve property transformed noteworthy analytic properties of the dym kruskal equation such as the weak painleve property and the pogrebkov property a second order pole at infinity are proven to be exactly the image of the usual painleve property under the ibragimov transformation the transformed painleve analysis is carried out for two infinite sets of evolution equations similar to the dym kruskal equation only four of the equations pass the test being known and integrable

integrability characteristics of two dimensional generalizations of nls type equations a recent procedure based on truncated painleve expansions is used to derive lax pairs darboux transformations and various soliton solutions for integrable 2 1 generalizations of nls type equations in particular diverse classes of solutions are found analogous to the dromion instanton lump and ring soliton solutions derived recently for 2 1 korteweg de vries type equations the nizhnik novikov veselov equation and the 2 1 broer kaup system

diffuse interface modeling of three phase contact line dynamics on curved boundaries we introduce a simple and efficient lattice boltzmann method for immiscible multiphase flows capable of handling large density and viscosity contrasts the model is based on a diffuse interface phase field approach within this context we propose a new algorithm for specifying the three phase contact angle on curved boundaries within the framework of structured cartesian grids the proposed method has superior computational accuracy compared with the common approach of approximating curved boundaries with stair cases we test the model by applying it to four benchmark problems i wetting and dewetting of a droplet on a flat surface and ii on a cylindrical surface iii multiphase flow past a circular cylinder at an intermediate reynolds number and iv a droplet falling on hydrophilic and superhydrophobic circular cylinders under differing conditions where available our results show good agreement with analytical solutions and or existing experimental data highlighting strengths of this new approach

performance evaluation of the general characteristics based off lattice boltzmann scheme and dugks for low speed continuum flows the general characteristics based off lattice boltzmann scheme proposed by bardow et al 1 hereafter bardow s scheme and the discrete unified gas kinetic scheme dugks 2 are two methods that successfully overcome the time step restriction by the collision time which is commonly seen in many other kinetic schemes in this work we first perform a theoretical analysis of the two schemes in the finite volume framework by comparing their numerical flux evaluations it is found that the effect of collision term is considered in the evaluation of the cell interface distribution function in both schemes which explains why they can overcome the time step restriction and can give accurate results even as the time step is much larger than the collision time the difference between the two schemes lies in the treatment of the integral of the collision term when evaluating the cell interface distribution function in which bardow s scheme uses the rectangular rule while dugks uses the trapezoidal rule the performance of the two schemes i e accuracy stability and efficiency are then compared by simulating several two dimensional flows including the unsteady taylorgreen vortex flow the steady lid driven cavity flow and the laminar boundary layer problem it is observed that dugks can give more accurate results than bardow s scheme with a same mesh size furthermore the numerical stability of bardow s scheme decreases as the courantfriedrichslewy cfl number approaches to 1 while the stability of dugks is not affected by the cfl number apparently as long as cfl 1 it is also observed that dugks is twice as expensive as the bardow s scheme with the same mesh size

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

painleve analysis of new higher dimensional soliton equation in this note we prove that the recently proposed new higher dimensional nonlinear partial differential equation admits the painleve property we briefly discuss the integrability properties of the equation

separation of variables for the constrained flows the separation of variables for some constrained flows of soliton hierarchies are shown the lax matrix and r matrix either constant or dynamical type are used to construct the separation variables and separation equation for the constrained flows

painleve analysis of the two dimensional burgers equation a painleve analysis of the two dimensional burgers equation is carried out and used to obtain a restricted backlund transformation that maps a subclass of the solutions of the 1 2d burgers equation onto a linear heat like equation alternatively the backlund transformation can be expressed as a map onto the derivative of the one dimensional burgers equation in appropriate dependent and independent variables the singularity analysis also yields a further class of solutions obtained by solving a schwarzian differential equation

the two singular manifold method ii classical boussinesq system for pt i see musette et al ibid vol 24 p 3895 1994 the two singular manifold method a generalization of the singular manifold method of weiss 1983 is applied to the classical boussinesq system also known as the broer kaup system from the point of view of its singularity analysis the important feature of this system is the existence of two principal families with opposite principal parts the usual singular manifold method takes into account only one of these families at a time our generalization takes into account both families and in this way we are able to derive the lax pair and darboux transformation and hence the auto backlund transformation for the classical boussinesq system from its painleve analysis

bifurcations of invariant manifolds in the generalized henon heiles system abstract the generalized henon heiles hamiltonian h 1 2 q dot 1 2 q dot 2 2 aq 1 2 bq 2 2 q 1 2 q 2 e 3 q 2 3 in the integrable case e 6 is studied from topological point of view the topology of the real level sets for all generic values of the constants of the motion and all generic bifurcations of the liouville tori and cylinders are described by making use of the reach algebraic structure of the problem

a perturbative painleve approach to nonlinear differential equations abstract we further improve the painleve test so that negative indices resonances can be treated we demand single valuedness not only for any pole like expansion as in the usual painleve test but also for every solution close to it represented as a perturbation series in a small parameter e order zero is the usual test order one already treated in a preliminary paper reduces to a linear fuchs analysis near a regular singularity and allows the introduction of all missing arbitrary coefficients higher orders lead to the analysis of a linear fuchsian type inhomogeneous equation we obtain an infinite sequence of necessary conditions for the absence of movable logarithmic branch points arising at every integer index whether positive or negative and at every order those arising at negative indices including 1 are new while some conditions may not arise before some high perturbation order we present several illustrative examples we discuss the understanding of negative indices and conclude that they are indistinguishable from positive indices just as in the fuchs theory in particular negative indices give rise to doubly infinite laurent series

isotropic finite differences new finite differences called isotropic finite differences where the lowest order error terms are without any directional bias are proposed an accurate simulation of 6 fold symmetric dendritic solidification is presented using a numerical scheme based on these finite differences