inverse problems for multiple invariant curves planar polynomial vector fields which admit invariant algebraic curves darboux integrating factors or darboux first integrals are of special interest in the present paper we solve the inverse problem for invariant algebraic curves with a given multiplicity and for integrating factors under generic assumptions regarding the multiple invariant algebraic curves involved in particular we prove in this generic scenario that the existence of a darboux integrating factor implies darboux integrability furthermore we construct examples where the genericity assumption does not hold and indicate that the situation is different for these

a high order iterative formula for simultaneous determination of zeros of a polynomial abstract we propose a hybrid method to determine all the zeros of a polynomial simultaneously by combining the single root method and the simultaneous one the present method has a high convergence order even for multiple roots by using the pade approximation

geometry of the modulational instability iii homoclinic orbits for the periodic sine gordon equation in this paper the homoclinic geometric structure of the integrable sine gordon equation under periodic boundary conditions is developed specifically focus is given to orbits homoclinic to n tori simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given a labeling is provided which identifies and catalogues all such orbits these orbits are related in a one to one manner to linearized instabilities explicit formulas for all homoclinic orbits are given in terms of backlund transformations

a note on determinantal representation of a schroder konig like simultaneous method for finding polynomial zeros abstract using pade approximation sakurai torii and sugiura derived in the paper sakurai et al 1991 the generalized iterative method of order n 2 for finding all zeros of a polynomial where n is the highest order of a polynomial derivative involved in the presented iterative formula in this note we give the determinantal representation of this method and analyze procedures for its implementation and some computational aspects

darboux polynomials and global phase portraits for the d2 vector field abstract in this study we consider a vector field of r 3 that is equivariant under the d 2 symmetry group which is called the d 2 field in previous studies we construct the complete list of darboux polynomials for this vector field and solve the partial differential equation that defines it we also use these polynomials to comment on its global qualitative behavior this is the first step in the comparison of vector fields based on the module generated by their darboux polynomials

rogue waves dissipation and downshifting abstract we investigate the effects of dissipation on the development of rogue waves and downshifting by adding nonlinear and linear damping terms to the one dimensional dysthe equation significantly rogue waves do not develop after the downshifting becomes permanent thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves using the inverse spectral theory of the nls equation simulations of the damped dysthe equation for sea states characterized by jonswap spectrum consistently show that rogue wave events are well predicted by proximity to homoclinic data as measured by the spectral splitting distance the cut off distance c u t o f f decreases as the strength of the damping increases indicating that for stronger damping the jonswap initial data must be closer to homoclinic data for rogue waves to occur

invariant algebraic curves and conditions for a centre conditions for the existence of a centre in two dimensional systems are considered along the lines of darboux we show how these methods can be used in the search for maximal numbers of bifurcating limit cycles we also extend the method to include more degenerate cases such as are encountered in less generic systems these lead to new classes of integrals in particular the kukles system is considered and new centre conditions for this system are obtained

chaotic and homoclinic behavior for numerical discretizations of the nonlinear schro dinger equation abstract certain conservative discretizations of the nls can produce irregular behavior we consider the diagonal discretization as a conservative perturbation of the integrable discretization and study the homoclinic crossings in its nonlinear spectrum we find that irregularity sets in when two homoclinic structures are present and in this case many and continual homoclinic crossings occur throughout the irregular time series we indicate a melnikov analysis to study the consequences of this homoclinic behavior

darboux integrating factors inverse problems abstract we discuss planar polynomial vector fields with prescribed darboux integrating factors in a nondegenerate affine geometric setting we establish a reduction principle which transfers the problem to polynomial solutions of certain meromorphic linear systems and show that the space of vector fields with a given integrating factor modulo a subspace of explicitly known standard vector fields has finite dimension for several classes of examples we determine this space explicitly

computer methodologies for comparison of computational efficiency of simultaneous methods for finding polynomial zeros abstract the study and analysis of iterative methods for the determination of zeros of algebraic polynomials the subject of this paper is one of the most actual tasks in the theory and practice our main goal is to examine and compare the quality of most frequently used methods for simultaneously finding all zeros of a polynomial by combining numerical experiments theoretical models and empirical cpu time methodology special attention is devoted to empirical average case model assuming that average cpu times are calculated performing the experiment i on several different computers ii running each method through many loops for a fixed example iii testing reasonably large number of polynomials of different structure body and iv the use of almost arbitrary initial approximations the empirical model ranks the tested methods on the base of the consumed cpu times needed for the fulfillment of the required accuracy of approximations under the same initial conditions our aim is to select a group of the most powerful simultaneous methods with globally convergent performance convergence behavior of two most powerful methods is illustrated by trajectories in the complex plane that consist of the point approximations starting with initial points and going towards the endpoints where the iterative process stops due to the termination criterion

a family of simultaneous zero finding methods abstract applying hansen patrick s formula for solving the single equation f z 0 to a suitable function appearing in the classical weierstrass method two one parameter families of interation functions for the simultaneous approximation of all simple and multiple zeros of a polynomial are derived it is shown that all the methods of these families have fourth order of convergence some computational aspects of the proposed methods and numerical examples are given

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

inverse problems for invariant algebraic curves explicit computations given an algebraic curve in the complex affine plane we describe how to determine all planar polynomial vector fields which leave this curve invariant if all finite singular points of the curve are non degenerate we give an explicit expression for these vector fields in the general setting we provide an algorithmic approach and as an alternative we discuss sigma processes

convergence analysis of sakurai torii sugiura iterative method for simultaneous approximation of polynomial zeros abstract in 1991 t sakurai t torii and h sugiura presented a fourth order iterative algorithm for finding all zeros of a polynomial simultaneously in this paper we provide a detailed convergence analysis local and semilocal of this method the new results improve and complement existing results due to petkovic et al 2003 and petkovic 2008 numerical examples are given to show the applicability of our semilocal convergence results

on the 16th hilbert problem for limit cycles on nonsingular algebraic curves abstract we give an upper bound for the maximum number n of algebraic limit cycles that a planar polynomial vector field of degree n can exhibit if the vector field has exactly k nonsingular irreducible invariant algebraic curves additionally we provide sufficient conditions in order that all the algebraic limit cycles are hyperbolic we also provide lower bounds for n

liouvillian first integrals of differential equations liouvillian functions are functions that are built up from rational functions using exponentiation integration and algebraic functions we show that if a system of differential equations has a generic solution that satisfies a liouvillian relation that is there is a liouvillian function of several variables vanishing on the curve defined by this solution then the system has a liouvillian first integral that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set we can refine this result in special cases to show that the first integral must be of a very special form

smoothing schemes for reaction diffusion systems with nonsmooth data cox and matthews s m cox p c matthews exponential time differencing for stiff systems j comput phys 176 2002 430 455 developed a class of exponential time differencing runge kutta schemes etdrk for nonlinear parabolic equations kassam and trefethen a k kassam ll n trefethen fourth order time stepping for stiff pdes siam j sci comput 26 2005 1214 1233 have shown that these schemes can suffer from numerical instability and they proposed a modified form of the fourth order etdrk4 scheme they use complex contour integration to implement these schemes in a way that avoids inaccuracies when inverting matrix polynomials but this approach creates new difficulties in choosing and evaluating the contour for larger problems neither treatment addresses problems with nonsmooth data where spurious oscillations can swamp the numerical approximations if one does not treat the problem carefully such problems with irregular initial data or mismatched initial and boundary conditions are important in various applications including computational chemistry and financial engineering we introduce a new version of the fourth order cox matthews kassam trefethen etdrk4 scheme designed to eliminate the remaining computational difficulties this new scheme utilizes an exponential time differencing runge kutta etdrk scheme using a diagonal pade approximation of matrix exponential functions while to deal with the problem of nonsmooth data we use several steps of an etdrk scheme using a sub diagonal pade formula the new algorithm improves computational efficiency with respect to evaluation of the high degree polynomial functions of matrices having an advantage of splitting the matrix polynomial inversion problem into a sum of linear problems that can be solved in parallel in this approach it is only required that several backward euler linear problems be solved in serial or parallel numerical experiments are described to support the new scheme

mel nikov analysis of numerically induced chaos in the nonlinear schro dinger equation abstract certain hamiltonian discretizations of the periodic focusing nonlinear schrodinger equation nls have been shown to be responsible for the generation of numerical instabilities and chaos in this paper we undertake a dynamical systems type of approach to modeling the observed irregular behavior of a conservative discretization of the nls using heuristic mel nikov methods the existence of a pair of isolated homoclinic orbits is indicated for the perturbed system the structure of the persistent homoclinic orbits that are predicted by the mel nikov theory possesses the same features as the wave form observed numerically in the perturbed system after the onset of chaotic behavior and appears to be the main structurally stable feature of this type of chaos the mel nikov analysis implemented in the pde context appears to provide relevant qualitative information about the behavior of the pde in agreement with the numerical experiments in a neighborhood of the persistent homoclinic orbits the existence of a horseshoe is investigated and related with the onset of chaos in the numerical study

darboux integrability and invariant algebraic curves for planar polynomial systems in this paper we study the normal forms of polynomial systems having a set of given generic invariant algebraic curves

on the poincare problem abstract we derive some restrictions on the possible degrees of algebraic invariant curves and on the possible form of algebraic integrating factors for plane polynomial vector fields whose stationary points at infinity satisfy a certain genericity condition the method is elementary and we show by example that it also yields strong results for certain non generic vector fields

simultaneous factorization of a polynomial by rational approximation in this note we present a numerical method to approximate some relatively prime factors of a polynomial simultaneously our approach gives methods of arbitrary order grau s method carstensen 1992 grau 1971 is obtained as the second order method which is durand kerner s method when we have linear factors for linear factors our approach yields the simultaneous methods introduced in sakurai et al 1991 we prove local convergence and estimate the r order of the total step version as well as the single step version of the methods we derive an algorithm and present numerical examples which confirm the convergence behavior theoretically predicted

an efficient higher order family of root finders a one parameter family of iterative methods for the simultaneous approximation of simple complex zeros of a polynomial based on a cubically convergent hansen patrick s family is studied we show that the convergence of the basic family of the fourth order can be increased to five and six using newton s and halley s corrections respectively since these corrections use the already calculated values the computational efficiency of the accelerated methods is significantly increased further acceleration is achieved by applying the gauss seidel approach single step mode one of the most important problems in solving nonlinear equations the construction of initial conditions which provide both the guaranteed and fast convergence is considered for the proposed accelerated family these conditions are computationally verifiable they depend only on the polynomial coefficients its degree and initial approximations which is of practical importance some modifications of the considered family providing the computation of multiple zeros of polynomials and simple zeros of a wide class of analytic functions are also studied numerical examples demonstrate the convergence properties of the presented family of root finding methods

the convergence of a family of parallelzero finding methods abstract one of the crucial problems in solving polynomial equations is to state some kind of quantitative conditions that provide both safe and fast convergence in the spirit of smale s point estimation theory introduced in 1981 we construct in this paper computationally verifiable initial conditions which guarantee the convergence of the recently proposed one parameter family of methods of the fourth order for the simultaneous finding polynomial zeros these conditions are computationally verifiable since they depend only on initial approximations polynomial coefficients and polynomial degree which is of significant practical importance

on planar polynomial vector fields with elementary first integrals abstract we show that under rather general conditions a polynomial differential system having an elementary first integral already must admit a darboux first integral and we explicitly characterize the vector fields in this class we also investigate some exceptional cases i e equations admitting an elementary first integral but not a darboux first integral in particular we provide a rather detailed discussion of exceptional elementary first integrals built from algebraic functions of prime degree

classical quasi steady state reduction a mathematical characterization abstract we discuss parameter dependent polynomial ordinary differential equations that model chemical reaction networks by classical quasi steady state qss reduction we understand the following familiar heuristically motivated mathematical procedure set the rate of change for certain a priori chosen variables equal to zero and use the resulting algebraic equations to obtain a system of smaller dimension for the remaining variables this procedure will generally be valid only for certain parameter ranges we start by showing that the reduction is accurate if and only if the corresponding parameter is what we call a qss parameter value and that the reduction is approximately accurate if and only if the corresponding parameter is close to a qss parameter value the qss parameter values can be characterized by polynomial equations and inequations hence parameter ranges for which qss reduction is valid are accessible in an algorithmic manner a defining characteristic of a qss parameter value is that the algebraic variety defined by the qss relations is invariant for the differential equation a closer investigation of the associated systems shows the existence of further invariant sets here singular perturbations enter the picture in a natural manner we compare qss reduction and singular perturbation reduction and show that while they do not agree in general they do up to lowest order in a small parameter for a quite large and relevant class of examples this observation in turn allows the computation of qss reductions even in cases where an explicit resolution of the polynomial equations is not possible