fully discrete entropy conservative schemes of arbitrary order we consider weak solutions of hyperbolic or hyperbolic elliptic systems of conservation laws in one space dimension and their approximation by finite difference schemes in conservative form the systems under consideration are endowed with an entropy entropy flux pair we introduce a general approach to construct second and third order accurate fully discrete in both space and time entropy conservative schemes in general these schemes are fully nonlinear implicit but in some important cases can be explicit or linear implicit furthermore semidiscrete entropy conservative schemes of arbitrary order are presented the entropy conservative schemes are used to construct a numerical method for the computation of weak solutions containing nonclassical regularization sensitive shock waves finally specific examples are investigated and tested numerically our approach extends the results and techniques by tadmor in numerical methods for compressible flows finite difference element and volume techniques asme new york 1986 pp 149 158 lefloch and rohde siam j numer anal 37 2000 pp 2023 2060

a roadmap for discretely energy stable schemes for dissipative systems based on a generalized auxiliary variable with guaranteed positivity abstract we present a framework for devising discretely energy stable schemes for general dissipative systems based on a generalized auxiliary variable the auxiliary variable a scalar number can be defined in terms of the energy functional by a general class of functions not limited to the square root function adopted in previous approaches the current method has another remarkable property the computed values for the generalized auxiliary variable are guaranteed to be positive on the discrete level regardless of the time step sizes or the external forces this property of guaranteed positivity is not available in previous approaches a unified procedure for treating the dissipative governing equations and the generalized auxiliary variable on the discrete level has been presented the discrete energy stability of the proposed numerical scheme and the positivity of the computed auxiliary variable have been proved for general dissipative systems the current method termed gpav generalized positive auxiliary variable requires only the solution of linear algebraic equations within a time step with appropriate choice of the operator in the algorithm the resultant linear algebraic systems upon discretization involve only constant and time independent coefficient matrices which only need to be computed once and can be pre computed several specific dissipative systems are studied in relative detail using the gpav framework ample numerical experiments are presented to demonstrate the performance of the method and the robustness of the scheme at large time step sizes

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

squared eigenfunction symmetries for soliton equations part i abstract nonlinear integrable evolution equations in 1 1 dimensions arise from constraints of the 2 1 dimensional hierarchies associated with the kadomtsev petviashvili kp equation the modified kp equation and the dym equation respectively links of backlund type and of reciprocal type are known to exist between the 2 1 dimensional systems the corresponding links between the constrained flows are discussed in a general framework in particular squared eigenfunction symmetries generated by solutions of the associated linear scattering problems are considered the links between the soliton hierarchies are extended to these symmetries

on the symmetric form of systems of conservation laws with entropy abstract this paper reviews the symmetrizability of systems of conservation laws which possess entropy functions symmetric formulations in conservation form for the equations of gas dynamics are presented

from semidiscrete to fully discrete stability of runge kutta schemes by the energy method the integration of semidiscrete approximations for time dependent problems is encountered in a variety of applications the runge kutta rk methods are widely used to integrate the ode systems which arise in this context resulting in large ode systems called methods of lines these methods of lines are governed by possibly ill conditioned systems with a growing dimension consequently the naive spectral stability analysis based on scalar eigenvalues arguments may be misleading instead we present here a stability analysis of rk methods for well posed semidiscrete approximations based on a general energy method we review the stability question for such rk approximations and highlight its intricate dependence on the growing dimension of the problem in particular we prove the strong stability of general fully discrete rk methods governed by coercive approximations we conclude with two nontrivial examples which demonstrate the versatility of our approach in the context of general systems of convection diffusion equations with variable coefficients a straightforward implementation of our results verify the strong stability of rk methods for local finite difference schemes as well as global spectral approximations since our approach is based on the energy method which is carried in the physical space and since it avoids the von neumann analysis which is carried in the dual fourier space we are able to easily adapt additional extensions due to nonperiodic boundary conditions general geometries etc

efficient second order unconditionally stable schemes for a phase field moving contact line model using an invariant energy quadratization approach we consider the numerical approximations for a phase field model consisting of incompressible navier stokes equations with a generalized navier boundary condition and the cahn hilliard equation

explicit solutions for the harry dym equation the well known link between korteweg de vries and hd harry dym equations is improved in this paper by the introduction of a suitable spectral parameter in the corresponding backlund relationship this additional spectral parameter allows the complete transfer of the group theoretical structure between the two equations as an application the authors give new solutions of the hd on finite intervals several plots are included the method of solution is also applied to the kawamoto equation

zero curvature conditions and conformal covariance two dimensional zero curvature conditions with special emphasis on conformal properties are investigated in detail and the appearance of covariant higher order differential operators constructed in terms of a projective connection is elucidated the analysis is based on the kostant decomposition of simple lie algebras in terms of representations with respect to their principal sl 2 subalgebra

backlund transformation of partial differential equations from the painleve gambier classification ii tzitzeica equation from the existing methods of singularity analysis only we derive the two equations which define the backlund transformation of the tzitzeica equation this is achieved by defining a truncation in the spirit of the approach of weiss et al so as to preserve the lorentz invariance of the tzitzeica equation if one assumes a third order scattering problem this truncation admits a unique solution thus leading to a matrix lax pair and a darboux transformation in order to obtain the backlund transformation bt which is the main new result of this paper one represents the lax pair by an equivalent two component riccati pseudopotential this yields two different bts the first one is a bt for the hirota satsuma equation while the second one is a bt for the tzitzeica equation one of the two equations defining the bt is the fifth ordinary differential equation of gambier

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

an efficient runge kutta 4 5 pair abstract a pair of explicit runge kutta formulas of orders 4 and 5 is derived it is significantly more efficient than the fehlberg and dormand prince pairs and by standard measures it is of at least as high quality there are two independent estimates of the local error the local error of the interpolant is to leading order a problem independent function of the local error at the end of the step

method for deriving rational solutions of some nonlinear evolution equations maps for nonlinear evolution equations are discussed an algorithmic method for deriving rational solutions is presented this approach is illustrated by examples which have solutions in the form of two truncated expansions

conservation laws and the numerical solution of odes ii abstract many physical systems described by an initial value problem for a system of ordinary differential equations odes conserve physical quantities such as the net charge or total energy as the system evolves typical codes for the numerical solution of the odes will not conserve these quantities and this can lead to solutions which are not even qualitatively correct one way to impose conservation laws is to perturb the numerical solution at each step of the integration a simple theory is developed in this paper which tells how this should be done so as to guarantee convergence of codes based on one step methods it is also easy to interpret the effect on the accuracy of the perturbations

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

an explicit finite difference scheme with exact conservation properties abstract a finite difference scheme for the numerical study of the korteweg de vries equation is constructed it is explicit and yet conserves exactly the energy of the computed solutions the underlying idea can also be applied to more general equations or systems numerical experiments are included

exponential collocation methods for conservative or dissipative systems abstract in this paper we propose and analyse a novel class of exponential collocation methods for solving conservative or dissipative systems based on exponential integrators and collocation methods it is shown that these novel methods can be of arbitrarily high order and exactly or approximately preserve first integrals or lyapunov functions we also consider order estimates of the new methods furthermore we explore and discuss the application of our methods in important stiff gradient systems and it turns out that our methods are unconditionally energy diminishing and strongly damped even for very stiff gradient systems practical examples of the new methods are derived and the efficiency and superiority are confirmed and demonstrated by three numerical experiments including a nonlinear schrodinger equation as a byproduct of this paper arbitrary order trigonometric rkn collocation methods are also presented and analysed for second order highly oscillatory general systems the paper is accompanied by numerical results that demonstrate the great potential of this work

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

convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints the equations of motion of mechanical multibody systems with algebraic constraints are of index 3 and therefore not directly solvable by standard ode or dae methods reducing the index by different

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

high order implicit time marching methods based on generalized summation by parts operators this article extends the theory of classical finite difference summation by parts fd sbp time marching methods to the generalized summation by parts gsbp framework dual consistent gsbp time marching methods are shown to retain a and l stability as well as superconvergence of integral functionals when integrated with the quadrature associated with the discretization this also implies that the solution approximated at the end of each time step is superconvergent in addition gsbp time marching methods constructed with a diagonal norm are bn stable this article also formalizes the connection between fd sbp gsbp time marching methods and implicit runge kutta methods through this connection the minimum accuracy of the solution approximated at the end of a time step is extended for nonlinear problems it is also exploited to derive conditions under which nonlinearly stable gsbp time marching methods can be constructed the gsbp approach to time marching can simplify the construction of high order f

backlund transformation and the painleve property when a differential equation possesses the painleve property it is possible for specific equations to define a backlund transformation by truncating an expansion about the singular manifold at the constant level term from the backlund transformation it is then possible to derive the lax pair modified equations and miura transformations associated with the completely integrable system under consideration in this paper completely integrable systems are considered for which backlund transformations as defined above may not be directly defined these systems are of two classes the first class consists of equations of toda lattice type e g sine gordon bullough dodd equations we find that these equations can be realized as the minus one equation of sequences of integrable systems although the backlund transformation may or may not exist for the minus one equation it is shown for specific sequences that the backlund transformation does exist for the positive equations of the sequence this in turn allows the derivation of lax pairs and the recursion operation for the entire sequence the second class of equations consists of sequences of harry dym type these equations have branch point singularities and thus do not directly possess the painleve property yet by a process similar to the uniformization of algebraic curves their solutions may be parametrically represented by meromorphic functions for specific systems this is shown to provide a natural extension of the painleve property

explicit methods based on a class of four stage fourth order runge kutta methods for preserving quadratic laws this paper concerns the study of explicit 4 stage fourth order runge kutta methods which preserve quadratic conservation laws when their weights are made solution dependent application to orthogonal and hamiltonian linear systems and numerical tests are also reported

strong stability preserving explicit linear multistep methods with variable step size strong stability preserving ssp methods are designed primarily for time integration of nonlinear hyperbolic pdes for which the permissible ssp step size varies from one step to the next we develop the first ssp linear multistep methods of order two and three with variable step size and prove their optimality stability and convergence the choice of step size for multistep ssp methods is an interesting problem because the allowable step size depends on the ssp coefficient which in turn depends on the chosen step sizes the description of the methods includes an optimal step size strategy we prove sharp upper bounds on the allowable step size for explicit ssp linear multistep methods and show the existence of methods with arbitrarily high order of accuracy the effectiveness of the methods is demonstrated through numerical examples

fully discrete explicit locally entropy stable schemes for the compressible euler and navier stokes equations abstract recently relaxation methods have been developed to guarantee the preservation of a single global functional of the solution of an ordinary differential equation here we generalize this approach to guarantee local entropy inequalities for finitely many convex functionals entropies and apply the resulting methods to the compressible euler and navier stokes equations based on the unstructured h p adaptive ssdc framework of entropy conservative or dissipative semidiscretizations using summation by parts and simultaneous approximation term operators we develop the first discretizations for compressible computational fluid dynamics that are primary conservative locally entropy stable in the fully discrete sense under a usual cfl condition explicit except for the parallelizable solution of a single scalar equation per element and arbitrarily high order accurate in space and time we demonstrate the accuracy and the robustness of the fully discrete explicit locally entropy stable solver for a set of test cases of increasing complexity

general relaxation methods for initial value problems with application to multistep schemes recently an approach known as relaxation has been developed for preserving the correct evolution of a functional in the numerical solution of initial value problems using runge kutta methods we generalize this approach to multistep methods including all general linear methods of order two or higher and many other classes of schemes we prove the existence of a valid relaxation parameter and high order accuracy of the resulting method in the context of general equations including but not limited to conservative or dissipative systems the theory is illustrated with several numerical examples

a method for the integration in time of certain partial differential equations abstract a method for the numerical solution of ordinary differential equations is analyzed that is explicit and yet can conserve the quadratic quantities conserved by the equations the method can be a useful alternative to the usual leapfrog technique in that it does not suffer from the occurrence of blowup phenomena numerical examples concerning the korteweg de vries equation and the nonlinear schrodinger equation are given

linear and unconditionally energy stable schemes for the binary fluid surfactant phase field model abstract in this paper we consider the numerical solution of a binary fluid surfactant phase field model in which the free energy contains a nonlinear coupling entropy a ginzburg landau double well potential and a logarithmic flory huggins potential the resulting system consists of two nonlinearly coupled cahn hilliard type equations we develop a first and a second order time stepping schemes for this system using the invariant energy quadratization approach in particular the system is transformed into an equivalent one by introducing appropriate auxiliary variables and all nonlinear terms are then treated semi explicitly both schemes are linear and lead to symmetric positive definite systems in space at each time step thus they can be efficiently solved we further prove that these schemes are unconditionally energy stable in the discrete sense various 2d and 3d numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes

efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model abstract in this paper we study efficient numerical schemes of the classical phase field elastic bending energy model that has been widely used to describe the shape deformation of biological lipid vesicles in which the free energy of the system consists of an elastic bending energy a surface area constraint and a volume constraint one major challenge in solving such model numerically is how to design appropriate temporal discretizations in order to preserve energy stability with large time step sizes at the semi discrete level we develop a first order and a second order time stepping scheme for this highly nonlinear and stiff parabolic pde system based on the invariant energy quadratization approach in particular the resulted semi discretizations lead to linear systems in space with symmetric positive definite operators at each time step thus can be efficiently solved in addition the proposed schemes are rigorously proved to be unconditionally energy stable various numerical experiments in 2d and 3d are presented to demonstrate the stability and accuracy of the proposed schemes

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

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

relaxation runge kutta methods fully discrete explicit entropy stable schemes for the compressible euler and navier stokes equations the framework of inner product norm preserving relaxation runge kutta methods d i ketcheson siam j numer anal 57 2019 pp 2850 2870 is extended to general convex quantities conservat

analytic bilinear approach to integrable hierarchies ii multicomponent kp and 2d toda lattice hierarchies an analytic bilinear approach for the construction and study of integrable hierarchies is discussed generalized multicomponent kp and 2d toda lattice hierarchies are considered this approach allows us to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations a generalized hierarchy incorporates basic hierarchy modified hierarchy singularity manifold equation hierarchy and corresponding linear problems different levels of generalized hierarchy are connected via invariants of combescure symmetry transformation the resolution of functional equations also leads to the function and addition formulas to it

lie symmetries and painleve test for explicitly space and time dependent nonlinear wave equations the authors investigate the lie symmetry vector fields of the wave equation square operator u f x1 xn u 0 where f is some nonlinear smooth function and n or 2 the painleve test is considered for the construction of explicitly space and time dependent integrable one space dimensional nonlinear wave equations

bootstrap tolerance and confidence limits for two variable reliability using independent and weakly dependent observations abstract tolerance and confidence limits are obtained for two variable reliability p y x using the bootstrap method x and y are two random variables with unspecified distributions both independent and weakly dependent data cases are considered moving block bootstrap mbb method is used for the later case minitab macros are provided for obtaining tolerance and confidence limits simulation studies indicate the accuracy of the proposed tolerance and confidence limits

estimation of reliability for exponential stress strength models with explanatory variables abstract confidence bounds for p y x z w are obtained where x and y are independent exponential variables with explanatory variables z and w respectively the fisher scoring method as an iteration method is used to obtain estimates of regression parameters simulation results are presented to verify the distributional assumptions of estimators

on discretely entropy conservative and entropy stable discontinuous galerkin methods abstract high order methods based on diagonal norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic pdes 1 2 these methods can also be interpreted as nodal discontinuous galerkin methods with diagonal mass matrices 3 4 5 6 in this work we describe how use flux differencing quadrature based projections and sbp like operators to construct discretely entropy conservative schemes for dg methods under more arbitrary choices of volume and surface quadrature rules the resulting methods are semi discretely entropy conservative or entropy stable with respect to the volume quadrature rule used numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible euler equations in one and two dimensions

the bi hamiltonian structure of some nonlinear fifth and seventh order differential equations and recursion formulas for their symmetries and conserved covariants using a bi hamiltonian formulation we give explicit formulas for the conserved quantities and infinitesimal generators of symmetries for some nonlinear fifth and seventh order nonlinear partial differential equations among them the caudrey dodd gibbon sawada kotera equation and the kupershmidt equation we show that the lie algebras of the symmetry groups of these equations are of a very special form among the c vector fields they are generated from two given commuting vector fields by a recursive application of a single operator furthermore for some higher order equations those multisoliton solutions which for t asymptotically decompose into traveling wave solutions are characterized as eigenvector decompositions of certain operators

confidence limits for stress strength models with explanatory variables a lower confidence bound is obtained for pr y x z 1 z 2 where x and y are independent normal variables with explanatory variables z 1 and z 2 respectively for equal residual variances an exact solution is obtained but for the unequal variance case an approximate lower confidence bound is developed examples of the use of these procedures are given

a general framework for deriving integral preserving numerical methods for pdes a general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented in particular linearly implicit methods preserving a time discretized version of the invariant are developed for systems of partial differential equations with polynomial nonlinearities the framework is rather general and allows for an arbitrary number of dependent and independent variables with derivatives of any order it is proved formally that second order convergence is obtained the procedure is applied to a test case and numerical experiments are provided

painleve analysis and new analytic solutions for variable coefficient kadomtsev petviashvili equation with symbolic computation abstract the variable coefficient kadomtsev petviashvili kp equation is hereby under investigation painleve analysis is given out and an auto backlund transformation is presented via the truncated painleve expansion based on the auto backlund transformation new analytic solutions are given including the soliton like and periodic solutions it is also reduced to a 1 1 dimensional partial differential equation via classical lie group method and the painleve i equation by ck direct method

entropy stable discontinuous interfaces coupling for the three dimensional compressible navier stokes equations non linear entropy stability and a summation by parts sbp framework are used to derive entropy stable interior interface coupling for the semi discretized three dimensional 3d compressible navier stokes equations a complete semidiscrete entropy estimate for the interior domain is achieved combining a discontinuous entropy conservative operator of any order 1 2 with an entropy stable coupling condition for the inviscid terms and a local discontinuous galerkin ldg approach with an interior penalty ip procedure for the viscous terms the viscous penalty contributions scale with the inverse of the reynolds number re so that for re their contributions vanish and only the entropy stable inviscid interface penalty term is recovered this paper extends the interface couplings presented 1 2 and provides a simple and automatic way to compute the magnitude of the viscous ip term the approach presented herein is compatible with any diagonal norm summation by parts sbp spatial operator including finite element finite volume finite difference schemes and the class of high order accurate methods which include the large family of discontinuous galerkin discretizations and flux reconstruction schemes this note relies on the formalism introduced in 1 3 and complements the new class of interior entropy stable sbp operators of any order for the 3d compressible navier stokes equations on unstructured grids that was proposed in 1 2 to keep the notation as simple as possible a uniform cartesian grid is considered in the derivation however the extension to generalized curvilinear coordinates and unstructured grids follows immediately if the transformation from computational to physical space preserves the semi discrete geometric conservation 4 the proposed interface coupling technique has been successfully combined with a high order entropy stable discretization for the simulation of two dimensional 2d and 3d viscous subsonic and supersonic flows presented in 1 5 6

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 jackknife and the bootstrap for general stationary observations we extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence we do not attempt a reduction to i i d values the jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of l consecutive data once in the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero under appropriate conditions consistency is obtained if l l n rightarrow infty and l n n rightarrow 0 general statistics are approximated by an arithmetic mean in regular cases this approximation determines the asymptotic behavior bootstrap replicates are constructed by selecting blocks of length l randomly with replacement among the blocks of observations the procedures are illustrated by using the sunspot numbers and some simulated data

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 tzitzeica equation a backlund transformation interpreted as truncated painleve expansion it is shown that the overdetermined system of equations obtained by truncating the painleve expansion for the classical tzitzeica equation admits a solution which defines a backlund transformation

an integrability test for differential difference systems the authors present a novel integrability criterion that is adapted to differential difference systems it is based on the recently introduced notion of singularity confinement and uses elements from the singularity analysis approach which has proved to be a most successful integrability criterion for continuous systems a study of the toda lattice illustrates the applicability of the method

numerical solution of isospectral flows in this paper we are concerned with the problem of solving numerically isospectral flows these flows are characterized by the differential equation l b l l l 0 l 0 where l 0 is a d x d symmetric matrix b l is a skew symmetric matrix function of l and b l is the lie bracket operator we show that standard runge kutta schemes fail in recovering the main qualitative feature of these flows that is isospectrality since they cannot recover arbitrary cubic conservation laws this failure motivates us to introduce an alternative approach and establish a framework for generation of isospectral methods of arbitrarily high order

entropy stable wall boundary conditions for the three dimensional compressible navier stokes equations non linear entropy stability and a summation by parts framework are used to derive entropy stable wall boundary conditions for the three dimensional compressible navier stokes equations a semi discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator the data at the boundary are weakly imposed using a penalty flux approach and a simultaneous approximation term penalty technique although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy the new boundary conditions are compatible with any diagonal norm summation by parts spatial operator including finite element finite difference finite volume discontinuous galerkin and flux reconstruction correction procedure via reconstruction schemes the proposed boundary treatment is tested for three dimensional subsonic and supersonic flows the numerical computations corroborate the non linear stability entropy stability and accuracy of the boundary conditions

the higher times approach to multisoliton solutions of the harry dym equation an n cusp soliton solution of the harry dym equation hde is constructed explicitly the proposed approach is based on the algebraic effectivization of the reciprocal links between the systems consisting of n higher order analogues of the hde and the corresponding analogues of the kdv equation an important role in the discussed scheme belongs to the reciprocal auto backlund transformation for the dym hierarchy

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

on the behavior of solutions of a class of nonlinear partial differential equations the behavior of the steady state or the traveling wave solutions for a class of nonlinear partial differential equations is studied the nonlinearity in these equations is expressed by the presence of the convective term it is shown that the steady state or the traveling wave solution may explode at a finite value of the spatial or the characteristic variable this holds whatever the order of the spatial derivative term in these equations furthermore new special solutions of a set of equations in this class are also found

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

relaxation runge kutta methods conservation and stability for inner product norms we further develop a simple modification of runge kutta methods that guarantees conservation or stability with respect to any inner product norm the modified methods can be explicit and retain th

accuracy and conservation properties in numerical integration the case of the korteweg de vries equation when numerically integrating time dependent differential equations it is often recommended to employ methods that preserve some of the invariant quantities mass energy etc of the problem being considered this recommendation is usually justified on the grounds that conservation of invariant quantities may ensure that the numerical solution possesses some important qualitative features however there are cases where schemes that preserve invariants are also advantageous in that they possess favourable error propagation mechanisms that render them superior from a quantitative point of view in the present paper we consider the korteweg de vries equation as a case study we show rigorously that for soliton problems and at leading order the error of conservative schemes consists of a phase error that grows linearly with time plus a complementary term that is bounded in the h 1 norm uniformly in time for general nonconservative schemes the error involves a linearly growing amplitude error a quadratically growing phase error and a complementary term that grows linearly in the h 1 norm numerical experiments are presented

algebraic structure of a generalized coupled dispersionless system we study a physical model of the o 3 invariant coupled integrable dispersionless equations that describes the dynamic of a focused system within the background of a plane gravitational field the investigation is carried out both numerically and analytically and realized beneath some assumptions superseding the structure constant with the structure function implemented in lie algebra and quasigroup theory respectively the energy density and topological structures such as loop soliton are examined

the weak painleve property as a criterion for the integrability of dynamical systems we investigate the validity of the weak painleve property as an integrability criterion we present an example of a time dependent hamiltonian system which possesses a weak painleve type expansion while presenting a chaotic behavior however this system presents also critical fixed singularities the importance of the latter as far as integrability is concerned is discussed here

preserving energy resp dissipation in numerical pdes using the average vector field method we give a systematic method for discretizing hamiltonian partial differential equations pdes with constant symplectic structure while preserving their energy exactly the same method applied to pdes with constant dissipative structure also preserves the correct monotonic decrease of energy the method is illustrated by many examples in the hamiltonian case these include the sine gordon korteweg de vries nonlinear schrodinger linear time dependent schrodinger and maxwell equations in the dissipative case the examples are the allen cahn cahn hilliard ginzburg landau and heat equations

factorization and painleve analysis of a class of nonlinear third order partial differential equations a class of fully nonlinear third order partial differential equations pdes is considered this class contains several examples which have recently appeared in the literature and for which rather unusual travelling wave solutions have been given these solutions consist essentially of sums of exponentials we trace the occurrence of these exponentials back to the existence of a linear subequation which appears as a factor in the travelling wave reduction in addition we consider the painleve analysis of this set of equations both for the original pde and also for reductions to ordinary differential equations odes no equation in the class considered survives the combination of pde and ode tests also an equation in the class considered which is known to be integrable is shown to possess only the weak painleve property our analysis therefore confirms the limitations of the painleve test as a test for complete integrability when applied to fully nonlinear pdes

soliton structure versus singularity analysis third order completely intergrable nonlinear differential equations in 1 1 dimensions abstract backlund transformations between all known completely integrable third order differential equations in 1 1 dimensions are established and the corresponding transformations formulas for their hereditary operators and hamiltonian formulations are exhibited some of these backlund transformations are not injective therefore additional non commutative symmetry groups are found for some equations these non commutative symmetry groups are classified as having a semisimple part isomorphic to the affine algebra a 1 1 new completely integrable third order integro differential equations some depending explicitly on x are given these new equations give rise to nonin equation connections between the singularity equations from the painleve analysis and the nonlinear equations for interacting solitons are established a common approach to singularity analysis and soliton structure is introduced the painleve analysis is modified in such a sense that it carries over directly and without difficulty to the time evolution of singularity manifolds of equations like the sine gordon and nonlinear schrodinger equation a method to recover the painleve series from its constant level term is exhibit the soliton singularity transform is recognized to be connected to the mobius group this gives rise to a darboux like result for the spectral properties of the recursion operator these connections are used in order to explain why poles of soliton equations move like trajectories of interacting solitons furthermore it is explicitly computed how solitons of singularity equations behave under the effect of this soliton singularity transform this then leads to the result that only for scaling degrees 1 and 2 the usual painleve analysis can be carried out a new invariance principle connected to kernels of differential operators is discovered this new invariance for example connects the explicit solutions of the liouville equation with the miura transform simple methods are exhibited which allow to compute out of n soliton solutions of the kdv bargman potentials explicit solutions of equations like the harry dym equation certain solutions are plotted

the two singular manifold method i modified korteweg de vries and sine gordon equations we address the problem of finding by painleve analysis only the backlund transformation of partial differential equations pdes having two families of movable singularities with opposite principal parts such as the modified korteweg de vries mkdv sine gordon or nonlinear schrodinger equations this first paper gives an almost algorithmic method which extends the singular manifold method of weiss 1986 that is unable to handle these equations first with only one singular manifold at a time we obtain the darboux transformation second we assume that the ratio of two functions defining the singular manifolds satisfies the most general projective riccati system with undetermined coefficients the darboux transformation then generates a very small number of determining equations admitting a unique solution equivalent to the lax pair of the zakharov shabat ablowitz kaup newell segur scheme by the canonical linearization of the riccati system the method is here applied to the mkdv and sine gordon equations

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

highly efficient invariant conserving explicit runge kutta schemes for nonlinear hamiltonian differential equations abstract a unified framework of invariant conserving explicit runge kutta schemes for the nonlinear hamiltonian odes and pdes are proposed by utilizing the invariant energy quadratization technique first the nonlinear hamiltonian differential equation is transformed into an equivalent reformulation which admits a quadratic energy invariant for the nonlinear ode the reformulation is then discretized using a class of relaxation runge kutta schemes for the nonlinear pde the reformulation is first discretized in the space direction by adopting the fourier pseudo spectral discretization which preserves the semi discrete quadratic conservation laws then the semi discrete system is integrated in the time direction using the explicit relaxation runge kutta method the obtained method can conserve different quadratic conservation laws to machine accuracy which improves the numerical stability during long time computations besides the proposed method keeps the same convergence rate of the standard runge kutta scheme when computed with time step size rescaling and gives a rate of convergence reduced by at most one when computed without step size rescaling numerical experiments for several odes and pdes are provided to illustrate the advantages of the proposed algorithms over long time and verify the theoretical analysis

high order linear multistep methods with general monotonicity and boundedness properties we consider linear multistep methods that possess general monotonicity and boundedness properties strict monotonicity in terms of arbitrary starting values for the multistep schemes is only valid for a small class of methods under very stringent step size restrictions this makes them uncompetitive with the strong stability preserving ssp runge kutta methods by relaxing these strict monotonicity requirements a larger class of methods can be considered including many methods of practical interest in this paper we construct linear multistep methods of high order up to six that possess relaxed monotonicity or boundedness properties with optimal step size conditions numerical experiments show that the new schemes perform much better than the classical monotonicity preserving multistep schemes moreover there is a substantial gain in efficiency compared to recently constructed ssp runge kutta ssprk methods

conservation laws and symmetries of generalized sine gordon equations we study some systems of non linear pde s eqs 1 1 below which can be regarded either as generalizations of the sine gordon equation or as two dimensional versions of the toda lattice equations we show that these systems have an infinite number of non trivial conservation laws and an infinite number of symmetries the second result is deduced from the first by a variant of the hamiltonian formalism for evolution equations we also consider some specializations of the systems

the analytic structure of dynamical systems and self similar natural boundaries abstract in this paper we investigate the analytic complex time structure of the movable singularities for several dynamical systems in general it is found that there exists a direct connection between the occurencce of a certain type of multiple valuedness of the singularities and the existence of a class of remarkable self similar natural boundaries for these systems an asymptotic description of the distribution of singularities in the natural boundary is developed this provides a description of the fine scale structure of these natural boundaries that agrees closely with the numerical calculations

strong stability of explicit runge kutta time discretizations motivated by studies on fully discrete numerical schemes for linear hyperbolic conservation laws we present a framework on analyzing the strong stability of explicit runge kutta rk time discret

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

numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method abstract the molecular beam epitaxial model is derived from the variation of a free energy that consists of either a fourth order ginzburg landau double well potential or a nonlinear logarithmic potential in terms of the gradient of a height function one challenge in solving the mbe model numerically is how to develop proper temporal discretization for the nonlinear terms in order to preserve energy stability at the time discrete level in this paper we resolve this issue by developing a first and second order time stepping scheme based on the invariant energy quadratization ieq method the novelty is that all nonlinear terms are treated semi explicitly and the resulted semi discrete equations form a linear system at each time step moreover the linear operator is symmetric positive definite and thus can be solved efficiently we then prove that all proposed schemes are unconditionally energy stable the semi discrete schemes are further discretized in space using finite difference methods and implemented on gpus for high performance computing various 2d and 3d numerical examples are presented to demonstrate stability and accuracy of the proposed schemes

geometric properties of kahan s method we show that kahan s discretization of quadratic vector fields is equivalent to a runge kutta method when the vector field is hamiltonian on either a symplectic vector space or a poisson vector space with constant poisson structure the map determined by this discretization has a conserved modified hamiltonian and an invariant measure a combination previously unknown amongst runge kutta methods applied to nonlinear vector fields this produces large classes of integrable rational mappings in two and three dimensions explaining some of the integrable cases that were previously known

aspects of the painleve property for partial differential equations abstract aspects of the painleve property are reviewed special attention is paid to the connection with hirota s method the formulation of a painleve type property suitable for differential difference equations and certain results concerning the self dual yang mills equations

painleve analysis soliton solutions and lump type solutions of the 3 1 dimensional generalized kp equation abstract the painleve analysis is applied and the multi soliton criterion is presented to test the integrability of the 3 1 dimensional generalized kp equation derived from a hirota bilinear equation it is shown that the considered equation does not pass the well known painleve test and it is only integrable in a conditional sense solitary wave solutions are shown to interact each other like solitons in multiple wave collisions unless some additional conditions are imposed moreover we analyze a class of analytical rational lump type solutions in detail which are generated from positive quadratic polynomial function and rationally localized in many directions in the space based upon the hirota bilinear form

generalised summation by parts operators and variable coefficients abstract high order methods for conservation laws can be highly efficient if their stability is ensured a suitable means mimicking estimates of the continuous level is provided by summation by parts sbp operators and the weak enforcement of boundary conditions recently there has been an increasing interest in generalised sbp operators both in the finite difference and the discontinuous galerkin spectral element framework however if generalised sbp operators are used the treatment of the boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely interpolating the product of two functions will in general result in a value different from the product of the interpolations thus desired properties such as conservation and stability are more difficult to obtain here new formulations are proposed allowing the creation of discretisations using general sbp operators that are both conservative and stable thus several shortcomings that might be attributed to generalised sbp operators are overcome cf nordstrom and ruggiu 2017 38 and manzanero et al 2017 39

periodic solutions of the non integrable convective fluid equation in the present paper we have obtained and analyzed a family of exact periodic solutions of the nonlinear evolution partial differential convecting fluid equation cfe by applying a modification of the bilinear transformation method this modification is used in view of the circumstance that cfe is a non integrable nonlinear equation a detailed consideration has been given to the quite important case of balance between the dispersion and nonlinear effects establishing that this balance changes the structure of the equation itself the exact periodic solutions of cfe have been also found in the important case when cfe is identical with the kuramoto sivashinsky equation

bicomplexes and backlund transformations a bicomplex is a simple mathematical structure in particular associated with completely integrable models the conditions defining a bicomplex are a special form of a parameter dependent zero curvature condition we generalize the concept of a darboux matrix to bicomplexes and use it to derive backlund transformations for several models the method also works for moyal deformed equations with a corresponding deformed bicomplex

estimation of p z y for correlated stochastic time series models let z and y represent two time series that are not necessarily independent and z n l y m k denote their values respectively at future times n l and m k where n l m k autoregressive ar moving average ma and autoregressive moving average arma models are employed both under stationary and non stationary conditions to estimate p z n l

on the preservation of invariants by explicit runge kutta methods a new strategy to preserve invariants in the numerical integration of initial value problems with explicit runge kutta methods is presented it is proved that this technique retains the order of the original method has an easy and cheap implementation and can be used in adaptive runge kutta codes some numerical experiments with the classical code of dormand and prince dopri5 4 based on a pair of embedded methods with orders 5 and 4 are presented to show the behavior of the new method for several problems which possess invariants