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

a new extended tanh function method and its application to the dispersive long wave equations in 2 1 dimensions in this paper a new extended tanh function method is suggested to constructing multiple exact solutions for nonlinear partial differential equations the validity and reliability of the method is tested by its application to the dispersive long wave equations in 2 1 dimensions more new explicit exact solutions are found

algorithm and mechanization to cauchy problem of parabolic equation the present paper discusses the algorithm and mechanization to the cauchy problem of parabolic equation the author uses the initial data to construct the solutions that satisfy the initial condition and obtain a brief algorithm to this problem thereafter the algorithm is used in maple to solve this cauchy problem automatically

traveling wave solutions for some factorized nonlinear pdes in this work some new special traveling wave solutions of the convective fisher equation the time delayed burgers fisher equation the burgers fisher equation and a nonlinear dispersive dissipative equation kakutani and kawahara 1970 j phys soc japan 29 1068 are obtained through the factorization technique all of them share the same type of factorization scheme which reduces the original equation to a riccati equation of the same kind whose general solution is given in terms of bessel and neumann functions in addition some novel particular solutions of the nonlinear dispersive dissipative equation are provided

challenges in climate science and contemporary applied mathematics this article discusses the challenges in climate science from the emerging viewpoint of stochastic statistical properties of turbulent dynamical systems the mathematical topics discussed here include empirical information theory fluctuation dissipation theorems reduced order stochastic modeling and the development of mathematically unambiguous exactly solvable test models for climate science that capture crucial features of vastly more complex scientific problems the applied mathematics topics include the emerging development of multiscale algorithms for filtering data assimilation and superparametrization for climate science and other problems in science and engineering as well as suitable unambiguous mathematical test problems for their behavior interesting contemporary research directions and specific open problems are mentioned throughout the article the perspective here should also be useful for applications to other complex dynamical systems arising in neural science material science and environmental mechanical engineering 2011 wiley periodicals inc

new multiple soliton like solutions to the generalized 2 1 dimensional kp equation a new generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations pdes in a unified way the main idea of this method is to take full advantage of the riccati equation which has more new solutions more new multiple soliton like solutions are obtained for the the generalized 2 1 dimensional kp equation

painleve analysis of the non linear schrodinger family of equations the authors apply the painleve test to the generalised derivative non linear schrodinger equation iut uxx iauu ux ibu2ux cu3u 2 where u denotes the complex conjugate of u and a b and c are real constants to determine under what conditions the equation might be completely integrable it is shown that apart from a trivial multiplicative factor this equation possesses the painleve property for partial differential equations as formulated by weiss tabor and carnevale 1983 only if c 1 4b 2b a when then this relation holds this is equivalent under a gauge transformation to the derivative non linear schrodinger equation dnls of kaup and newell which is known to be completely integrable or else to a linear equation

generalized riccati equation expansion method and its application to the 3 1 dimensional jumbo miwa equation based on the computerized symbolic system maple and a riccati equation a generalized riccati equation expansion method for constructing soliton like solutions of non linear evolution equations nees is presented by a new general ansatz the proposed method is more powerful than most of the existing tanh methods the extended tanh function method the modified extended tanh function method and generalized hyperbolic function method by use of the method we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some nees making use of the method we study the the 3 1 dimensional jumbo miwa equation and obtain rich new families of the exact solutions including the non travelling wave and coefficient functions soliton like solutions singular soliton like solutions periodic form solutions

the prolongation structures of quasi polynomial flows we look closely at the process of finding a wahlquist estabrook prolongation structure for a given system of nonlinear evolution equation s there are two main steps in this calculation the first to reduce the problem to the investigation of a finitely generated free lie algebra with constraints the second to find a finite dimensional linear representation of these generators we discuss some of the difficulties that arise in this calculation for quasi polynomial flows defined later we give an algorithm for the first step we do not totally solve the problems of the second step but do give an algebraic framework and a number of techniques that are quite generally applicable we illustrate these methods with many examples several of which are new

dynamically orthogonal field equations for continuous stochastic dynamical systems abstract in this work we derive an exact closed set of evolution equations for general continuous stochastic fields described by a stochastic partial differential equation spde by hypothesizing a decomposition of the solution field into a mean and stochastic dynamical component we derive a system of field equations consisting of a partial differential equation pde for the mean field a family of pdes for the orthonormal basis that describe the stochastic subspace where the stochasticity lives as well as a system of stochastic differential equations that defines how the stochasticity evolves in the time varying stochastic subspace these new evolution equations are derived directly from the original spde using nothing more than a dynamically orthogonal condition on the representation of the solution if additional restrictions are assumed on the form of the representation we recover both the proper orthogonal decomposition equations and the generalized polynomial chaos equations we apply this novel methodology to two cases of two dimensional viscous fluid flows described by the navier stokes equations and we compare our results with monte carlo simulations

painleve analysis and integrability properties of a 2 1 nonrelativistic field theory a model for planar phenomena introduced by jackiw and pi and described by a lagrangian including a chern simons term is considered the associated equations of motion among which there is a 2 1 gauged nonlinear schrodinger equation are rewritten into a gauge independent form involving the modulus of the matter field application of a painleve analysis as adapted to partial differential equations by weiss tabor and carnevale shows up resonance values that are all integer however compatibility conditions need be considered which cannot be satisfied consistently in general such a result suggests that the examined equations are not integrable but provides tools for the investigation of the integrability of different reductions this in particular puts forward the familiar integrable liouville and 1 1 nonlinear schrodinger equations

attractor local dimensionality nonlinear energy transfers and finite time instabilities in unstable dynamical systems with applications to two dimensional fluid flows we examine the geometry of the finite dimensional attractor associated with fluid flows described by navier stokes equations and relate its nonlinear dimensionality to energy exchanges between dynamical components modes of the flow specifically we use a stochastic framework based on the dynamically orthogonal equations to perform efficient order reduction and describe the stochastic attractor in the reduced order phase space in terms of the associated probability measure we introduce the notion of local fractal dimensionality to describe the geometry of the attractor and we establish a connection with the number of positive finite time lyapunov exponents subsequently we illustrate in specific fluid flows that the low dimensionality of the stochastic attractor is caused by the synergistic activity of linearly unstable and stable modes as well as the action of the quadratic terms in particular we illustrate the connection of the low dimensionality of the attractor with the circulation of energy i from the mean flow to the unstable modes due to their linearly unstable character ii from the unstable modes to the stable ones due to a nonlinear energy transfer mechanism and iii from the stable modes back to the mean due to the linearly stable character of these modes

an approximate sine gordon equation and its traveling wave solution in n 1 dimensional space in this paper we apply the painleve analysis to the study of an approximate sine gordon equation and obtain the traveling solitary wave solution in n 1 dimensional space explicitly which is significantly different from one which is found in our last work the technique described herein can also be applied to other nonlinear evolution equations such as the kdv type equation and the nonlinear schrodinger equation in higher dimensional space

blended reduced subspace algorithms for uncertainty quantification of quadratic systems with a stable mean state abstract order reduction schemes have been used successfully for the analysis and simplification of high dimensional systems exhibiting low dimensional dynamics in this work we first focus on presenting generic limitations of order reduction techniques in systems with stable mean state that exhibit irreducible high dimensional features such as non normal dynamics wide energy spectra or strong energy cascades between modes the reduced order framework that we consider to illustrate these limitations is the dynamically orthogonal do field equations this framework is applied to a series of examples with stable mean state including a linear non normal system and a nonlinear triad system in various dynamical configurations after illustrating the weaknesses and generic limitations of order reduction we develop a novel two way coupled blended approach based on the quasilinear gaussian qg closure and the do field equations the new method qg do overcomes the limitations of its two ingredients and achieves exceptional performance in the examples described previously as well as in other configurations with strongly transient character without using any tuned or adjustable parameters

explicit exact solutions for new general two dimensional kdv type and two dimensional kdv burgers type equations with nonlinear terms of any order in this paper we have improved the tanh method by means of a proper transformation and a general ansatz applying the improved method and direct assumption method with symbolic computation we consider two kinds of equations general two dimensional kdv type equations with nonlinear terms

exact solutions to the two dimensional korteweg de vries burgers equation two methods are described for obtaining exact travelling monotone shock wave solutions to the two dimensional korteweg de vries burgers equation they are compared with two methods that have been reported recently by other authors

travelling wave solutions and proper solutions to the two dimensional burgers korteweg de vries equation in this paper we study the two dimensional burgers korteweg de vries 2d bkdv equation by analysing an equivalent two dimensional autonomous system which indicates that under some particular conditions the 2d bkdv equation has a unique bounded travelling wave solution then by using a direct method a travelling solitary wave solution to the 2d bkdv equation is expressed explicitly which appears to be more efficient than the existing methods proposed in the literature at the end of the paper the asymptotic behaviour of the proper solutions of the 2d bkdv equation is established by applying the qualitative theory of differential equations

linearizability for third order evolution equations the problem of linearization for third order evolution equations is considered criteria for testing equations for linearity are presented a class of linearizable equations depending on arbitrary functions is obtained by requiring presence of an infinite dimensional symmetry group linearizing transformations for this class are found using symmetry structure and local conservation laws a number of special cases as examples are discussed their transformation to equations within the same class by differential substitutions and connection with kdv and mkdv equations is also reviewed in this framework

travelling wave solutions of two dimensional korteweg de vries burgers and kadomtsev petviashvili equations the travelling wave solutions of the two dimensional korteweg de vries burgers and kadomtsev petviashvili equations are studied from two complementary points of view the first one is an adaptation of the factorization technique that provides particular as well as general solutions the second one applies the painleve analysis to both equations throwing light on some aspects of the first method and giving an explanation to some restriction on the coefficients as well as the relation between factorizations and integrals of motion

solution of nonlinear volterra fredholm hammerstein integral equations via rationalized haar functions abstract rationalized haar functions are developed to approximate of the nonlinear volterra fredholm hammerstein integral equations the properties of rationalized haar functions are first presented and the operational matrix of integration together with the product operational matrix are utilized to reduce the computation of integral equations into some algebraic equations the method is computationally attractive and applications are demonstrated through illustrative examples

models for stochastic climate prediction there has been a recent burst of activity in the atmosphere ocean sciences community in utilizing stable linear langevin stochastic models for the unresolved degree of freedom in stochastic climate prediction here several idealized models for stochastic climate modeling are introduced and analyzed through unambiguous mathematical theory this analysis demonstrates the potential need for more sophisticated models beyond stable linear langevin equations the new phenomena include the emergence of both unstable linear langevin stochastic models for the climate mean and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances the strategy for stochastic climate modeling that emerges from this analysis is illustrated on an idealized example involving truncated barotropic flow on a beta plane with topography and a mean flow in this example the effect of the original 57 degrees of freedom is well represented by a theoretically predicted stochastic model with only 3 degrees of freedom

algorithm and mechanization to cauchy problem of wave equation with linear damping and source terms this paper discusses the cauchy problem of wave equation with linear damping and source terms the author reduces the problem to a parabolic one and uses the corresponding initial value to construct the solutions of the resulting parabolic problem thereafter he obtains a brief algorithmization formula to this problem with mechanization in maple

on the integrability of nonlinear dirac equations the integrability of nonlinear dirac equations is discussed applying recent results in soliton theory using the lie point transformation groups of the nonlinear dirac equations we reduce these partial differential equations to systems of ordinary differential equations and study whether these systems are integrable we also discuss whether lie backlund vector fields exist we conclude that the nonlinear dirac equations are not integrable

a generalized cole hopf transformation for nonlinear odes we introduce a hybrid cole hopf darboux transformation to relate the solutions of nonlinear and linear second order differential equations and derive classification and sufficient condition for this correspondence we explore physical applications of this correspondence to nonlinear oscillations the duffing equation and a nonlinear form of the schr dinger equation for the nonrelativistic hydrogen atom in addition we show that solutions of some nonlinear second order equations are related to the special functions of mathematical physics through this transformation these nonlinear equations can be viewed as the class of special nonlinear equations which correspond to the linear differential equations which define the special functions of mathematical physics

travelling wave solutions to the korteweg de vries burgers equation the existence and certain qualitative properties of travelling wave solutions to the korteweg de vries burgers equation are established the limiting behaviour of these waves when e tends to zero and when tends to zero is examined together with a singular limit wherein both e and tend to zero

symmetries and invariant solutions of the two dimensional variable coefficient burgers equation we discuss symmetries and reductions of the two dimensional burgers equation with variable coefficient we classify one dimensional and two dimensional subalgebras of the burgers symmetry algebra which is infinite dimensional into conjugacy classes under the adjoint action of the symmetry group invariance under one dimensional subalgebras provides reductions to lower dimensional partial differential equations further reductions of these equations to second order ordinary differential equations are obtained through invariance under two dimensional subalgebras the reduced odes are then analysed and shown that they belong to the polynomial class of second order equations which can be linearized only for particular values of parameters figuring in the coefficient

exact solutions to kdv equations with variable coefficients and or nonuniformities abstract a class of kdv equations with variable coefficients and or nonuniformities is discussed we derive easily the constraint conditions which must apply to this class in order to obtain exact solutions by a direct method we obtain solutions in a number of generic as well as specific cases

snapshot based balanced truncation for linear time periodic systems we introduce an algorithm based on a method of snapshots for computing approximate balanced truncations for discrete time linear time periodic systems the algorithm is applicable to stable including neutrally stable systems with very high dimension allowing either the output or the input to be very high dimensional as well this algorithm is illustrated with a numerical example

new travelling wave solutions to the ostrovsky equation in this work new travelling wave solutions to the ostrovsky equation are studied by employing the improved tanh function method with this method the ostrovsky equation is reduced to the nonlinear ordinary differential equation and then the different types of exact solutions are derived based on the solutions of the riccati equation we will compare our solutions with those gained by the other methods

dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty abstract we estimate and study the evolution of the dominant dimensionality of dynamical systems with uncertainty governed by stochastic partial differential equations within the context of dynamically orthogonal do field equations transient nonlinear dynamics irregular data and non stationary statistics are typical in a large range of applications such as oceanic and atmospheric flow estimation to efficiently quantify uncertainties in such systems it is essential to vary the dimensionality of the stochastic subspace with time an objective here is to provide criteria to do so working directly with the original equations of the dynamical system under study and its do representation we first analyze the scaling of the computational cost of these do equations with the stochastic dimensionality and show that unlike many other stochastic methods the do equations do not suffer from the curse of dimensionality subsequently we present the new adaptive criteria for the variation of the stochastic dimensionality based on instantaneous i stability arguments and ii bayesian data updates we then illustrate the capabilities of the derived criteria to resolve the transient dynamics of two 2d stochastic fluid flows specifically a double gyre wind driven circulation and a lid driven cavity flow in a basin in these two applications we focus on the growth of uncertainty due to internal instabilities in deterministic flows we consider a range of flow conditions described by varied reynolds numbers and we study and compare the evolution of the uncertainty estimates under these varied conditions

a new truncation in painleve analysis the author observes that when the expansion variable in painleve analysis satisfies a system of riccati equations truncation at a level higher than constant level is allowed this extends the range of exact solutions to nonlinear partial differential equations that one is able to obtain using truncated painleve expansions

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

an exact solution to two dimensional korteweg de vries burgers equation by applying a special solution of square hopf cole type to an ordinary differential equation the authors propose a bounded travelling wave solution u x y t v xi v kx ly omega t to the two dimensional korteweg de vries burgers equation which is monotonic and possesses an inflection point with respect with xi

a mathematical framework for stochastic climate models there has been a recent burst of activity in the atmosphere ocean sciences community in utilizing stable linear langevin stochastic models for the unresolved degrees of freedom in stochastic climate prediction here a systematic mathematical strategy for stochastic climate modeling is developed and some of the new phenomena in the resulting equations for the climate variables alone are explored the new phenomena include the emergence of both unstable linear langevin stochastic models for the climate mean variables and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances all of these phenomena are derived from a systematic self consistent mathematical framework for eliminating the unresolved stochastic modes that is mathematically rigorous in a suitable asymptotic limit the theory is illustrated for general quadratically nonlinear equations where the explicit nature of the stochastic climate modeling procedure can be elucidated the feasibility of the approach is demonstrated for the truncated equations for barotropic flow with topography explicit concrete examples with the new phenomena are presented for the stochastically forced three mode interaction equations the conjecture of smith and waleffe phys fluids 11 1999 1608 1622 for stochastically forced three wave resonant equations in a suitable regime of damping and forcing is solved as a byproduct of the approach examples of idealized climate models arising from the highly inhomogeneous equilibrium statistical mechanics for geophysical flows are also utilized to demonstrate self consistency of the mathematical approach with the predictions of equilibrium statistical mechanics in particular for these examples the reduced stochastic modeling procedure for the climate variables alone is designed to reproduce both the climate mean and the energy spectrum of the climate variables c 2001 john wiley sons inc

factorization of the classical boussinesq system the scalar equation equivalent to the classical boussinesq or broer kaup system is shown to factorize into a differential operator acting on burgers equation the linearizability of this latter equation provides a very simple explanation for the recently found property of fusion and fission of a class of solitons for this system

darboux transformation and solutions for an equation in 2 1 dimensions painleve analysis and the singular manifold method are the tools used in this paper to perform a complete study of an equation in 2 1 dimensions this procedure has allowed us to obtain the lax pair darboux transformation and functions in such a way that a plethora of different solutions with solitonic behavior can be constructed iteratively

the action angle transformation for soliton equations abstract for third order completely integrable equations in 1 1 dimensions canonical transformations which map in the multi soliton case gradients of action variables to gradients of angle variables are constructed furthermore this action angle map is shown always to be an infinitesimal symmetry group generator of the interacting soliton equation

traveling wave solutions of the bbm like equations in this work we apply the factorization technique to the benjamin bona mahony like equations b m n in order to get traveling wave solutions we will focus on some special cases for which m n and we will obtain these solutions in terms of the special forms of weierstrass functions

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

backlund transformations for fourth painleve hierarchies abstract backlund transformations bts for ordinary differential equations odes and in particular for hierarchies of odes are a topic of great current interest here we give an improved method of constructing bts for hierarchies of odes this approach is then applied to fourth painleve p iv hierarchies recently found by gordoa et al publ res inst math sci kyoto 37 2001 327 347 we show how the known pattern of bts for p iv can be extended to our p iv hierarchies remarkably the bts required to do this are precisely the miura maps of the dispersive water wave hierarchy we also obtain the important result that the fourth painleve equation has only one nontrivial fundamental bt and not two such as is frequently stated

on an algorithmic construction of lump solutions in a 2 1 integrable equation the singular manifold method is used to generate lump solutions of a generalized integrable nonlinear schrodinger equation in 2 1 dimensions we present several essentially different types of lump solutions the connection between this method and the ablowitz villarroel scheme is also analysed

applications of the combined tanh function method with symmetry method to the nonlinear evolution equations based on the symbolic computation we combined the tanh function method with the symmetry method to construct new type of solutions of the nonlinear evolution equations for the first time with the combined method some new types of solutions of the coupled 2 1 dimensional nonlinear system of schrodinger equations were obtained and their properties were studied by analyzing their figures

representation of exact solution for the nonlinear volterra fredholm integral equations abstract in this paper the representation of the exact solution for the nonlinear volterra fredholm integral equations will be obtained in the reproducing kernel space the exact solution is given by the form of series its approximate solution is obtained by truncating the series and a new numerical approximate method is obtained the error of the approximate solution is monotone deceasing in the sense of w 2 1 a b the intrinsic merit of the method given in the paper lies in its speedy convergence in addition it must be pointed out that additional conditions as linearization or unjustified assumptions are not required for nonlinear cases

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

factorization of a class of almost linear second order differential equations a general type of almost linear second order differential equations which are directly related to several interesting physical problems is characterized the solutions of these equations are obtained using the factorization technique and their non autonomous invariants are also found by means of scale transformations

korteweg de vries burgers equation and the painleve property it is shown that the korteweg de vries burgers equation possesses the painleve property conditionally using an algorithmic approach a travelling wave solution is reproduced

numerical solution of nonlinear volterra fredholm hammerstein integral equations via collocation method based on radial basis functions abstract a numerical technique based on the spectral method is presented for the solution of nonlinear volterra fredholm hammerstein integral equations this method is a combination of collocation method and radial basis functions rbfs with the differentiation process drbf using zeros of the shifted legendre polynomial as the collocation points different applications of rbfs are used for this purpose the integral involved in the formulation of the problems are approximated based on legendre gauss lobatto integration rule the results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme

a probabilistic decomposition synthesis method for the quantification of rare events due to internal instabilities we consider the problem of the probabilistic quantification of dynamical systems that have heavy tailed characteristics these heavy tailed features are associated with rare transient responses due to the occurrence of internal instabilities systems with these properties can be found in a variety of areas including mechanics fluids and waves here we develop a computational method a probabilistic decomposition synthesis technique that takes into account the nature of internal instabilities to inexpensively determine the non gaussian probability density function for any arbitrary quantity of interest our approach relies on the decomposition of the statistics into a non extreme core typically gaussian and a heavy tailed component this decomposition is in full correspondence with a partition of the phase space into a stable region where we have no internal instabilities and a region where non linear instabilities lead to rare transitions with high probability we quantify the statistics in the stable region using a gaussian approximation approach while the non gaussian distribution associated with the intermittently unstable regions of phase space is inexpensively computed through order reduction methods that take into account the strongly nonlinear character of the dynamics the probabilistic information in the two domains is analytically synthesized through a total probability argument the proposed approach allows for the accurate quantification of non gaussian tails at more than 10 standard deviations at a fraction of the cost associated with the direct monte carlo simulations we demonstrate the probabilistic decomposition synthesis method for rare events for two dynamical systems exhibiting extreme events a two degree of freedom system of nonlinearly coupled oscillators and in a nonlinear envelope equation characterizing the propagation of unidirectional water waves

he s variational iteration method for solving nonlinear mixed volterra fredholm integral equations in this paper he s variational iteration method is used to give the approximate solution of nonlinear mixed volterra fredholm integral equations the method constructs a convergent sequence of functions which approximates the exact solution with few iterations numerical results and a comparison with the exact solution are given which reveal its efficiency

a reliable treatment for mixed volterra fredholm integral equations the main goal of this paper is to demonstrate the use of the modified decomposition method for mixed nonlinear volterra fredholm integral equations the modified method combined with the noise terms phenomena may provide the exact solution by using two iterations only two numerical illustrations are given to show the pertinent features of the technique

blended particle methods with adaptive subspaces for filtering turbulent dynamical systems abstract it is a major challenge throughout science and engineering to improve uncertain model predictions by utilizing noisy data sets from nature hybrid methods combining the advantages of traditional particle filters and the kalman filter offer a promising direction for filtering or data assimilation in high dimensional turbulent dynamical systems in this paper blended particle filtering methods that exploit the physical structure of turbulent dynamical systems are developed non gaussian features of the dynamical system are captured adaptively in an evolving in time low dimensional subspace through particle methods while at the same time statistics in the remaining portion of the phase space are amended by conditional gaussian mixtures interacting with the particles the importance of both using the adaptively evolving subspace and introducing conditional gaussian statistics in the orthogonal part is illustrated here by simple examples for practical implementation of the algorithms finding the most probable distributions that characterize the statistics in the phase space as well as effective resampling strategies is discussed to handle realizability and stability issues to test the performance of the blended algorithms the forty dimensional lorenz 96 system is utilized with a five dimensional subspace to run particles the filters are tested extensively in various turbulent regimes with distinct statistics and with changing observation time frequency and both dense and sparse spatial observations in real applications perfect dynamical models are always inaccessible considering the complexities in both modeling and computation of high dimensional turbulent system the effects of model errors from imperfect modeling of the systems are also checked for these methods the blended methods show uniformly high skill in both capturing non gaussian statistics and achieving accurate filtering results in various dynamical regimes with and without model errors

taylor polynomial solutions of nonlinear volterra fredholm integral equations the kanwal and liu method for the solution of fredholm integral equations is applied to certain nonlinear volterra fredholm integral equations in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed

an application of the decompositionmethod for the two dimensional kdv burgers equation a relatively new decomposition method is presented to find the explicit and numerical solutions of the two dimensional korteweg de vries burgers equation tdkdvb for short for the initial conditions some numerical results are also given for this equation by using the decomposition method

integrable nonlocal hirota equations we construct several new integrable systems corresponding to nonlocal versions of the hirota equation which is a particular example of higher order nonlinear schrodinger equations the integrability of the new models is established by providing their explicit forms of lax pairs or zero curvature conditions the two compatibility equations arising in this construction are found to be related to each other either by a parity transformation p by a time reversal t or a p t transformation possibly combined with a conjugation we construct explicit multisoliton solutions for these models by employing hirota s direct method as well as darboux crum transformations the nonlocal nature of these models allows for a modification of these solution procedures as the new systems also possess new types of solutions with different parameter dependence and different qualitative behavior the multisoliton solutions are of varied type being for instance nonlocal in space nonlocal in time of time crystal type regular with local structures either in time space or of rogue wave type we construct several new integrable systems corresponding to nonlocal versions of the hirota equation which is a particular example of higher order nonlinear schrodinger equations the integrability of the new models is established by providing their explicit forms of lax pairs or zero curvature conditions the two compatibility equations arising in this construction are found to be related to each other either by a parity transformation p by a time reversal t or a p t transformation possibly combined with a conjugation we construct explicit multisoliton solutions for these models by employing hirota s direct method as well as darboux crum transformations the nonlocal nature of these models allows for a modification of these solution procedures as the new systems also possess new types of solutions with different parameter dependence and different qualitative behavior the multisoliton solutions are of varied type being for instance nonlocal in space nonlocal in time of time crystal type regular

travelling wave solutions to the two dimensional kdv burgers equation travelling wave solutions for the two dimensional kdv burgers equation are obtained by using mathematica they are shown to be sums of a shock wave and solitary wave

new exact solutions for solving the initial value problem of the kdv kp equation via the lie group method the traveling wave solutions of the two dimensional kdv kp equation are studied via the lie group method the kdv kp equation is a nonlinear partial differential equation in two spatial and one temporal coordinate which describes the evolution of nonlinear long waves of small amplitude with slow dependence on the transverse coordinate the investigation of exact solutions of the kdv kp via the application of lie group and the extended f expansion methods plays the main role in understanding the nonlinear physical phenomena

algorithmization formula and mechanization to cauchy problem of high dimensional wave equation this paper discusses the constructive method and mechanization to the cauchy problem of high dimensional wave equations the author uses the initial value to construct the solutions that satisfy the initial conditions and obtain a brief algorithmization formula to this problem thereafter the algorithmization formula is used in maple to solve this cauchy problem automatically

explicit traveling wave solutions of five kinds of nonlinear evolution equations abstract first of all by using bernoulli equations we develop some technical lemmas then we establish the explicit traveling wave solutions of five kinds of nonlinear evolution equations nonlinear convection diffusion equations including burgers equations nonlinear dispersive wave equations including korteweg de vries equations nonlinear dissipative dispersive wave equations including ginzburg landau equation korteweg de vries burgers equation and benjamin bona mahony burgers equation nonlinear hyperbolic equations including sine gordon equation and nonlinear reaction diffusion equations including belousov zhabotinskii system of reaction diffusion equations

single and multi solitary wave solutions to a class of nonlinear evolution equations abstract in this paper an effective discrimination algorithm is presented to deal with equations arising from physical problems the aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations many examples are given to illustrate the algorithm at the same time some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations such as camassa holm equation kolmogorov petrovskii piskunov equation and so on then a direct constructive method called multi auxiliary equations expansion method is described to derive the multi solitary wave solutions of nonlinear evolution equations finally a class of novel multi solitary wave solutions of the 2 1 dimensional asymmetric version of the nizhnik novikov veselov equation are given by three direct methods the algorithm proposed in this paper can be steadily applied to some other nonlinear problems

numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows the quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial and boundary conditions such situations are common in multiscale intermittent and non homogeneous fluid and ocean flows the dynamically orthogonal do field equations provide an adaptive methodology to predict the probability density functions of such flows the present work derives efficient computational schemes for the do methodology applied to unsteady stochastic navier stokes and boussinesq equations and illustrates and studies the numerical aspects of these schemes semi implicit projection methods are developed for the mean and for the do modes and time marching schemes of first to fourth order are used for the stochastic coefficients conservative second order finite volumes are employed in physical space with new advection schemes based on total variation diminishing methods other results include i the definition of pseudo stochastic pressures to obtain a number of pressure equations that is linear in the subspace size instead of quadratic ii symmetric advection schemes for the stochastic velocities iii the use of generalized inversion to deal with singular subspace covariances or deterministic modes and iv schemes to maintain orthonormal modes at the numerical level to verify our implementation and study the properties of our schemes and their variations a set of stochastic flow benchmarks are defined including asymmetric dirac and symmetric lock exchange flows lid driven cavity flows and flows past objects in a confined channel different reynolds number and grashof number regimes are employed to illustrate robustness optimal convergence under both time and space refinements is shown as well as the convergence of the probability density functions with the number of stochastic realizations

non isospectral problem in 2 1 dimensions in this paper the singular manifold method allows us to obtain a non isospectral lax pair darboux transformations and miura transformations for an equation in 2 1 dimensions and its modified version in this way we can iteratively build different kinds of solutions with solitonic behaviour

a priori tests of a stochastic mode reduction strategy several a priori tests of a systematic stochastic mode reduction procedure recently devised by the authors proc natl acad sci 96 1999 14687 commun pure appl math 54 2001 891 are developed here in this procedure reduced stochastic equations for a smaller collections of resolved variables are derived systematically for complex nonlinear systems with many degrees of freedom and a large collection of unresolved variables while the above approach is mathematically rigorous in the limit when the ratio of correlation times between the resolved and the unresolved variables is arbitrary small it is shown here on a systematic hierarchy of models that this ratio can be surprisingly big typically the systematic reduced stochastic modeling yields quantitatively realistic dynamics for ratios as large as 1 2 the examples studied here vary from instructive stochastic triad models to prototype complex systems with many degrees of freedom utilizing the truncated burgers hopf equations as a nonlinear heat bath systematic quantitative tests for the stochastic modeling procedure are developed here which involve the stationary distribution and the two time correlations for the second and fourth moments including the resolved variables and the energy in the resolved variables in an important illustrative example presented here the nonlinear original system involves 102 degrees of freedom and the reduced stochastic model predicted by the theory for two resolved variables involves both nonlinear interaction and multiplicative noises even for large value of the correlation time ratio of the order of 1 2 the reduced stochastic model with two degrees of freedom captures the essentially nonlinear and non gaussian statistics of the original nonlinear systems with 102 modes extremely well furthermore it is shown here that the standard regression fitting of the second order correlations alone fails to reproduce the nonlinear stochastic dynamics in this example 2002 elsevier science b v all rights reserved

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

symbolic computation and a uniform direct ansatze method for constructing travelling wave solutions of nonlinear evolution equations on the basis of the computer symbolic system maple and the tanh method the riccati equation method as well as all kinds of improved versions of these methods we present a further uniform direct ansatze method for constructing travelling wave solutions of nonlinear evolution equations compared with the existing methods the presented method can be used to construct more new general solutions and we give some examples to illustrate the key step of our method

menelaus theorem clifford configurations and inversive geometry of the schwarzian kp hierarchy it is shown that the integrable discrete schwarzian kp dskp equation which constitutes an algebraic superposition formula associated with for instance the schwarzian kp hierarchy the classical darboux transformation and quasi conformal mappings encapsulates nothing but a fundamental theorem of ancient greek geometry thus it is demonstrated that the connection with menelaus theorem and more generally clifford configurations renders the dskp equation a natural object of inversive geometry on the plane the geometric and algebraic integrability of dskp lattices and their reductions to lattices of menelaus darboux schwarzian kdv schwarzian boussinesq and schramm type are discussed the dskp and discrete schwarzian boussinesq equations are shown to represent discretizations of families of quasi conformal mappings

painleve analysis lie symmetries and invariant solutions of potential kadomstev petviashvili equation with time dependent coefficients in this paper the painleve analysis of 2 1 dimensional variable coefficients potential kadomstev petviashvili vcpkp equation s x email protected t s email protected x email protected t s x x x email protected t s y y 0 has been performed to study the painleve properties the vcpkp equation is also reduced to a 1 1 dimensional partial differential equation via classical lie group method and reduced equations are further studied to obtain certain general solutions

an efficient algorithm for solving nonlinear volterra fredholm integral equations in this work we develop a new effective method for solving nonlinear volterra fredholm integral equation the existence of any e approximate solution is proved at the same time an effective method for obtaining the e approximate solution is established the final numerical examples illustrate that our approach is valid not only for weakly nonlinear problems but also for strongly nonlinear problems

numerical solutions of the nonlinear volterra fredholm integral equations by using homotopy perturbation method abstract a numerical method for solving the nonlinear volterra fredholm integral equations is presented the method is based upon homotopy perturbation theory the results reveal that the homotopy perturbation method is very effective and simple and gives the exact solution

predictive turbulence modeling by variational closure we show that a variational implementation of probability density function pdf closures has the potential to make predictions of general turbulence mean statistics for which a priori knowledge of the incorrectness is possible this possibility exists because of realizability conditions on effective potential functions for general turbulence statistics these potentials measure the cost for fluctuations to occur away from the ensemble mean value in empirical time averages of the given variable and their existence is a consequence of a refined ergodic hypothesis for the governing dynamical system navier stokes dynamics approximations of the effective potentials can be calculated within pdf closures by an efficient rayleigh ritz algorithm the failure of realizability within a closure for the approximate potential of any chosen statistic implies a priori that the closure prediction for that statistic is not converged the systematic use of these novel realizability conditions within pdf closures is shown in a simple 3 mode system of lorenz to result in a statistically improved predictive ability in certain cases the variational method allows an a priori optimum choice of free parameters in the closure to be made

traveling waves to a burgers korteweg de vries type equation with higher order nonlinearities abstract in this paper first we survey some recent advances in the study of traveling wave solutions to the burgers korteweg de vries equation and some comments are given then we study a burgers korteweg de vries type equation with higher order nonlinearities a qualitative analysis to a two dimensional autonomous system which is equivalent to the burgers kdv type equation is presented and indicates that under certain conditions the burgers korteweg de vries type equation has neither nontrivial bell profile solitary waves nor periodic waves finally a solitary wave solution is obtained by means of the first integral method which is based on the ring theory of commutative algebra

seiberg witten equations in r4 lie symmetries particular solutions integrability it is shown that the 11 parameter automorphism group of lie algebra of four dimensional euclidean group is a maximal lie symmetries group of the seiberg witten equations in r4 particular explicit solutions which are invariant under so 3 subgroups of the maximal lie symmetries group are constructed it is established that seiberg witten equations do not possess the painleve property nevertheless so 3 invariant solutions obtained are turned to admit a characteristic singularity structure

analytic study on two nonlinear evolution equations by using the g g expansion method abstract the validity and reliability of the so called g g expansion method is tested by applying it to two nonlinear evolutionary equations solutions in more general forms are obtained when the parameters are taken as special values it is observed that the previously known solutions can be recovered new rational function solutions are also presented being concise and less restrictive the method can be applied to many nonlinear partial differential equations