matrix kadomtsev petviashvili equation matrix identities and explicit non singular solutions a new version of the backlund darboux transformation for the matrix kadomtsev petviashvili kp equation is used to construct and study explicit multi parameter solutions and wavefunctions in terms of the matrix exponents a class of the self adjoint non singular solutions of kp i is introduced using the controllability notion from the system theory a subclass of the rationally decaying self adjoint non singular solutions is studied in particular several results prove new in the scalar case also

an efficient algorithm for the multivariable adomian polynomials abstract in this article the sum of the series of multivariable adomian polynomials is demonstrated to be identical to a rearrangement of the multivariable taylor expansion of an analytic function of the decomposition series of solutions u 1 u 2 u m about the initial solution components u 1 0 u 2 0 u m 0 of course the multivariable adomian polynomials were developed and are eminently practical for the solution of coupled nonlinear differential equations the index matrices and their simplified forms of the multivariable adomian polynomials are introduced we obtain the recurrence relations for the simplified index matrices which provide a convenient algorithm for rapid generation of the multivariable adomian polynomials another alternative algorithm for term recurrence is established in these algorithms recurrence processes do not require complicated operations such as parametrization expanding and regrouping derivatives etc as practiced in prior art the mathematica program generating the adomian polynomials based on the algorithm in this article is designed

invariance of white noise for kdv on the line we consider the korteweg de vries equation with white noise initial data posed on the whole real line and prove the almost sure existence of solutions moreover we show that the solutions obey the group property and follow a white noise law at all times past or future as an offshoot of our methods we also obtain a new proof of the existence of solutions and the invariance of white noise measure in the torus setting

lasalle type theorem and its applications to infinite horizon optimal control of discrete time nonlinear stochastic systems based on discrete martingale theory the lasalle type theorem for general discrete time stochastic systems is obtained and the almost sure stability is in turn discussed as applications infinite horizon nonlinear optimal regulator is investigated and a dynamical programming equation called the hamilton jacobi bellman equation is also derived for discrete time nonlinear stochastic optimal control

relative index theorems and supersymmetric scattering theory we discuss supersymmetric scattering theory and employ krein s theory of spectral shift functions to investigate supersymmetric scattering systems this is the basis for the derivation of relative index theorems on some classes of open manifolds as an example we discuss the de rham complex for obstacles in n and asymptotically flat manifolds it is shown that the absolute or relative euler characteristic of an obstacle in n may be obtained from scattering data for the laplace operator on forms with absolute or relative boundary conditions respectively in the case of asymptotically flat manifolds we obtain the chern gauss bonnet theorem for thel2 euler characteristic

on the solutions of the time delayed burgers equation the time delayed burgers equation is introduced and the improved tanh function method is used to construct exact multiple soliton and triangular periodic solutions for an understanding of the nature of the exact solutions that contained the time delay parameter we calculated the numerical solutions of this equation by using the adomian decomposition method and the variational iteration method ivm to the boundary value problem

on the toda and kac van moerbeke systems given a solution of the toda lattice we explicitly construct a solution of the kac van moerbeke system related to each other by a miura type transformation as an illustration of our method we derive the n soliton solutions of the kac van moerbeke lattice

a fundamental mean square convergence theorem for sdes with locally lipschitz coefficients and its applications a version of the fundamental mean square convergence theorem is proved for stochastic differential equations sdes in which coefficients are allowed to grow polynomially at infinity and which satisfy a one sided lipschitz condition the theorem is illustrated on a number of particular numerical methods including a special balanced scheme and fully implicit methods the proposed special balanced scheme is explicit and its mean square order of convergence is 1 2 some numerical tests are presented

piecewise homotopy methods for nonlinear ordinary differential equations abstract piecewise homotopy perturbation methods are developed for the solution of nonlinear ordinary differential equations these methods are based on the introduction of an artificial or book keeping parameter and the expansion of the solution in a power series of this parameter and provide analytical solutions in open intervals which are smooth everywhere three piecewise adaptive homotopy perturbation methods based on the use of either a fixed number of approximants and a variable step size a variable number of approximants and a fixed step size or a variable number of approximants and a variable step size are presented and applied to eight nonlinear ordinary differential equations it is shown that piecewise adaptive homotopy perturbation methods predict essentially the same solutions as matlab s variable step variable order solvers and variable order transition matrix techniques provided that five term approximations of the decomposition method are applied to both the displacement and the velocity it is also shown that piecewise homotopy perturbation techniques that use three term approximations to both the displacement and the velocity provide essentially the same results as those obtained with a second order accurate time linearization technique when the same step is employed in both schemes

modified kdv solitons with non zero vacuum parameter obtainable from the zs akns inverse method the authors obtain a new n soliton solution to the modified kdv equation with non zero vacuum parameter b using the inverse scattering method see zakharov and shabat 1972 and ablowitz et al 1973 under a non vanishing condition kawata and inoue 1977 as demonstrations they study the special features of the one soliton and two solitons in detail the amplitude width and magnitude of the velocity of these solitons are all dependent on b they all move in the positive direction such a characteristic is different from that of the kdv solitons

relation between the gelfand levitan procedure and the method of supersymmetric partners it is shown that for one dimensional quantum mechanical systems the gelfand levitan procedure based on an inverse scattering approach is equivalent to two successive applications of the method based on construction of supersymmetric partners

application of a discrete ito formula to determine stability instability of the equilibrium of a scalar linear stochastic difference equation we apply a variant of a discretised ito formula to develop sharp conditions for the global a s asymptotic stability of the equilibrium solution of a particular linear stochastic difference equation the difference equation relies on a parameter h which can be interpreted as the stepsize of an euler maruyama discretisation of a 1 dimensional linear stochastic differential equation which has constant drift and diffusion a natural consequence of using the discretised ito formula is that h must be sufficiently small in order for the stability instability conditions to be valid however the version of the formula developed here enables us to impose a bound on h which can be expressed explicitly in terms of the equation parameters and which is therefore computable

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

an algorithm for solving boundary value problems in this paper a numerical algorithm based on the decomposition technique is presented for solving a class of nonlinear boundary value problems the method is implemented for well known examples including troesch s and bratu s problems which have been extensively studied the scheme is shown to be highly accurate and only a few terms are required to obtain accurate computable solutions

commutation methods applied to the mkdv equation an explicit construction of solutions of the modified korteweg de vries equation given a solution of the ordinary korteweg de vries equation is provided our theory is based on commutation methods i e n 1 supersymmetry underlying miura s transformation that links solutions of the two evolution equations

the damped string problem revisited abstract we revisit the damped string equation on a compact interval with a variety of boundary conditions and derive an infinite sequence of trace formulas associated with it employing methods familiar from supersymmetric quantum mechanics we also derive completeness and riesz basis results with parentheses for the associated root functions under less smoothness assumptions on the coefficients than usual using operator theoretic methods rather than detailed eigenvalue and root function asymptotics only

geometry of kdv 2 three examples let q be a 1 dimensional schroedinger operator with spectrum bounded from infinity by addition i mean a map of the form q q q 2d sup 2 1g e with qe lambdae lambda to the left of spec q and either integral sub infinity sup 0 e sup 2 or integral sub 0 sup infinity e sup 2 finite the additive class of q is obtained by composite addition and a subsequent closure it is a substitute for the kdv invariant manifold even if the individual kdv flows have no existence kdv 1 mckean 1987 suggested that the additive class of q is the same as its unimodular spectral class defined in terms of the 2 x 2 spectral weight df by fixing a the measure class of df and b the value of sqrt det df the present paper verifies this for 1 the scattering case 2 hill s case and 3 when the additive class is finite dimensional neumann case

a note on the adomian decomposition method for the generalized huxley equation in a recent paper ismail et al adomian decomposition method for burgers huxley and burgers fisher equations appl math comput 159 2004 291 301 employed the adomian decomposition method adm to solve in particular a generalized nonlinear huxley equation the purpose of this note is to correct ismail et al s numerical solutions of the generalized huxley equation in addition a convergence proof of the adm applied to the generalized huxley equation is also given

comparison between adomian s method and he s homotopy perturbation method in this paper it is revealed that modified form of he s homotopy perturbation method corresponds to adomian s decomposition method for certain nonlinear problems

a maple package of new adm pade approximate solution for nonlinear problems abstract based on the new definition of four classes of adomian polynomials proposed by rach in 2008 a maple package of new adomian pade approximate solution for solving nonlinear problems is presented this package combines the merits of the adomian decomposition method and the diagonal pade technique and may give more accurate solutions of nonlinear problems with strong nonlinearity besides the package is user friendly and efficient one only needs to input the initial conditions governing equation and four optional parameters then our package will output the analytic approximate solution within a few seconds where the equation is decomposed into three parts they are the linear term r nonlinear term nn and source function g which are all in functional form meanwhile several graphs generated from the above solutions are displayed and demonstrate a favorable comparison in this paper several different types of examples are given to illustrate the validity and promising flexibility of the package this package provides us with a convenient and useful tool for dealing with nonlinear problems as well as its electronic version is free to download via the journal website

mean square and asymptotic stability of the stochastic theta method stability analysis of numerical methods for ordinary differential equations odes is motivated by the question for what choices of stepsize does the numerical method reproduce the characteristics of the test equation we study a linear test equation with a multiplicative noise term and consider mean square and asymptotic stability of a stochastic version of the theta method we extend some mean square stability results in saito and mitsui siam j numer anal 33 1996 pp 2254 2267 in particular we show that an extension of the deterministic a stability property holds we also plot mean square stability regions for the case where the test equation has real parameters for asymptotic stability we show that the issue reduces to finding the expected value of a parametrized random variable we combine analytical and numerical techniques to get insights into the stability properties for a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region this allows us to prove a number of results the technique introduced is widely applicable and we use it to show that a fully implicit method suggested by kloeden and platen numerical solution of stochastic differential equations springer verlag 1992 has an asymptotic stability extension of the deterministic a stability property we also use the approach to explain some numerical results reported in milstein platen and schurz siam j numer anal 35 1998 pp 1010 1019

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

explicit construction of solutions of the modified kadomtsev petviashvili equation abstract given a solution of the kadomtsev petviashvili equation we explicitly construct a solution of the modified kadomtsev petviashvili equation related to one another by a generalized miura transformation the construction is modeled after a previous treatment of the modified korteweg de vries case as an illustration of our method we derive the soliton solutions of the modified kadomtsev petviashvili equation

the zero dispersion limit of the korteweg de vries equation with periodic initial data on etudie le probleme aux valeurs initiales pour l equation de korteweg de vries u t 6uu x e 2 u xxx 0 dans la limite e 0 on calcul la limite faible de la solution pour des donnees initiales periodiques quand e 0

gauge transformation and the higher order korteweg de vries equation to the family of higher order korteweg de vries kdv equations a new family of higher order mkdv equations depending on a parameter is constructed they are connected by an 2 dependent miura transformation a backlund transformation is also established furthermore a new gauge transformation is associated to this backlund transformation it enables one to derive auto backlund transformations for the families of higher order 2 mkdv and kdv equations respectively for illustration four generations of explicit solutions of the second order kdv equation are presented

preserving positivity in solutions of discretised stochastic differential equations we consider the euler discretisation of a scalar linear test equation with positive solutions and show for both strong and weak approximations that the probability of positivity over any finite interval of simulation tends to unity as the step size approaches zero although a s positivity in an approximation is impossible to achieve we develop for the strong maruyama approximation an asymptotic estimate of the number of mesh points required for positivity as our tolerance of non positive trajectories tends to zero and examine the effectiveness of this estimate in the context of practical numerical simulation we show how this analysis generalises to equations with a drift coefficient that may display a high level of nonlinearity but which must be linearly bounded from below i e when acting towards zero and a linearly bounded diffusion coefficient finally in the linear case we develop a refined asymptotic estimate that is more useful as an a priori guide to the number of mesh points required to produce positive approximations with a given probability

constructing solutions of the mkdv equation abstract using commutation methods i e n 1 supersymmetry underlying miura s transformation an explicit construction of solutions of the modified korteweg de vries equation given a solution of the ordinary korteweg de vries equation is provided

completion problems and scattering problems for dirac type differential equations with singularities abstract a completion problem to recover a rational matrix function which is j unitary on the line is treated a dirac type system with singularities on the semi axis is recovered explicitly from its left reflection coefficient the close relation between these two problems is discussed

scattering theory for one dimensional systems with dx v x 0 abstract low energy scattering for schrodinger operators of the type h v in l2 r with rdx v x 0 is considered the possibility of zero energy eigenstates of h is taken into account explicitly in particular a laurent expansion for the transition operator and recursion relations for its coefficients are provided and the leading behaviour of the scattering operator is given in all possible cases

adaptive timestepping for pathwise stability and positivity of strongly discretised nonlinear stochastic differential equations abstract we consider the use of adaptive timestepping to allow a strong explicit euler maruyama discretisation to reproduce dynamical properties of a class of nonlinear stochastic differential equations with a unique equilibrium solution and non negative non globally lipschitz coefficients solutions of such equations may display a tendency towards explosive growth countered by a sufficiently intense and nonlinear diffusion we construct an adaptive timestepping strategy which closely reproduces the almost sure a s asymptotic stability and instability of the equilibrium and which can ensure the positivity of solutions with arbitrarily high probability our analysis adapts the derivation of a discrete form of the ito formula from appleby et al 2009 in order to deal with the lack of independence of the wiener increments introduced by the adaptivity of the mesh we also use results on the convergence of certain martingales and semi martingales which influence the construction of our adaptive timestepping scheme in a way proposed by liu mao 2017

recurrence triangle for adomian polynomials in this paper a recurrence technique for calculating adomian polynomials is proposed the convergence of the series for the adomian polynomials is discussed and the dependence of the convergent domain of the solution s decomposition series n 0 u n on the initial component function u 0 is illustrated by introducing the index vectors of the adomian polynomials the recurrence relations of the index vectors are discovered and the recurrence triangle is given the method simplifies the computation of the adomian polynomials in order to obtain a solution s decomposition series with larger domain of convergence we illustrate by examples that the domain of convergence can be changed by choosing a different u 0 and a modified iteration

trace formulas and borg type theorems for matrix valued jacobi and dirac finite difference operators abstract borg type uniqueness theorems for matrix valued jacobi operators h and supersymmetric dirac difference operators d are proved more precisely assuming reflectionless matrix coefficients a b in the self adjoint jacobi operator h as a s b with s the right left shift operators on the lattice z and the spectrum of h to be a compact interval e e e e we prove that a and b are certain multiples of the identity matrix an analogous result which however displays a certain novel nonuniqueness feature is proved for supersymmetric self adjoint dirac difference operators d with spectrum given by e 1 2 e 1 2 e 1 2 e 1 2 0 e e our approach is based on trace formulas and matrix valued exponential herglotz representation theorems as a by product of our techniques we obtain the extension of flaschka s borg type result for periodic scalar jacobi operators to the class of reflectionless matrix valued jacobi operators

new method for calculating adomian polynomials in this paper we introduce a new method for calculating adomian polynomials implementing the new method is very similar to using derivative operator which is familiar to everyone

almost sure instability of the equilibrium solution of a milstein type stochastic difference equation we derive a condition guaranteeing the almost sure instability of the equilibrium of a stochastic difference equation with a structure motivated by the euler milstein discretisation of an ito stochastic differential equation our analysis relies upon the convergence of non negative martingale sequences coupled with a discrete form of the ito formula and requires a distinct variant of this formula for each of the linear and nonlinear cases the conditions developed in this article appear to be quite sharp

matrix korteweg de vries and modified korteweg de vries hierarchies noncommutative soliton solutions the present work continues work on kdv type hierarchies presented by s carillo and c schiebold noncommutative korteweg de vries and modified korteweg de vries hierarchies via recursion methods j math phys 50 073510 2009 10 1063 1 3155080 general solution formulas for the kdv and mkdv hierarchies are derived by means of banach space techniques both in the scalar and matrix case a detailed analysis is given of solitons breathers their countable superpositions as well as of multisoliton solutions for the matrix hierarchies

almost sure and moment exponential stability in the numerical simulation of stochastic differential equations relatively little is known about the ability of numerical methods for stochastic differential equations sdes to reproduce almost sure and small moment stability here we focus on these stability properties in the limit as the timestep tends to zero our analysis is motivated by an example of an exponentially almost surely stable nonlinear sde for which the euler maruyama em method fails to reproduce this behavior for any nonzero timestep we begin by showing that em correctly reproduces almost sure and small moment exponential stability for sufficiently small timesteps on scalar linear sdes we then generalize our results to multidimensional nonlinear sdes we show that when the sde obeys a linear growth condition em recovers almost surely exponential stability very well under the less restrictive condition that the drift coefficient of the sde obeys a one sided lipschitz condition where em may break down we show that the backward euler method maintains almost surely exponential stability

almost periodicity in time of solutions of the kdv equation we study the cauchy problem for the kdv equation tu 6u xu xu 0 with almost periodic initial data u x 0 v x we consider initial data v for which the associated schrodinger operator is absolutely continuous and has a spectrum that is not too thin in a sense we specify and show the existence uniqueness and almost periodicity in time of solutions this establishes a conjecture of percy deift for this class of initial data the result is shown to apply to all small analytic quasiperiodic initial data with diophantine frequency vector

convergence of adomian s method applied to differential equations this paper deals with a new proof of convergence of adomian s method applied to differential equations we also give new formulae and properties and we suggest a simple computational form for adomian s polynomials

nonuniqueness for solutions of the korteweg de vries equation des variantes de la methode de diffusion inverse donnent des exemples de non unicite pour le probleme de cauchy pour l equation de korteweg de vries

a modified adomian method for system of nonlinear differential equations in this paper a modified adomian method is introduced to solve some systems of nonlinear differential equations the method can be mechanized in maple and a procedure is written to solve the approximate analytic solution of the systems some examples will be presented to illustrate the effectiveness of the method and the procedure a comparison between the program and the original adomian method s program is drawn

decay and regularity in the inverse scattering problem abstract the regularity and decay properties for the potential q x in the schrodinger equation q k 2 on the line are characterized in terms of the decay and regularity of the reflection coefficients r and their fourier transforms

an alternate algorithm for computing adomian polynomials in special cases in this article we introduce an alternate algorithm for computing adomian polynomials and present some examples to show the simplicity and efficiency of the new method

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

large time behavior of the lp norm of schrodinger semigroups abstract letv 0 v c0 rv with v 3 be such that h 1 2 v 0 but for any e 0 1 2 1 e v is not positive we determine the exact rate of divergence of the norm of e th as a map from l to l a number of related problems are discussed

solutions of the korteweg devries equation with steplike initial data no differentiability of u is assumed at all the goal is to find the range of n such that 1 1 and 1 2 has a classical solution it is easy to check that if u x t t 0 solves kdv so does u x t u x 6ht t h where h is in 02 similarly one verifies that if u x t solves kdv for t 0 with initial data u x 0 u x then w x t u x t solves kdv for t 0 and t 0 it is convenient to write b c2 c 0 our existence theorem is based on a construction suggested by the inverse scattering method if t 0 we prove existence of regular solutions for n 2 3 regularity and decay as x cc depend on n by using a new l inverse scattering result 9 we are also able to get control over our solution as x p co at least for initial data satisfying a rather mild 306 0022 0396186 3 00

the index formula and the spectral shift function for relatively trace class perturbations we compute the fredholm index index da of the operator da d dt a on l2 r h associated with the operator path a t t where af t a t f t for a e t r and appropriate f l2 r h via the spectral shift function a a associated with the pair a a of asymptotic operators a a on the separable complex hilbert space h in the case when a t is generally an unbounded relatively trace class perturbation of the unbounded self adjoint operator a we derive a formula an extension of a formula due to pushnitski relating the spectral shift function a a for the pair a a and the corresponding spectral shift function h2 h1 for the pair of operators h2 h1 dada da da in this relative trace class context h2 h1 1 1 2 1 2 a a d 2 1 2for a e 0 this formula is then used to identify the fredholm index of da with 0 a a in addition we prove that index da coincides with the spectral flow spflow a t t of the family a t t r and also relate it to the fredholm perturbation determinant for the pair a a index da spflow a t t 0 a a 1lime 0im ln deth a iei a iei 1 0 h2 h1 with the choice of the branch of ln deth on c such that limim z ln deth a zi a zi 1 0 we also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta invariant

dispersive estimate for the 1d schrodinger equation with a steplike potential abstract we prove a sharp dispersive estimate p a c u t x c t 1 2 u 0 l 1 r for the one dimensional schrodinger equation i u t u x x v x u v 0 x u 0 where 1 x 2 v l 1 r and v 0 is a step function real valued and constant on the positive and negative real axes

the lax representation for an integrable class of relativistic dynamical systems we exhibit the lax pair for the class of relativistic dynamical systems recently introduced by ruijsenaars and schneider whose equations of motion read whereb is an arbitrary constant and the weierstrass elliptic function

non hermitian matrix schrodinger equation backlund darboux transformation weyl functions and script pscript t symmetry the non hermitian matrix schrodinger equation obtained by the backlund darboux transformation bdt is treated the potentials fundamental solutions and weyl functions are constructed explicitly a symmetric reduction of the bdt is introduced and this case is studied in greater detail including potentials fundamental solutions bound states the reality of the discrete spectrum and spontaneous break of the symmetry the sign indefinite scalar product and examples

solitary wave solutions for time fractional third order modified kdv equation using two reliable techniques g g expansion method and improved g g expansion method abstract in the present paper we construct the analytical exact solutions of a nonlinear evolution equation in mathematical physics namely time fractional modified kdv equation by using g g expansion method and improved g g expansion method as a result new types of exact analytical solutions are obtained

reduction technique for matrix nonlinear evolution equations solvable by the spectral transform the main purpose of this paper is to describe a technique of reduction whereby from the class of evolution equations for matrices of order n solvable via the spectral transform associated to the matrix linear schrodinger eigenvalue problem one derives subclasses of nonlinear evolution equations involving less than n2 fields to illustrate the method from the equations for matrices of order 2 two subclasses of equations for 2 fields rather than 4 are obtained the first class coincides or rather includes that solvable via the spectral transform associated to the generalized zakharov shabat spectral problem further reduction to nonlinear evolution equations for a single field reproduces a number of well known equations but also yields a novel one highly nonlinear the second class also yields highly nonlinear equations some examples are given including another novel evolution equation for a single field

gauge theory of ba cklund transformations ii abstract a class of gauge transformations is constructed for hamiltonian hierarchies of completely integrable systems on semi simple lie algebras these transformations create or annihilate poles in the scattering data hence create or annihilate solitons in the potential q in that sense they generalize backlund transformations to which they reduce precisely for the 2 2 akns systems to each backlund gauge transformation is associated an automorphism in the weyl group of the lie algebra two proofs are given to show that these backlund gauge transformations preserve the nonlinear evolution equation one proof uses the transformation of the scattering data under the backlund transformation the second proof is purely gauge theoretic

stabilization and destabilization of nonlinear differential equations by noise this paper considers the stabilization and destabilization by a brownian noise perturbation that preserves the equilibrium of the ordinary differential equation x t f x t in an extension of earlier work we lift the restriction that f obeys a global linear bound and show that when f is locally lipschitz a function g can always be found so that the noise perturbation g x t db t either stabilizes an unstable equilibrium or destabilizes a stable equilibrium when the equilibrium of the deterministic equation is nonhyperbolic we show that a nonhyperbolic perturbation suffices to change the stability properties of the solution

structure of positive solutions to δ v u 0 in rn on etudie la structure de toutes les solutions positives d une equation de schrodinger stationnaire ou le potentiel v est suppose etre une fonction a valeur reelle appartenant a l p loc r n avec p n 2 pour n 2 et p 1 pour n 1 on considere une formule de representation explicite pour des solutions positives leurs asymptotes a l infini et le probleme de dirichlet avec une condition aux limites a l infini

τ functions and zero curvature equations of toda akns type the connection between functions and zero curvature equations for the homogeneous construction of the basic module l 0 over the simplest affine kac moody algebra a 1 1 is studied

a new algorithm for the decomposition solution of nonlinear differential equations in this paper we introduce a new algorithm for applying the adomian decomposition method to nonlinear differential and partial differential equations the proposed algorithm does not require the explicit evaluation of adomian polynomials to approximate the nonlinearity term the numerical experiments show that the proposed algorithm is more accurate at larger values for time though at the cost of more computations

a new algorithm for calculating adomian polynomials for nonlinear operators in this paper a reliable technique for calculating adomian polynomials for nonlinear operators will be developed the new algorithm offers a promising approach for calculating adomian polynomials for all forms of nonlinearity the algorithm will be illustrated by studying suitable forms of nonlinearity a nonlinear evolution model will be investigated

first order strong approximations of scalar sdes defined in a domain we are interested in strong approximations of one dimensional sdes which have non lipschitz coefficients and which take values in a domain under a set of general assumptions we derive an implicit scheme that preserves the domain of the sdes and is strongly convergent with rate one moreover we show that this general result can be applied to many sdes we encounter in mathematical finance and bio mathematics we will demonstrate flexibility of our approach by analyzing classical examples of sdes with sublinear coefficients cir cev models and wright fisher diffusion and also with superlinear coefficients 3 2 volatility ait sahalia model our goal is to justify an efficient multilevel monte carlo method for a rich family of sdes which relies on good strong convergence properties

one dimensional schrodinger operators with random or deterministic potentials new spectral types abstract we prove sufficient conditions involving only potential asymptotic near one of the infinities in order to have purely absolutely continuous components in the spectrum these deterministic results are then applied to random cases and exhibit classes of models for which with probability one one component of the spectrum is purely absolutely continuous and the rest is dense pure point with exponentially decaying eigenfunctions

adm pade solutions for generalized burgers and burgers huxley systems with two coupled equations we present an approximate numerical solution to generalized burgers and burgers huxley systems with two coupled equations using the adm pade technique which is a combination of the adomian decomposition method and pade approximation the pade technique is used to enhance the accuracy and speed up the convergence rate of the truncated series solution produced by the adm results are given in graphical form for explicit examples of the two systems

commutation methods for jacobi operators abstract we offer two methods of inserting eigenvalues into spectral gaps of a given background jacobi operator the single commutation method which introduces eigenvalues into the lowest spectral gap of a given semi bounded background jacobi operator and the double commutation method which inserts eigenvalues into arbitrary spectral gaps moreover we prove unitary equivalence of the commuted operators restricted to the orthogonal complement of the eigenspace corresponding to the newly inserted eigenvalues with the original background operator in addition we compute the matrix valued weyl m functions of the commuted operator in terms of the background weyl m functions finally we show how to iterate the above methods and give explicit formulas for various quantities such as eigenfunctions and spectra of the iterated operators in terms of the corresponding background quantities and scattering matrix concrete applications include an explicit realization of the isospectral torus for algebro geometric finite gap jacobi operators and the n soliton solutions of the toda and kac van moerbeke lattice equations with respect to arbitrary background solutions

the inverse periodic problem for hill s equation with a finite gap potential let h q d d q x be the hill s operator with a periodic potential q of period one consider the following inverse problem find all potentials q x with the same periodic spectrum as a given potential p x two explicit solutions are known for finite gap potentials its and matveev 6 give an explicit formula in terms of theta functions their solution was generalized by mckean and trubowitz lo to infinite gap potentials finkel et al 5 give a second formula which expresses q in terms of the base potential p the dirichlet eigenfunctions of p and the floquet solutions of p the formula involves no algebraic geometry in this paper we represent finite gap potentials q as rational functions of the base potential p and the derivatives ofp the representation makes use of the finkel isaacson trubowitz fit formula

on critical and subcritical sturm liouville operators we study critical and subcritical sturm liouville operators k 1 ddx p ddx v in l2 a b k dx a b r defined as friedrichs extensions under most general conditions on k p and v in particular we prove the equivalence of the notions of criticality as introduced by simon and murata for an important class of potentials v

factorisation of energy dependent schrodinger operators miura maps and modified systems we consider the energy dependent schrodinger operator mathbb l sum limits i 0 n lambda i varepsilon i partial 2 u i which we have previously shown to be associated with multi hamiltonian structures 2 in this paper we use an unusual form of the lax approach to derive by asingle construction the time evolutions of the eigenfunctions of mathbb l the associated hamiltonian operators and the hamiltonian functionals we then generalise the well known factorisation of standard lax operators to the case of energy dependent operators the simple product of linear factors is replaced by a dependent quadratic form we thus generalise the resulting construction of miura maps and modified equations we show that for some of our systems there exists a sequence ofn such modifications therth modification possessing n r 1 hamiltonian structures

treibich verdier potentials and the stationary m kdv hierarchy here x x col co3 denotes the elliptic weierstrass function with fundamental periods f p 2a l 2co3 im co3 wl 5 0 see 1 ch 18 in the special case where col is real and co3 is purely imaginary the potential q is real valued and ince s striking result 31 in modern spectral theoretic terminology yields the fact that the self adjoint operator l associated with the differential expression d 2 q in l2 ir has a finite gap spectrum of the type