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

solution of the dispersionless hirota equations the dispersionless differential pay identity is shown to be equivalent to a kernel expansion providing a universal algebraic characterization and solution of the dispersionless hirota equations some calculations based on d bar data of the action are also indicated

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

quasigroup construction and first class constraints a generalization of the lie group construction is proposed wherein the composition law depends apart from the parameters of the transformations composed also on the transformed variables this construction is met in particular on the hypersurfaces specified by the first class constraints in phase space

a stochastic version of center manifold theory random dynamical systems arise naturally if the influence of white or real noise on the parameters of a nonlinear determinstic dynamical system is studied in this situation lyapunov exponents attached to the linearized flow replace the real parts of the eigenvalues and describe the stability behavior of the linear system if at least one of them vanishes then it is possible to prove the existence of a stochastic analogue of the deterministic center manifold the asymptotic behavior of the entire system can then be derived from the lower dimensional system restricted to this stochastic center manifold a dynamical characterization of the stochastic center manifold is given and approximation results are proved

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

finite band solutions of the coupled dispersionless hierarchy the coupled dispersionless hierarchy is derived with the help of the zero curvature equation based on the lax matrix we introduce an algebraic curve of arithmetic genus n from which we establish the corresponding meromorphic function the baker akhiezer function and dubrovin type equations the straightening out of all the flows is given under the abel jacobi coordinates using the asymptotic properties of and we obtain the explicit theta function representations of the meromorphic function the baker akhiezer function and of solutions for the whole hierarchy

stochastic calculus with anticipating integrands we study the stochastic integral defined by skorohod in 24 of a possibly anticipating integrand as a function of its upper limit and establish an extended ito formula we also introduce an extension of stratonovich s integral and establish the associated chain rule in all the results the adaptedness of the integrand is replaced by a certain smoothness requirement

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

nonlinear schrodinger equation painleve test backlund transformation and solutions the authors demonstrate that the nonlinear schrodinger equation passes the painleve test and construct a backlund transformation and solutions within this approach the connection with the hirota bilinear formalism is discussed also the heisenberg ferromagnet equation which is gauge equivalent to the nonlinear schrodinger equation is studied

the structures and interactions of solitary waves in a 2 1 dimensional coupled nonlinear extension of the reaction diffusion equation the complete integrability of a 2 1 dimensional coupled nonlinear extension of the reaction diffusion cnlerd equation using the technique of painlev p analysis is investigated using the formalism of weiss et al 1983 j math phys 24 522 the arbitrariness of the expansion coefficients are proved besides following the hirota s formalism hirota r 1980 direct methods in soliton theory soliton berlin springer combined to weiss et al s methodology weiss et al 1984 j math phys 25 13 the consistency of the truncation is shown thus without the use of kruskal s simplification the bcklund transformation bt of the equations is obtained via the truncation procedure taking into account the arbitrariness of the expansion coefficients in the foregoing truncation a typical spectrum of localized coherent structures may be unearthed the scattering behavior of such structures is also investigated

the stable manifold theorem for stochastic differential equations type equations are treated starting with the existence of a stochastic flow for a sde we introduce the notion of a hyperbolic stationary trajectory we prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution for stratonovich sdes the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde the proof of the stable manifold theorem is based on ruelle oseledec multiplicative ergodic theory

state classification of time nonhomogeneous markov chains and average reward optimization of multi chains in a discrete time nonhomogeneous markov chain tnhmc the states spaces transition probabilities and reward functions at different times may be different in this paper with the confluencity previously introduced we show that the states of a tnhmc can be classified into the branching states and a number of classes of confluent states versus the transient and recurrent states in the time homogeneous case the optimization of average reward in tnhmc s consisting of a single confluent class uni chain have been addressed in a previous paper by the author in this paper we show that with confluencity and the state classification and under some bound conditions we can obtain the necessary and sufficient conditions for optimal policies of the average reward of tnhmcs consisting of multiple confluent classes multi chains just like in the uni chain tnhmc case the sufficient condition does not need to hold in any zero frequently visited time sequence this under selectivity makes the problem not amenable to dynamic programming a direct comparison based approach is used to prove the results the results enhance our understanding of state classification and performance optimization with the notion of confluencity

existence theorem for gauge algebra a general gauge action is defined by postulating a minimum of its properties necessary for the existence of loop expansion in the quantum theory the structure of the general gauge algebra is derived from these postulates the proof of existence of the structure functions lacking in previous works is given the question of uniqueness is also completely cleared up some unsolved problems are discussed in connection with the covariant quantization conjecture

on the spatial asymptotic behavior of stochastic flows in euclidean space we study asymptotic growth rates of stochastic flows on r d and their derivatives with respect to the spatial parameter under lipschitz conditions on the local characteristics of the generating semimartingales in a first step these conditions are seen to imply moment inequalities for the flow of the form e sup 0t x 0t y p 1 0 t t in a second step we deduce the growth rates from an integrated version of these moment inequalities using the continuity lemma of garsia rodemich and rumsey we provide two examples to show that our results are sharp

the τ function of the universal whitham hierarchy matrix models and topological field theories the universal witham hierarchy is considered from the point of view of topological field theories the function for this hierarchy is defined it is proved that the algebraic orbits of whitham hierarchy can be identified with various topological matter models coupled with topological gravity

painleve analysis and integrability of coupled non linear schrodinger equations considering a system of coupled non linear schrodinger nls equations the authors discuss the integrability properties through painleve p analysis for the two coupled nls equations they show that there exists a pair of parametric values possessing the p property for which the associated backlund transformation bt and the hirota bilinearisation are constructed these parametric choices are identical to those of zakharov and schulman 1982 who established the integrability in terms of motion invariants finally they extend the p analysis to the n coupled nls system and identify two parametric choices possessing the p property which are natural generalisations of the two coupled nls cases

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

painleve conjecture revisited the discovery of new integrable two dimensional hamiltonian systems is reported the analytic structure of the solutions makes necessary the generalization of the painlev e conjecture a widely used integrability criterion such a generalization is presented which the authors believe should replace the usual conjecture for two dimensional hamiltonian systems it is indeed compatible with all the systems already found and in addition leads to still new integrable cases