liouville integrability of the finite dimensional hamiltonian systems derived from principal chiral field for finite dimensional hamiltonian systems derived from 1 1 dimensional integrable systems if they have lax representations then the lax operator creates a set of conserved integrals when these conserved integrals are in involution it is believed quite popularly that there will be enough functionally independent ones among them to guarantee the liouville integrability of the hamiltonian systems at least for those derived from physical problems in this article we give a counterexample based on the u 2 principal chiral field it is proved that the finite dimensional hamiltonian systems derived from the u 2 principal chiral field are liouville integrable moreover their lax operator gives a set of involutive conserved integrals but they are not enough to guarantee the integrability of the hamiltonian systems

periodic wave solutions of the boussinesq equation the boussinesq equation usually arises in a physical problem as a long wave equation the present work extends the search of periodic wave solutions for it the hirota bilinear method and riemann theta function are employed in the process we also analyse the asymptotic property of periodic waves in detail furthermore it is of interest to note that well known soliton solutions can be reduced from the periodic wave solutions

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

approximate bayes factors and accounting for model uncertainty in generalised linear models summary ways of obtaining approximate bayes factors for generalised linear models are described based on the laplace method for integrals we propose a new approximation which uses only the output of standard computer programs for estimating generalised linear models this appears to be quite accurate a reference set of proper priors is suggested both to represent the situation where there is not much prior information and to assess the sensitivity of the results to the prior distribution the methods can be used when the dispersion parameter is unknown when there is overdispersion to compare link functions and to compare error distributions and variance functions the methods can be used to implement the bayesian approach to accounting for model uncertainty we describe an application to inference about relative risks in the presence of control factors where model uncertainty is large and important software to implement the methods is available at no cost from statlib

an algebro geometric solution for a hamiltonian system with application to dispersive long wave equation by using an iterative algebraic method we derive from a spectral problem a hierarchy of nonlinear evolution equations associated with dispersive long wave equation it is shown that the hierarchy is integrable in liouville sense and possesses bi hamiltonian structure two commutators with zero curvature and lax representations for the hierarchy are constructed respectively by using two different systematic methods under a bargmann constraint the spectral is nonlinearized to a completely integrable finite dimensional hamiltonian system by introducing the abel jacobi coordinates an algebro geometric solution for the dispersive long wave equation is derived by resorting to the riemann theta function

asynchronously switched control of discrete impulsive switched systems with time delays this paper is concerned with the stabilization problem for a class of uncertain discrete impulsive switched delay systems under asynchronous switching the so called asynchronous switching means that the switches between the candidate controllers and system modes are asynchronous by using the average dwell time adt approach sufficient conditions for the existence of an asynchronously switched controller is derived such that the resulting closed loop system is exponentially stable the desired controller gains and the admissible switching signals are obtained in terms of a set of matrix inequalities a numerical example is given to illustrate the effectiveness of the proposed method

symbolic computation and families of jacobi elliptic function solutions of the 2 1 dimensional nizhnik novikov veselov equation in this paper with the aid of the symbolic computation we have improved the extended f expansion method in chaos solitons and fractals 22 2004 111 and proposed the further improved f expansion method by this method we have gotten many new exact solutions which we have never seen before within our knowledge of the 2 1 dimensional nizhnik novikov veselov equations in addition the solutions we get are more general than the solutions the extended f expansion method gets at the same time our method is more convenient than the method in chaos solitons and fractals 18 2003 299 309 and the solutions we get are more abundant than the solutions in h q zh appl math e notes 1 2001 139 in other word the solutions in h q zh appl math e notes 1 2001 139 are part of our solutions our method can also apply to other partial differential equations and also get many new exact solutions

finite genus solution of a new 2 1 burgers equation with a discrete variable a new 2 1 burgers equation with a discrete variable is given as a compatible condition of the lax pair which are transformed into two hamilton flows and a discrete symplectic flow these flows are integrable in liouville sense and share the same integrals by the abel jacobi coordinates the flows are straightened based on the riemann jacobi inverse theorem a finite genus solution is obtained

nonlinearization of the lax pairs for discrete ablowitz ladik hierarchy abstract the discrete ablowitz ladik hierarchy with four potentials and the hamiltonian structures are derived under a constraint between the potentials and eigenfunctions the nonlinearization of the lax pairs associated with the discrete ablowitz ladik hierarchy leads to a new symplectic map and a class of finite dimensional hamiltonian systems the generating function of the integrals of motion is presented by which the symplectic map and these finite dimensional hamiltonian systems are further proved to be completely integrable in the liouville sense each member in the discrete ablowitz ladik hierarchy is decomposed into a hamiltonian system of ordinary differential equations plus the discrete flow generated by the symplectic map

stability analysis for stochastic hybrid systems this survey addresses stability analysis for stochastic hybrid systems shs which are dynamical systems that combine continuous change and instantaneous change and that also include random effects we re emphasize the common features found in most of the models that have appeared in the literature which include stochastic switched systems markov jump systems impulsive stochastic systems switching diffusions stochastic impulsive systems driven by renewal processes diffusions driven by levy processes piecewise deterministic markov processes general stochastic hybrid systems and stochastic hybrid inclusions then we review many of the stability concepts that have been studied including lyapunov stability lagrange stability asymptotic stability and recurrence next we detail lyapunov based sufficient conditions for these properties and additional relaxations of lyapunov conditions many other aspects of stability theory for shs like converse lyapunov theorems and robustness theory are not fully developed hence we also formulate some open problems to serve as a partial roadmap for the development of the underdeveloped pieces

from the special 2 1 toda lattice to the kadomtsev petviashvili equation the nonlinearization of the eigenvalue problems associated with the toda hierarchy and the coupled korteweg de vries kdv hierarchy leads to an integrable symplectic map s and an integrable hamiltonian system h0 respectively it is proved that s and h0 have the same integrals hk the quasi periodic solution of the 2 1 dimensional kadomtsev petviashvili equation is split into three hamiltonian systems h0 h1 h2 while that of the special 2 1 dimensional toda equation is separated into h0 h1 plus the discrete flow generated by the symplectic map s a clear evolution picture of various flows is given through the window of abel jacobi coordinates the explicit theta function solutions are obtained by resorting to this separation technique

input to state stability of efficient robust h mpc scheme for nonlinear systems propose an efficient constrained h nmpc with guaranteed iss w r t bounded disturbances use piecewise affine h control and augmented pldi models to decrease conservatism of nmpc use pldi method to turn the non convex min max problem into a semi definite programming present a sufficient condition in form of lmis to ensure the iss of the proposed nmpc scheme apply the proposed nmpc to some fast systems such as mass carts and vehicular platoon in this work we present an efficient h model predictive control mpc scheme with guaranteed input to state stability iss for constrained nonlinear systems subject to unknown but bounded disturbances the nonlinear system is represented by using some given inclusion polytope the notion of iss is introduced to derive the sufficient condition in terms of linear matrix inequalities lmis which ensures the iss of the closed loop system in order to reduce the online computational demand of mpc the non convex min max optimization problem is converted to a minimization problem in form of lmis a suboptimal mpc algorithm is then addressed to further decrease the computational demand the applicability of the proposed results is illustrated by two numerical examples of a mass cart and a vehicular platoon

l2 l filter design for discrete time singular markovian jump systems with time varying delays the problem of l 2 l filter design for discrete time singular systems with markovian jump and time varying delays is investigated in this paper by using the delay partitioning technique a delay dependent condition is established to guarantee the filtering error systems to be stochastically admissible and achieve a prescribed l 2 l performance index based on the derived condition the full order and reduced order filters with mode independent characterization are designed in a unified framework the corresponding filter parameters can be obtained by solving a set of linear matrix inequalities the reported results not only depend upon the delay but also depend upon the partitioning which aims at reducing the conservatism two numerical examples are provided to illustrate the effectiveness of the proposed methods

on the decomposition of the modified kadometsev petviashvili equation and the connection between the kn equation and a completely integrable finite dimensional hamiltonian system the 2 1 dimensional modified kadomtsev petviashvili mkp equation is decomposed into the known 1 1 dimensional kaup newell kn equation by using the nonlinearization of the lax pair a classically integrable hamiltonian system in the liouville sense and the involutive solution of the mkp equation 1 1 are obtained from the first two nontrivial kn equations

inverse groundwater modeling for hydraulic conductivity estimation using bayesian model averaging and variance window 1 this study proposes a bayesian model averaging bma method to address parameter estimation uncertainty arising from nonuniqueness in parameterization methods bma is able to incorporate multiple parameterization methods for prediction through the law of total probability and to obtain an ensemble average of hydraulic conductivity estimates two major issues in applying bma to hydraulic conductivity estimation are discussed the first problem is using occam s window in usual bma applications to measure approximated posterior model probabilities occam s window only accepts models in a very narrow range tending to single out the best method and discard other good methods we propose a variance window to replace occam s window to cope with this problem the second problem is the kashyap information criterion kic in the approximated posterior model probabilities which tends to prefer highly uncertain parameterization methods by considering the fisher information matrix with sufficient amounts of observation data the bayesian information criterion bic is a good approximation and is able to avoid controversial results from using kic this study adopts multiple generalized parameterization gp methods such as the bma models to estimate spatially correlated hydraulic conductivity numerical examples illustrate the issues of using kic and occam s window and show the advantages of using bic and the variance window in bma application finally we apply bma to the hydraulic conductivity estimation of the 1500 foot sand in east baton rouge parish louisiana

information criteria for selecting possibly misspecified parametric models abstract we consider penalized likelihood criteria for selecting models of dependent processes the models may be strictly nested overlapping or nonnested linear or nonlinear and correctly specified or misspecified we provide sufficient conditions on the penalty to guarantee the selection with probability one or with probability approaching one of the model attaining the lower average kullback leibler information criterion klic or when both have the same klic the more parsimonious model as special cases our results describe the akaike schwarz and hannan quinn information criteria as examples we consider selection of armax garch and star models

the likelihood dominance criterion a new approach to model selection abstract the dominance ordering and the ldc provide a new approach to model selection the dominance ordering is described in terms of a fictive experiment in which the two competing hypotheses are nested in a composite if estimating the composite would lead to accepting one hypothesis and rejecting the other then which would be accepted and which rejected is determined by the two adjusted likelihood values and does not require estimating the composite although it does require specifying its parametric size the ldc generalizes the dominance ordering by considering a range of admissible composite parametric sizes this range will usually include all sizes of practical interest

stochastic input to state stability and h filtering for a class of stochastic nonlinear systems in this paper an h filtering problem is considered for a class of stochastic nonlinear systems with time delay and exogenous disturbances sufficient conditions for stochastic input to state stability siss of the stochastic nonlinear systems are developed via lyapunov function method and linear matrix inequalities lmis technique the purpose of this work is to design a nonlinear filter to ensure both the siss and a prescribed h attenuation level for the corresponding filtering error system by using the siss results a set of sufficient conditions for the existence of h filter is given in the form of lmi when the lmi is feasible an explicit expression of a desired filter is presented finally some numerical and practical examples are employed to demonstrate the effectiveness of the proposed approaches

input to state stable nonlinear filtering for a class of continuous time delayed nonlinear systems this paper investigates the input to state stable iss nonlinear filtering problem for a class of continuous time delayed nonlinear systems with external disturbance a new delay dependent nonlinear iss filter is established through available measurements to estimate the states of delayed nonlinear systems such that the filtering error system is both exponentially and input to state stable for any bounded external disturbance the design of the nonlinear iss filter for these nonlinear systems is achieved by solving a linear matrix inequality lmi which can be easily facilitated by using standard numerical packages an illustrative example is given to demonstrate the effectiveness of the proposed filter

on the relation of the stationary toda equation and the symplectic maps it is proved that every solution of the stationary toda equation in the generic case is represented as a discrete orbit of the symplectic map obtained through nonlinearization of the toda eigenvalue problem as an application the calculation of the finite band solution of the toda lattice equation is reduced to the solution of a system of odes plus a simple iterative process of the symplectic map

explicit solutions for a hierarchy of differential difference equations a new discrete isospectral problem and the corresponding hierarchy of nonlinear differential difference equations are proposed on the basis of the theory of algebraic curves the continuous flow and discrete flow related to the hierarchy of differential difference equations are straightened out using the abel jacobi coordinates the meromorphic function and the baker akhiezer function are introduced on the hyperelliptic curve quasi periodic solutions of the corresponding hierarchy of differential difference equations are constructed with the help of the asymptotic properties and the algebro geometric characters of the meromorphic function the baker akhiezer function and the hyperelliptic curve

lax pair backlund transformation and multi soliton solutions for the boussinesq burgers equations from shallow water waves abstract under investigation in this paper is the set of the boussinesq burgers bb equations which can be used to describe the propagation of shallow water waves based on the binary bell polynomials hirota method and symbolic computation the bilinear form and soliton solutions for the bb equations are derived backlund transformations bts in both the binary bell polynomial and bilinear forms are obtained through the bt in the binary bell polynomial form a type of solutions and lax pair for the bb equations are presented as well propagation characteristics and interaction behaviors of the solitons are discussed through the graphical analysis shock wave and bell shape solitons are respectively obtained for the horizontal velocity field u and height v of the water surface in both the head on and overtaking collisions the shock waves for the u profile change their shapes which denotes that the collisions for the u profile are inelastic however the collisions for the v profile are proved to be elastic through the asymptotic analysis our results might have some potential applications for the harbor and coastal design

sliding mode control of switched hybrid systems with stochastic perturbation this paper is concerned with the sliding mode control smc of a continuous time switched stochastic system a sufficient condition for the existence of reduced order sliding mode dynamics is derived and an explicit parametrization of the desired sliding surface is also given then a sliding mode controller is then synthesized for reaching motion moreover the observer based smc problem is also investigated some sufficient conditions are established for the existence and the solvability of the desired observer and the observer based sliding mode controller is synthesized finally numerical examples are provided to illustrate the effectiveness of the proposed theory

model selection when there is minimal prior information a number of authors argue that a bayesian posterior odds criterion is appropriate for model selection 2 this paper considers how to derive this criterion when there is minimal prior information we propose minimizing measures of prior information relative to the models in question rather than relative to the parameters of the particular models in so doing we obtain an expression for the odds that is invariant to the parameterization of the particular models and overcomes certain well known finite sample limiting problems we illustrate this procedure using two popular measures of information derived from the well known shannon 26 measure by minimizing these measures with the sample size held fixed we obtain the same model selection criterion that schwarz 25 derived asymptotically for large sample sizes this expression has a number of desirable properties and is computationally no more

constraints of the 2 1 dimensional integrable soliton systems the authors show that the linear systems associated with some integrable hierarchies of the soliton equations in 2 1 dimensions can be constrained to integrable hierarchies in 1 1 dimensions such that submanifolds solutions of the given systems in 2 1 can be obtained by solving the resulting integrable systems in 1 1 dimensions the constraints of the kp hierarchy to the akns and burgers hierarchies respectively are shown in detail and the results of these for the modified kp and 2 1 dimensional analogue of the caudrey dodd gibbon kotera sawata equations to several integrable systems in 1 1 are given

mixed hierarchy of soliton equations the mixed hierarchy of soliton equations in 1 1 dimensions is introduced it contains nonholonomic deformations of soliton equations such the kdv6 equation and the kupershmidt deformations of soliton equations as special members based on the commutator representation method a recipe for constructing zero curvature representations of mixed hierarchy is proposed as applications we obtain the mixed hierarchies and their zero curvature representations for the korteweg de vries hierarchy the ablowitz kaup newell segur hierarchy the modified korteweg de vries hierarchy the toda lattice hierarchy and the volterra lattice hierarchy

how to construct finite dimensional bi hamiltonian systems from soliton equations jacobi integrable potentials a systematic method of constructing finite dimensional integrable systems starting from a bi hamiltonian hierarchy of soliton equations is introduced the existence of two hamiltonian structures of the hierarchy leads to a bi hamiltonian formulation of the resulting finite dimensional systems the case of coupled kdv hierarchies is studied in detail a surprising connection with separable jacobi potentials is uncovered and described

brief paper input to state stability and integral input to state stability of nonlinear impulsive systems with delays this paper is concerned with analyzing input to state stability iss and integral iss iiss for nonlinear impulsive systems with delays razumikhin type theorems are established which guarantee iss iiss for delayed impulsive systems with external input affecting both the continuous dynamics and the discrete dynamics it is shown that when the delayed continuous dynamics are iss iiss but the discrete dynamics governing the impulses are not the iss iiss property of the impulsive system can be retained if the length of the impulsive interval is large enough conversely when the delayed continuous dynamics are not iss iiss but the discrete dynamics governing the impulses are the impulsive system can achieve iss iiss if the sum of the length of the impulsive interval and the time delay is small enough in particular when one of the delayed continuous dynamics and the discrete dynamics are iss iiss and the others are stable for the zero input the impulsive system can keep iss iiss no matter how often the impulses occur our proposed results are evaluated using two illustrative examples to show their effectiveness

explicit quasi periodic solutions of the vakhnenko equation the trigonal curve associated with the vakhnenko equation is introduced by using the lax matrix for the nth stationary positive flow based on the theory of the trigonal curve and the properties of the three kinds of abel differentials the riemann theta function representation for the baker akhiezer function is derived from which the straightening out of flows in the jacobian variety is exactly given through the abel maps we finally arrive at quasi periodic solutions of the vakhnenko equation

algebro geometric solution of the 2 1 dimensional burgers equation with a discrete variable the quasiperiodic solution of the 2 1 dimensional burgers equation with a discrete variable is obtained through three steps a decomposition into a symplectic map plus two finite dimensional hamiltonian systems b straightening out of both the discrete and the continuous flows in the jacobian variety c inversion into the original variables inner relation with the modified kadomtsev petviashvili equation is presented the explicit theta function solutions for these two 2 1 integrable models are given

information and posterior probability criteria for model selection in local likelihood estimation local likelihood estimation has proven to be an effective method for obtaining estimates of parameters that vary with a covariate to obtain useful estimates of such parameters approximating models are used in such cases it is useful to consider window based estimates we may need to choose between competing approximating models in this article we propose a modification to the methods used to motivate many information and posterior probability criteria for the weighted likelihood case we derive weighted versions for two of the most widely known criteria namely the aic and bic via a simple modification the criteria are also made useful for window span selection the usefulness of the weighted version of these criteria is demonstrated through a simulation study and an application to three datasets

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

benchmark priors for bayesian model averaging in contrast to a posterior analysis given a particular sampling model posterior model probabilities in the context of model uncertainty are typically rather sensitive to the speciflcation of the prior in particular difiuse priors on model speciflc parameters can lead to quite unexpected consequences here we focus on the practically relevant situation where we need to entertain a large number of sampling models and we have or wish to use little or no subjective prior information we aim at providing an automatic or benchmark prior structure that can be used in such cases we focus on the normal linear regression model with uncertainty in the choice of re gressors we propose a partly noninformative prior structure related to a natural conjugate g prior speciflcation where the amount of subjective information requested from the user is limited to the choice of a single scalar hyperparameter g0j the consequences of difierent choices for g0j are examined we investigate theoretical properties such as consistency of the implied bayesian procedure links with classical information criteria are provided in addition we examine the flnite sample implications of several choices of g0j in a simulation study the use of the mc 3 algorithm of madigan and york 1995 combined with e cient coding in fortran makes it feasible to conduct large simulations in addition to posterior criteria we shall also compare the predictive performance of difierent priors a classic exam ple concerning the economics of crime will also be provided and contrasted with results in the literature the main flndings of the paper will lead us to propose a benchmark prior speciflcation in a linear regression context with model uncertainty

restricted flows of soliton hierarchies coupled kdv and harry dym case restricted flows of soliton hierarchies associated with the energy dependent schrodinger spectral problem are determined explicitly a remarkable connection with separable potentials is used for proving complete integrability of the restricted flows a previously unknown lagrangian and hamiltonian formulation of the neumann system is found whole families of generalizations of the neumann and garnier systems are given

dissipativity based sliding mode control of switched stochastic systems this technical brief is concerned with dissipativity analysis and dissipativity based sliding mode control smc of continuous time switched stochastic systems firstly a sufficient condition is proposed to guarantee the mean square exponential stability and strict dissipativity for the switched stochastic system then an integral type sliding surface function is designed for establishing a sliding mode dynamics which can be formulated by a switched stochastic system with an external disturbance uncertainty dissipativity analysis and synthesis are both investigated for the sliding mode dynamics and consequently sufficient conditions are derived which pave the way for solving the dissipativity analysis and control problems moreover a smc law is synthesized to drive the system trajectories onto the predefined sliding surface in a finite time finally the efficiency of the theoretical findings is demonstrated by an illustrative example

stabilization of singular markovian jump systems with generally uncertain transition rates this note is devoted to investigating the stability and stabilization problems for continuous time singular markovian jump systems smjss with generally uncertain transition rates gutrs in this gutr singular model each transition rate can be completely unknown or only its estimate value is known in terms of a set of coupled linear matrix inequalities lmis a sufficient condition is established to ensure the systems to be regular impulse free and stochastically stable moreover the corresponding sufficient condition on the existence of a mode dependent state feedback controller is derived to guarantee the closed loop systems stochastically admissible by applying the lmi technique finally a numerical example is presented to illustrate the effectiveness and efficiency of the proposed method

a 2 1 dimensional derivative toda equation in the context of the kaup newell spectral problem a 2 1 dimensional derivative toda equation is derived from the lax triad composed of the kaup newell the negative kaup newell and a discrete spectral problem two integrable hamiltonian systems and an integrable symplectic map are derived from the lax triad they are straightened out in the jacobi variety of the associated hyperelliptic curve a finite genus solution is obtained through a technique based on the riemann jacobi inversion theorem besides explicit solutions are calculated for other associated integrable models including the mkp equation and the nkn equation

input to state stability for discrete time nonlinear switched singular systems discrete time nonlinear switched singular systems ssss are investigated the input to state stability iss problems for discrete time nonlinear ssss are concerned the iss criteria are obtained via average dwell time approach and iterative algorithm of discrete time systems the switching rules are optimized and designed this paper investigates the input to state stability iss problems for a class of discrete time nonlinear switched singular systems ssss two novel iss criteria are proposed based on average dwell time adt approach and iterative algorithm of discrete time systems iads in particular the following two cases are considered for the underlying systems the first case is that all the sub systems are exponentially stable and the other case is that only a few sub systems are exponentially stable the corresponding iss criteria are obtained to guarantee that the closed loop systems are input to state stable via adt approach and iads we neither construct a specific lyapunov function for iss in the proof process of the stability criteria nor design a specific structure of the control inputs the design deficit of the switching controllers is optimized and the design difficulty of the switching controllers is reduced via the proposed criteria finally two numerical examples are provided to illustrate the feasibility of the results obtained

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

new rational solitary wave solutions of 2 1 dimensional burgers equation abstract based on the computer symbolic system maple and the rational solitary wave solutions and rational periodic solutions obtained by others previously we present a new direct ansatz in terms of the hyperbolic functions and periodic function to solve the nonlinear evolution equations that is to say we assume the equations have the solution in the following form u i a i 0 j 1 n i g j 1 a i j f b i j g c i j h d i j r where f tanh sech a b tanh c sech 2 g sech a b tanh c sech 2 h sech 2 a b tanh c sech 2 r tanh a b tanh c sech 2 and a 2 b 2 c 2 0 compared with the solutions obtained by the existing tanh methods riccati equation method and all kinds of their improved methods the solutions obtained by the proposed direct method not only include some obtained solutions but also include some new and general rational solutions and we choose the 2 1 dimensional burgers equation to illustrate the key step as a result we obtain abundant families of rational solitary wave solutions and rational periodic solutions furthermore the direct method is proved to be efficient

rational solutions in grammian form for the 3 1 dimensional generalized shallow water wave equation abstract in this paper the 3 1 dimensional generalized shallow water wave equation is investigated using the hirota bilinear method and kadomtsev petviashvili hierarchy reduction the explicit rational solutions for such an equation have been presented in the grammian form based on the grammian form solution for the equation the one rational two rational and three order rational solutions are obtained when complex parameters p i with nonzero real and imaginary parts are chosen the lump soliton solutions which are localized in all directions for the 3 1 dimensional generalized shallow water wave equation can be derived from the corresponding rational solutions as the figures illustrate the one lump soliton solution with one peak and one trough propagates stably on the x y plane the two lump solitons with different velocities interact with each other and separate with their original shapes and propagation directions different from the case of two lump solitons the propagation directions of the third order lump solitons change after the interaction

c neumann and bargmann systems associated with the coupled kdv soliton hierarchy under two different constraints between the potentials and the eigenfunctions the eigenvalue problem associated with the coupled kdv hierarchy is nonlinearised to be a completely integrable c neumann system on the tangent bundle of sphere tsn 1 with the hamiltonian h 1 2 lambda q p 1 2 q q p p 1 2 q p 2 1 2 q p lambda p p and a completely integrable bargmann system r2n dpdq h with the hamiltonian h 1 2 lambda q p 1 2 p p q p 1 2 q q respectively the involutive solutions of the coupled kdv equation associated with the two systems are given

the finite band solution of the jaulent miodek equation a method to construct the finite band solution to the soliton equation is presented we take the jaulent miodek equation as an example with the use of the nonlinearization of lax pair the jaulent miodek equation is decomposed into two finite dimensional integrable systems whose properties are studied from the view of an r matrix then through solving these finite dimensional integrable systems the finite band solution of the jaulent miodek equation is obtained

quasi periodic wave solutions for the 2 1 dimensional generalized calogero bogoyavlenskii schiff cbs equation abstract the hirota bilinear method is implemented to directly construct quasi periodic wave solutions in terms of theta functions for a hirota bilinear equation the result is applied to the 2 1 dimensional generalized calogero bogoyavlenskii schiff cbs equation thereby yielding their quasi periodic wave solutions under the backlund transformation u y 3 l n f x y t x

algebro geometric constructions of the discrete ablowitz ladik flows and applications resorting to the finite order expansion of the lax matrix the elliptic coordinates are introduced from which the discrete ablowitz ladik equations and the 2 1 dimensional toda lattice are decomposed into solvable ordinary differential equations the straightening out of the continuous flow and the discrete flow is exactly given through the abel jacobi coordinates as an application explicit quasiperiodic solutions for the 2 1 dimensional toda lattice are obtained

algebro geometric solutions for the gerdjikov ivanov hierarchy this paper is dedicated to provide theta function representations of algebro geometric solutions and related crucial quantities for the gerdjikov ivanov gi hierarchy our main tools include the polynomial recursive formalism the hyperelliptic curve with finite number of genus the baker akhiezer functions the meromorphic function the dubrovin type equations for auxiliary divisors and the associated trace formulas with the help of these tools the explicit representations of the baker akhiezer functions the meromorphic function and the algebro geometric solutions are obtained for the entire gi hierarchy