canonical property of the miura maps between the mkp and kp hierarchies continuous and discrete apart from the case of the kp hierarchy all known miura maps between integrable hamiltonian systems had been proven to be canonical the remaining kp case is settled below as a corollary it is shown that the kp hierarchy is a factor hierarchy of the mkp one with the kernel consisting of a single scalar field a discrete mkp hierarchy and the associated miura map are constructed and the latter is shown to be canonical as well as in the continuous case this implies that one can extend the discrete kp hierarchy by a single new field into an extended discrete kp hierarchy in such a way that the extended discrete miura map mkp ekp is a canonical isomorphism

two component kp hierarchy and the classical boussinesq equation it has been shown that solutions to the kp equation and other equations in the kp hierarchy take a wronskian form the authors extend these ideas to two component wronskians and the hirota equations 1986 which they satisfy by a reduction procedure they show how dependence on some of the variables can be removed and look in detail at the classical boussinesq equation and its modifications and show how these equations fit into the two component kp hierarchy

a study on modeling the dynamics of statistically dependent returns abstract this paper develops a method to characterize the dynamic behavior of statistically dependent returns of assets via a scenario set the proposed method uses heteroskedastic time series to model serial correlations of returns as well as cholesky decomposition to generate the set of scenarios such that the statistical dependence of different asset returns is preserved in addition this scenario generation method preserves marginal distributions of returns to demonstrate the performance of the proposed method a multi period portfolio optimization model is presented then the method is implemented through a number of stocks selected from new york stock exchange nyse computational results show a high performance of the proposed method from the statistical point of view also results confirm sufficiency and in sample stability of the generated scenario set besides out of sample simulations for both risk and return illustrate a good performance of the proposed method

lagrangean decomposition a model yielding stronger lagrangean bounds given a mixed integer programming problem with two matrix constraints it is possible to define a lagrangean relaxation such that the relaxed problem decomposes additively into two subproblems each having one of the two matrices of the original problem as its constraints there is one lagrangean multiplier per variable we prove that the optimal value of this new lagrangean dual dominates the optimal value of the lagrangean dual obtained by relaxing one set of constraints and give a necessary condition for a strict improvement we show on an example that the resulting bound improvement can be substantial we show on a complex practical problem how lagrangean decomposition may help uncover hidden special structures and thus yield better solution methodology

second and fourth painleve hierarchies and jimbo miwa linear problems the relations between the different linear problems for painleve equations is an intriguing open problem here we consider our previously given second and fourth painleve hierarchies publ res inst math sci kyoto 37 327 347 2001 and show that they could alternatively have been derived using the linear problems of jimbo and miwa that is we give a gauge transformation of our linear problems for these two hierarchies which maps those of the second and fourth painleve equations themselves onto those of jimbo and miwa

constrained kp hierarchy and bi hamiltonian structures the kakomtsev petviashvili kp hierarchy is considered together with the evolutions of eigenfunctions and adjoint eigenfunctions constraining the kp flows in terms of squared eigenfunctions one obtains 1 1 dimensional integrable equations with scattering problems given by pseudo differential lax operators the bi hamiltonian nature of these systems is shown by a systematic construction of two general poisson brackets on the algebra of associated lax operators gauge transformations provide miura links to modified equations these systems are constrained flows of the modified kp hierarchy for which again a general description of their bi hamiltonian nature is given the gauge transformations are shown to be poisson maps relating the bi hamiltonian structures of the constrained kp hierarchy and the modified kp hierarchy the simplest realization of this scheme yields the akns hierarchy and its miura link to the kaup broer hierarchy

ergodic primal convergence in dual subgradient schemes for convex programming lagrangean dualization and subgradient optimization techniques are frequently used within the field of computational optimization for finding approximate solutions to large structured optimization prob lems the dual subgradient scheme does not automatically produce primal feasible solutions there is an abundance of techniques for computing such solutions via penalty functions tangential approximation schemes or the solution of auxiliary primal programs all of which require a fair amount of computational effort we consider a subgradient optimization scheme applied to a lagrangean dual formulation of a convex program and construct at minor cost an ergodic sequence of subproblem solutions which converges to the primal solution set numerical experiments performed on a traffic equilibrium assignment problem under road pricing show that the computation of the ergodic sequence results in a considerable improvement in the quality of the primal solutions obtained compared to those generated in the basic subgradient scheme

a parametric study of the term structure dynamics abstract we present an analysis of the dynamics of the term structure of interest rates based on the study of the time evolution of the parameters of a variation of the nelson siegel model the results show that it is extremely difficult to find a relation between the evolution of the term structure and the behavior of macroeconomic variables different from the official interest rate

geodesic flows bi hamiltonian structure and coupled kdv type systems abstract we show that all the antonowicz fordy type coupled kdv equations have the same symmetry group and similar bi hamiltonian structures it turns out that their configuration space is diff s 1 c s 1 where diff s 1 is the bott virasoro group of orientation preserving diffeomorphisms of the circle and all these systems can be interpreted as equations of a geodesic flow with respect to l 2 metric on the semidirect product space diff s 1 c s 1

constraints of the kp hierarchy and multilinear forms we consider the trilinear form of the kaup broer system which gives rise to solutions in wronskian form the kaup broer system is connected with akns system through a gauge transformation the akns hierarchy can be understood as a generalized 1 constraint of the kp hierarchy imposing that constraint on sato s equation we obtain the basic trilinear form and moreover a hierarchy of trilinear equations governing the akns flows similary hierarchies of multilinear forms are derived in the case of generalized k constraints

application of the variational iteration method to the whitham broer kaup equations explicit traveling wave solutions including blow up and periodic solutions of the whitham broer kaup equations are obtained by the variational iteration method moreover the results are compared with those obtained by the adomian decomposition method revealing that the variational iteration method is superior to the adomian decomposition method

lie symmetry analysis and some new exact solutions of the wu zhang equation the lie symmetry analysis and the basic similarity reductions are performed for the wu zhang equation a 2 1 dimensional nonlinear dispersive wave equation some new exact solutions generated from the similarity transformation are provided they demonstrate some new three dimensional features of a single solitary wave and two interacting solitary waves

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

nonclassical analysis and painleve property for the kupershmidt equations by means of the nonclassical lie approach and the direct method all the similarity reductions of the kupershmidt equations are found two reduction equations which can also be obtained by the classical lie approach are the painleve type ii and iv respectively the other three reduction equations which cannot be obtained by the classical lie approach are only linear or riccati type

bifurcation of travelling wave solutions for the modified dispersive water wave equation abstract the method of bifurcation of planar dynamical systems and method of numerical simulation of differential equations are employed to investigate the modified dispersive water wave equation we obtain the parameter bifurcation sets that divide the parameter space into different regions which correspond to qualitatively different phase portraits in different regions different types of travelling solutions including solitary wave solutions shock wave solutions and periodic wave solutions are simulated furthermore with a generalized projective riccati equation method several new explicit exact solutions are obtained

nonisospectral scattering problems and similarity reductions abstract we give mappings between hierarchies having nonisospectral scattering problems and hierarchies having isospectral scattering problems special cases of these changes of variables involve similarity variables and similarity solutions of a hierarchy are seen to correspond to time independent solutions of an equivalent hierarchy we thus explain why the use of nonisospectral scattering problems and similarity reductions yield the same painleve hierarchies as examples we consider the korteweg de vries hierarchy and the dispersive water wave hierarchy

coupled harry dym equations with multi hamiltonian structures the authors consider the isospectral flows of delta 2 sigma 1n upsilon i lambda i psi alpha psi using an unusual form of the lax approach they derive in a particularly simple manner a the second half of the lax pair b the associated hamiltonian structures c an infinite hierarchy of poisson commuting hamiltonians in this way they show that these equations possess n 1 compatible purely differential hamiltonian structures the case n 1 is just the bi hamiltonian harry dym hierarchy they thus extend the recent results on multi hamiltonian coupled kdv equations

classical liouville completely integrable systems associated with the solutions of boussinesq burgers hierarchy two new finite dimensional completely integrable systems in the liouville sense are obtained the solutions of boussinesq burgers hierarchy are generated by using involutive solutions of the commutable flows in the completely integrable systems

a cholesky dual method for proximal piecewise linear programming a quadratic programming method is given for minimizing a sum of piecewise linear functions and a proximal quadratic term subject to simple bounds on variables it may be used for search direction finding in nondifferentiable optimization algorithms an efficient implementation is described that updates a cholesky factorization of active constraints and provides good accuracy via iterative refinement numerical experience is reported for some large problems

r matrices and higher poisson brackets for integrable systems the tri hamiltonian nature of lax equations is revealed starting with an r matrix on an associative algebra g equipped with a trace form there are g compatible poisson brackets with linear quadratic and cubic dependence on the coordinates the invariant functions casimir functions on g are in involution relative to these brackets they yield a hierarchy of integrable tri hamiltonian lax equations the results can be applied to solvable pde s such as the korteweg de vries equation as well as to finite integrable systems such as the toda lattice in these cases the poisson structures considered here turn out to be abstract versions of the first 3 hamiltonian operators of these equations obtained by their well known recursion operators

the integrable hierarchy constructed from a pair of kdv type hierarchies and its associated w algebra for any two arbitrary positive integers n and m using them th kdv hierarchy and the n m th kdv hierarchy as building blocks we are able to construct another integrable hierarchy referred to as the n m th kdv hierarchy thew algebra associated to the second hamiltonian structure of the n m th kdv hierarchy calledw n m algebra is isomorphic via a miura map to the direct sum of aw m algebra aw n m algebra and an additionalu 1 current algebra in turn from the latter we can always construct a representation of aw algebra

nonstandard drinfeld sokolov reduction subject to some conditions the input data for the drinfeld sokolov construction of kdv type hierarchies is a quadruplet where the are gradations of a loop algebra and is a semisimple element of the nonzero grade a new sufficient condition on the quadruplet under which the construction works is proposed and examples are presented the proposal relies on splitting the grade zero part of into a vector space direct sum of two subalgebras this permits one to interpret certain gelfand dickey type systems associated with a nonstandard splitting of the algebra of pseudodifferential operators in the drinfeld sokolov framework

tri hamiltonian formulation for certain integrable lattice equations a systematic investigation of integrable differential difference equations with two independent variables admitting multi hamiltonian structure is presented considering the volterra vl toda tl relativistic toda rt belov chaltikian bc and blaszak marciniak both three bm3 and four bm4 coupled lattice equations it is shown that they admit a sequence of operators out of which only three are hamiltonian ones and so they are tri hamiltonian systems only it is observed that the constructed third operator for vl and bc lattice equations is hamiltonian only if the field variable is periodic with even period

string theory and water waves we uncover a remarkable role that an infinite hierarchy of nonlinear differential equations plays in organizing and connecting certain string theories non perturbatively we are able to embed the type 0a and 0b a a minimal string theories into this single framework the string theories arise as special limits of a rich system of equations underpinned by an integrable system known as the dispersive water wave hierarchy we observe that there are several other string like limits of the system and conjecture that some of them are type iia and iib a d minimal string backgrounds we explain how these and several string like special points arise and are connected in some cases the framework endows the theories with a non perturbative definition for the first time notably we discover that the painlev iv equation plays a key role in organizing the string theory physics joining its siblings painlev i and ii whose roles have previously been identified in this minimal string context

dirac s theory of constraints in field theory and the canonical form of hamiltonian differential operators a simple algorithm for constructing the canonical form of hamiltonian systems of evolution equations with constant coefficient hamiltonian differential operators is given the result of the construction is equivalent to the canonical system derived using dirac s theory of constraints from the corresponding degenerate lagrangian

korteweg de vries modified korteweg de vries systems and darboux transforms in 1 1 and 2 1 dimensions the jaulent miodek hierarchy m jaulent and i miodek lett math phys 1 243 1976 and its two dimensional analog are considered from the point of view of darboux transforms structures a new form of those transforms as well as old fashioned forms are presented the determinant form of solitons is obtained the combination of different transforms in the 2 1 case allows the buildup of localized solutions in both spatial dimensions including the two dimensional korteweg de vries kdv equation

the kaup broer system and trilinear odes of painleve type the kaup broer hierarchy can be considered as a certain reduction of the kp hierarchy upon introducing the function the kaup broer system can be formulated in trilinear form giving rise to wronskian solutions in this paper several odes of the painleve type obtained through reductions of the kaup broer system are considered those odes take a trilinear form which allows a linearization

semi discrete fourier spectral approximations of infinite dimensional hamiltonian systems and conservation laws abstract in this paper we discuss the semi discrete fourier spectral approximation for infinite dimensional hamiltonian systems and pay more attention to the hamiltonian structure in this manner we consider further its preservation of conservation laws of the original hamiltonian systems

a geometric study of duality gaps with applications lagrangian relaxation is often an efficient tool to solve large scale optimization problems even nonconvex however it introduces a duality gap which should be small for the method to be really efficient here we make a geometric study of the duality gap given a nonconvex problem we formulate in a first part a convex problem having the same dual this formulation involves a convexification in the product of the three spaces containing respectively the variables the objective and the constraints we apply our results to several relaxation schemes especially one called lagrangean decomposition in the combinatorial optimization community or operator splitting elsewhere we also study a specific application highly nonlinear the unit commitment problem

non perturbative string theory from water waves we use a combination of a t hooft limit and numerical methods to find non perturbative solutions of exactly solvable string theories showing that perturbative solutions in different asymptotic regimes are connected by smooth interpolating functions our earlier perturbative work showed that a large class of minimal string theories arise as special limits of a painleve iv hierarchy of string equations that can be derived by a similarity reduction of the dispersive water wave hierarchy of differential equations the hierarchy of string equations contains new perturbative solutions some of which were conjectured to be the type iia and iib string theories coupled to 4 4k 2 superconformal minimal models of type a d our present paper shows that these new theories have smooth non perturbative extensions we also find evidence for putative new string theories that were not apparent in the perturbative analysis

a new derivation of painleve hierarchies abstract we give a new derivation of two painleve hierarchies this is done by extending the accelerating wave reductions of the korteweg de vries and dispersive water wave equations to their respective hierarchies we also consider the extension of this reduction of burgers equation to the burgers hierarchy

generalized scaling reductions and painleve hierarchies we give an alternative derivation of two painleve hierarchies this is done by constructing generalized scaling reductions of the korteweg de vries and dispersive water wave hierarchies we also construct a generalized scaling reduction of burgers hierarchy

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

variable metric bundle methods from conceptual to implementable forms to minimize a convex function we combine moreau yosida regularizations quasi newton matrices and bundling mechanisms first we develop conceptual forms using reversal quasi newton formulae and we state their global and local convergence then to produce implementable versions we incorporate a bundle strategy together with a curve search no convergence results are given for the implementable versions however some numerical illustrations show their good behaviour even for large scale problems

multi lagrangians for integrable systems we propose a general scheme to construct multiple lagrangians for completely integrable nonlinear evolution equations that admit multi hamiltonian structure the recursion operator plays a fundamental role in this construction we use a conserved quantity higher lower than the hamiltonian in the potential part of the new lagrangian and determine the corresponding kinetic terms by generating the appropriate momentum map this leads to some remarkable new developments we show that nonlinear evolutionary systems that admit n fold first order local hamiltonian structure can be cast into variational form with 2n 1 lagrangians which will be local functionals of clebsch potentials this number increases to 3n 2 when the miura transformation is invertible furthermore we construct a new lagrangian for polytropic gas dynamics in 1 1 dimensions which is a free local functional of the physical field variables namely density and velocity thus dispensing with the necessity of introducing clebsch potentials entirely this is a consequence of bi hamiltonian structure with a compatible pair of first and third order hamiltonian operators derived from sheftel s recursion operator

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

some aspects of hamiltonian systems and symplectic algorithms abstract symplectic algorithms have been shown to be a right formalism for numerical computation of hamiltonian systems they are suitable to long time computation and of good qualitative properties these properties are ensured by the fact that a symplectic algorithm approximating to a time independent hamiltonian system can be regarded as a perturbed time dependent hamiltonian system of the original one that is a solution of a symplectic algorithm is a solution of a certain perturbed time dependent hamiltonian system evaluated at time discrete points moreover a linear symplectic algorithm approximating to a linear time independent hamiltonian system can be regarded as a perturbed time independent hamiltonian system so it has all properties that a linear hamiltonian system has based on these results stochastic webs and chaos appeared in symplectic simulation for hamiltonian systems are also discussed

on the rth dispersionless toda hierarchy factorization problem additional symmetries and some solutions for a family of poisson algebras parametrized by and an associated lie algebraic splitting we consider the factorization of given canonical transformations in this context we rederive the recently found rth dispersionless modified kp hierarchies and rth dispersionless dym hierarchies giving a new miura map among them we also found a new integrable hierarchy which we call the rth dispersionless toda hierarchy moreover additional symmetries for these hierarchies are studied in detail and new symmetries depending on arbitrary functions are explicitly constructed for the rth dispersionless kp rth dispersionless dym and rth dispersionless toda equations some solutions are derived by examining the imposition of a time invariance to the potential rth dispersionless dym equation for which a complete integral is presented and therefore an appropriate envelope leads to a general solution symmetries and miura maps are applied to get new solutions and solutions of the rth dispersionless modified kp equation

wronskian determinants the kp hierarchy and supersymmetric polynomials by using the wronskian representation of the solutions of the bilinear kp hierarchy a connection between hirota derivatives and supersymmetric polynomials is brought to light this correspondence is used in order to give an alternative construction of the hierarchy

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

the n m th korteweg de vries hierarchy and the associated w algebra a differential integrable hierarchy which is called the n m th korteweg de vries kdv hierarchy whose lax operator is obtained by properly adding m pseudo differential terms to the lax operator of the nth kdv hierarchy is discussed herein this new hierarchy contains both the higher kdv hierarchy and multifield representation of the kadomtsev petviashvili kp hierarchy as subsystems and naturally appears in multimatrix models the n 2m 1 coordinates or fields of this hierarchy satisfy two algebras of compatible poisson brackets which are local and polynomial each poisson structure generates an extended w1 and w algebras respectively w n m is called the generating algebra of the extended w algebra this algebra which corresponds with the second poisson structure shares many features of the usual wn algebra it is shown that there exist m distinct reductions of the n m th kdv hierarchy which are obtained by imposing suitable second class constraints the most drastic reduction corresponds

application of exp function method to the whitham broer kaup shallow water model using symbolic computation in this letter the exp function method is applied to the whitham broer kaup shallow water model with the help of symbolic computation several kinds of new solitary wave solutions are formally derived

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

a lie algebraic setting for miura maps related to an energy dependent linear problem we study the affine semi direct product vect s 1 s 1 1 and show how the coadjoint action on its extended dual leads immediately to a generalisation of the miura map and also to the existence of several compatible hamiltonian structures

on a primal proximal heuristic in discrete optimization lagrangian relaxation is useful to bound the optimal value of a given optimization problem and also to obtain relaxed solutions to obtain primal solutions it is conceivable to use a convexification procedure suggested by d p bertsekas in 1979 based on the proximal algorithm in the primal space the present paper studies the theory assessing the approach in the framework of combinatorial optimization our results indicate that very little can be expected in theory even though fairly good practical results have been obtained for the unit commitment problem

quasiperiodic solutions of the kadomtsev petviashvili equation via the multidimensional baker akhiezer function generated by the broer kaup hierarchy abstract the problem of constructing quasiperiodic solutions of the kadomtsev petviashvili equation is revisited following the idea of krichever and the technique developed by gesztesy and holden we shall explicitly construct the baker akhiezer function from the compatible solutions which satisfy infinite numbers of equations in the broer kaup hierarchy analytic and asymptotic properties will also be studied by introducing a special meromorphic function 1 p p from which we derive theta function representations for the baker akhiezer function and quasiperiodic solutions of the kadomtsev petviashvili equation

on the non local generalised symmetries of the benny kaup system the authors have extended the formalism of vinogradov and krasil shchik 1984 for the coupled system of benny kaup equations it is explicitly demonstrated that it is possible to generate new non local symmetries for this coupled equation if they consider d 1u integral u dx to be also in the space of independent variables

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

the algebra of q pseudodifferential symbols and the q wkp n algebra in this paper we continue with the program to explore the topography of the space of w type algebras in the present case the starting point is the work of khesin lyubashenko and roger on the algebra of q deformed pseudodifferential symbols and their associated integrable hierarchies the analysis goes on by studying the associated hamiltonian structures for which compact expressions are found the fundamental poisson brackets yield q deformations of wkp and related w type algebras which in specific cases coincide with the ones constructed by frenkel and reshetikhin the construction underlies a continuous correspondence between the hamiltonian structures of the toda lattice and the kp hierarchies

on a new class of completely integrable nonlinear wave equations ii multi hamiltonian structure the multi hamiltonian structure of a class of nonlinear wave equations governing the propagation of finite amplitude waves is discussed infinitely many conservation laws had earlier been obtained for these equations starting from a primary hamiltonian formulation of these equations the necessary and sufficient conditions for the existence of bi hamiltonian structure are obtained and it is shown that the second hamiltonian operator can be constructed solely through a knowledge of the first hamiltonian function the recursion operator which first appears at the level of bi hamiltonian structure gives rise to an infinite sequence of conserved hamiltonians it is found that in general there exist two different infinite sequences of conserved quantities for these equations the recursion relation defining higher hamiltonian structures enables one to obtain the necessary and sufficient conditions for the existence of the k 1 st hamiltonian operator which depends on the kth hamiltonian function the infinite sequence of conserved hamiltonians are common to all the higher hamiltonian structures the equations of gas dynamics are discussed as an illustration of this formalism and it is shown that in general they admit tri hamiltonian structure with two distinct infinite sets of conserved quantities the isothermal case of 1 is an exceptional one that requires separate treatment this corresponds to a specialization of the equations governing the expansion of plasma into vacuum which will be shown to be equivalent to poisson s equation in nonlinear acoustics

on the miura map between the dispersionless kp and dispersionless modified kp hierarchies the miura map between the dispersionless kp and dispersionless modified kp hierarchies is investigated it is shown that the miura map is canonical with respect to their bi hamiltonian structures moreover inspired by the works of takasaki and takebe the twistor construction of solution structure for the dispersionless modified kp hierarchy is given

the two component kaup kupershmidt equation the kaup kupershmidt equation is generalized to a system of equations in the same manner as the korteweg de vries equation is generalized to the hirota satsuma equation the gelfand dikii lax and hamiltonian formulation for this generalization is given the same construction is repeated for the constrained kadomtsev pietviashvili lax operator which leads to the four component kaup kupershmidt equation the modified version of the two component kaup kupershmidt equation is presented and analysed

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

dual applications of proximal bundle methods including lagrangian relaxation of nonconvex problems we exhibit useful properties of proximal bundle methods for finding min sf where f and s are convex we show that they asymptotically find objective subgradients and constraint multipliers involved in optimality conditions multipliers of objective pieces for max type functions and primal and dual solutions in lagrangian decomposition of convex programs when applied to lagrangian relaxation of nonconvex programs they find solutions to relaxed convexified versions of such programs numerical results are presented for unit commitment in power production scheduling

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

divide to conquer decomposition methods for energy optimization modern electricity systems provide a plethora of challenging issues in optimization the increasing penetration of low carbon renewable sources of energy introduces uncertainty in problems traditionally modeled in a deterministic setting the liberalization of the electricity sector brought the need of designing sound markets ensuring capacity investments while properly reflecting strategic interactions in all these problems hedging risk possibly in a dynamic manner is also a concern the fact of representing uncertainty and or competition of different companies in a multi settlement power market considerably increases the number of variables and constraints for this reason usually a trade off needs to be found between modeling and numerical tractability the more details are brought into the model the harder becomes the optimization problem for structured optimization and generalized equilibrium problems we explore some variants of solution methods based on lagrangian relaxation and on benders decomposition throughout we keep as a leading thread the actual practical value of such techniques in terms of their efficiency to solve energy related problems

on a non standard algebraic description of integrable nonlinear systems abstract we present a non standard algebraic description of some classical integrable systems in 1 1 dimensions resulting from interchanging the role of space and time variables bi or multi hamiltonian hierarchies obtained this way reduce naturally to the stationary subspaces giving some insight into the recursive structure of stationary flows

occupation times of the ornstein uhlenbeck process functional pca and evidence from electricity prices abstract we discuss the functional principal component analysis fpca of the occupation times of the ornstein uhlenbeck process for the eigenvalue problem of the covariance operator of the occupation times we derive the corresponding integral equation in the large time limit and we solve numerically for the principal components the formulation applies the path integral approach of feynman and kac the principal components are compared with those from empirical electricity price processes on energy markets the results indicate that fpca of the occupation times is a suitable tool in stochastic energy modeling to generate moderately sized scenario trees

multi hamiltonian structure of equations of hydrodynamic type the discussion of the hamiltonian structure of two component equations of hydrodynamic type is completed by presenting the hamiltonian operators for euler s equation governing the motion of plane sound waves of finite amplitude and another quasilinear second order wave equation there exists a doubly infinite family of conserved hamiltonians for the equations of gas dynamics that degenerate into one namely the benney sequence for shallow water waves infinite sequences of conserved quantities for these equations are also presented in the case of multicomponent equations of hydrodynamic type it is shown that kodama s generalization of the shallow water equations admits bi hamiltonian structure

classical extended conformal algebras associated with constrained kp hierarchy the conformal property of the second hamiltonian structure of constrained kp hierarchy derived by oevel and strampp is examined it is found that it naturally gives a family of nonlocal extended conformal algebras two examples of such algebras are given and it is found that they are similar to bilal s v algebra by taking a gauge transformation one can map the constrained kp hierarchy to kupershmidt s nonstandard lax hierarchy the second hamiltonian structure in this representation is considered it is shown that after mapping the lax operator to a pure differential operator the second structure becomes the sum of the second and the third gelfand dickey brackets defined by this differential operator the corresponding conformally covariant form of this differential operator is also given

generating scenario trees for multistage decision problems in models of decision making under uncertainty we often are faced with the problem of representing the uncertainties in a form suitable for quantitative models if the uncertainties are expressed in terms of multivariate continuous distributions or a discrete distribution with far too many outcomes we normally face two possibilities either creating a decision model with internal sampling or trying to find a simple discrete approximation of the given distribution that serves as input to the model this paper presents a method based on nonlinear programming that can be used to generate a limited number of discrete outcomes that satisfy specified statistical properties users are free to specify any statistical properties they find relevant and the method can handle inconsistencies in the specifications the basic idea is to minimize some measure of distance between the statistical properties of the generated outcomes and the specified properties we illustrate the method by single and multiple period problems the results are encouraging in that a limited number of generated outcomes indeed have statistical properties that are close to or equal to the specifications we discuss how to verify that the relevant statistical properties are captured in these specifications and argue that what are the relevant properties will be problem dependent

two well known properties of subgradient optimization the subgradient method is both a heavily employed and widely studied algorithm for non differentiable optimization nevertheless there are some basic properties of subgradient optimization that while well known to specialists seem to be rather poorly known in the larger optimization community this note concerns two such properties both applicable to subgradient optimization using the divergent series steplength rule the first involves convergence of the iterative process and the second deals with the construction of primal estimates when subgradient optimization is applied to maximize the lagrangian dual of a linear program the two topics are related in that convergence of the iterates is required to prove correctness of the primal construction scheme

new exact travelling wave solutions for the shallow long wave approximate equations based upon a generally projective riccati equation method which is a direct and unified algebraic method for constructing more general form travelling wave solutions of nonlinear partial differential equations and implemented in a computer algebraic system we consider the shallow long wave approximate equations new and more general form solutions are obtained including kink shaped solitons bell shaped solitons singular solitons and periodic solutions the properties of the new formal solitary wave solutions are shown by some figures

approximations for kdv equation as a hamiltonian system abstract kdv equation is one example of infinite dimensional hamiltonian systems in this paper we consider a fourier spectral approximation and a collocation approximation for the kdv equation with emphasis on its hamiltonian nature in particular we analyze the relations between the fourier spectral method and the collocation method and in this manner we naturally show that several conservation laws of the kdv equation can be preserved for the collocation method

rational solutions of the classical boussinesq hierarchy the classical boussinesq system and its associated higher order flows in the bilinear form are considered an infinite family of homogeneous rational solutions of this hierarchy is verified by the use of the wronskian technique developed by freeman and nimmo 1983

recursion operators and bi hamiltonian formulations of the landau lifshitz equation hierarchies using a polynomial pencil of lax pairs we give a new bi hamiltonian formulation for the landau lifshitz equation hierarchy and find a new recursion operator related to it

asynchronous level bundle methods in this paper we consider nonsmooth convex optimization problems with additive structure featuring independent oracles black boxes working in parallel existing methods for solving these distributed problems in a general form are synchronous in the sense that they wait for the responses of all the oracles before performing a new iteration in this paper we propose level bundle methods handling asynchronous oracles these methods require original upper bounds using upper models or scarce coordinations to deal with asynchronicity we prove their convergence using variational analysis techniques and illustrate their practical performance on a lagrangian decomposition problem

deformed harry dym and hunter zheng equations we study the deformed harry dym and hunter zheng equations with two arbitrary deformation parameters these reduce to various other known models in appropriate limits we show that both these systems are bi hamiltonian with the same hamiltonian structures they are integrable and belong to the same hierarchy corresponding to positive and negative flows we present the lax pair description for both the systems and construct the conserved charges of negative order from the lax operator for the deformed harry dym equation we construct the nonstandard lax representation for two special classes of values of the deformation parameters in general we argue that a nonstandard description will involve a pseudodifferential operator of infinite order

a primal proximal heuristic applied to the french unit commitment problem this paper is devoted to the numerical resolution of unit commitment problems with emphasis on the french model optimizing the daily production of electricity the solution process has two phases first a lagrangian relaxation solves the dual to find a lower bound it also gives a primal relaxed solution we then propose to use the latter in the second phase for a heuristic resolution based on a primal proximal algorithm this second step comes as an alternative to an earlier approach based on augmented lagrangian i e a dual proximal algorithm we illustrate the method with some real life numerical results a companion paper is devoted to a theoretical study of the heuristic in the second phase

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

super extensions of energy dependent schrodinger operators we consider the energy dependent super schrodinger linear problem sum limits i 0 n lambda i varepsilon i partial 2 u i psi eta i phi 0 sum limits i 0 n lambda i varepsilon i partial phi eta i psi 0 which is a direct generalization of the purely even energy dependent schrodinger equation discussed in 1 we show that the isospectral flows of that problem possess n 1 compatible hamiltonian structures we also extend a generalised factorisation approach of 2 to this case and derive a sequence ofn modifications for the 2n component systems thenth such modification possesses n n 1 compatible hamiltonian structures