a second wronskian formulation of the boussinesq equation a wronskian formulation leading to rational solutions is presented for the boussinesq equation it involves third order linear partial differential equations whose representative systems are systematically solved the resulting solutions formulas provide a direct but powerful approach for constructing rational solutions positon solutions and complexiton solutions to the boussinesq equation various examples of exact solutions of those three kinds are computed the newly presented wronskian formulation is different from the one previously presented by li et al which does not yield rational solutions

local torus actions modeled on the standard representation abstract we introduce the notion of a local torus action modeled on the standard representation for simplicity we call it a local torus action it is a generalization of a locally standard torus action and also an underlying structure of a locally toric lagrangian fibration for a local torus action we define two invariants called a characteristic pair and an euler class of the orbit map and prove that local torus actions are classified topologically by them as a corollary we obtain a topological classification of locally standard torus actions which includes the topological classifications of quasi toric manifolds by davis and januszkiewicz and of effective t 2 actions on four dimensional manifolds without nontrivial finite stabilizers by orlik and raymond we discuss locally toric lagrangian fibrations from the viewpoint of local torus actions we also investigate the topology of a manifold equipped with a local torus action when the euler class of the orbit map vanishes

the boussinesq equation revisited the continuous spectrum and soliton solutions for the boussinesq equation are investigated using the dressing method solitons demonstrate quite extraordinary behavior they may decay or form a singularity in a finite time formation of singularity collapse of solitons for the boussinesq equation was discovered several years ago systematic study of the solitonic sector is presented

symplectic surgery and the spinc dirac operator abstract letgbe a compact connected lie group and m a compact hamiltoniang space with moment mapj m g under the assumption that these data are pre quantizable one can construct an associated spinc dirac operator formula whose equivariant index yields a virtual representation ofg we prove a conjecture of guillemin and sternberg that if 0 is a regular value ofj the multiplicityn 0 of the trivial representation in the index space formula is equal to the index of the spinc dirac operator for the symplectic quotientm0 j 1 0 g this generalizes previous results for the case thatg tis abelian i e a torus

an extended noncommutative kp hierarchy introducing squared eigenfunctions in the moyal deformed lax equations generates an extended noncommutative kp nckp hierarchy the compatibility equations between the nckp flows and the extended nckp flows and the compatibility equations among the extended nckp flows themselves are constructed the n reductions and the k constraints are analyzed and thus the nckp hierarchy with self consistent sources the constrained nckp hierarchy and the constrained nckp hierarchy with self consistent sources are presented various illustrative examples are worked out together with their lax representations involving pseudodifferential lax operators

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

two new wronskian conditions for the 3 1 dimensional jimbo miwa equation abstract in this paper we obtain two kinds of sufficient conditions consisting of systems of linear partial differential equations which guarantee that the corresponding wroskian determinant solves the 3 1 dimensional jimbo miwa equation in the hirota bilinear form our results suggest that more general conditions could be derived by further study

negaton and positon solutions of the kdv and mkdv hierarchy we give a systematic classification and a detailed discussion of the structure motion and scattering of the recently discovered negaton and positon solutions of the korteweg de vries hierarchy there are two distinct types of negaton solutions which we label and where n 1 is the order of the wronskian used in the derivation for negatons the number of singularities and zeros is finite and they show very interesting time dependence the general motion is in the positive x direction except for certain negatons which exhibit one oscillation around the origin in contrast there is just one type of positon solution which we label for positons one gets a finite number of singularities for n odd but an infinite number for even values of n the general motion of positons is in the negative x direction with periodic oscillations negatons and positons retain their identities in a scattering process and their phase shifts are discussed we obtain a simple explanation of all phase shifts by generalizing the notions of mass and centre of mass to singular solutions finally it is shown that negaton and positon solutions of the kdv hierarchy can be used to obtain corresponding new solutions of the modified kdv hierarchy

construction of exact solutions to the jimbo miwa equation through backlund transformation and symbolic computation based on a simple transformation and with the aid of symbolic computation a backlund transformation relating the jimbo miwa equation and a system of linear partial differential equations is obtained which enables us to construct exact solutions of the jimbo miwa equation through the wronskian determinants of independent solutions of the linear system particularly explicit wronskian form n soliton solutions for the jimbo miwa equation are presented moreover the introduced transformation also helps to construct bi soliton like solutions of the jimbo miwa equation due to the arbitrary functions they contain the bi soliton like solutions can represent various waves such as classical cross line bi solitons curved bi solitons and bi soliton like breathers

torus fibrations and localization of index iii equivariant version and its applications this paper is the third of the series concerning the localization of the index of dirac type operators in our previous papers we gave a formulation of index of diractype operators on open manifolds under some geometric setting whose typical example was given by the structure of a torus fiber bundle on the ends of the open manifolds we introduce two equivariant versions of the localization as an application we give a proof of guillemin sternberg s quantization conjecture in the case of torus action

rational solutions to a kdv like equation two classes of rational solutions to a kdv like nonlinear differential equation are constructed the basic object is a generalized bilinear differential equation based on a prime number p 3 a conjecture is made that the two presented classes of rational solutions contain all rations solutions to the considered kdv like equation which are generated from polynomial solutions to the corresponding generalized bilinear equation

soliton solution to bkp equation in wronskian form we find out a group of identities named wronskian identities of bilinear kp hierarchy here as well as two useful properties of d operator this makes it possible to easily search for some new wronskian solutions for pde which owns bilinear form and to simplify the process of the proof as an application of this new method we propose the first wronskian condition for the bkp i equation and for the bkp ii equation respectively in addition we also discuss several generalized bkp equations and obtain their wronskian conditions

solitonic solutions for a variable coefficient variant boussinesq system in the long gravity waves variable coefficient variant boussinesq vcvb system is able to describe the nonlinear and dispersive long gravity waves traveling in two horizontal directions with varying depth in this paper with symbolic computation a lax pair associated with the vcvb system under some constraints for variable coefficients is derived and based on the lax pair two sorts of basic darboux transformations are presented by applying the darboux transformations some solitonic solutions are obtained with the relevant constraints given in the text in addition the vcvb system is transformed to a variable coefficient broer kaup system solitonic solutions and procedure of getting them could be helpful to solve the nonlinear and dispersive problems in fluid dynamics

complexiton solutions to integrable equations abstract complexiton solutions or complexitons for short are exact solutions newly introduced to integrable equations are newly introduced starting with the solution classification for a linear differential equation the korteweg de vries equation and the toda lattice equation are considered as examples to exhibit complexiton structures of nonlinear integrable equations the crucial step in the solution process is to apply the wronskian and casoratian techniques for hirota s bilinear equations correspondence between complexitons of the korteweg de vries equation and complexitons of the toda lattice equation is provided

the structure of the rational solutions to the boussinesq equation rational solutions of the one dimensional boussinesq equation both with zero and nonzero boundary conditions at the infinity are obtained by reducing the known solutions of the kadomtsev petviashvili equation the structure of the found solutions generalizes a family of rational solutions to the korteweg de vries equation to the case of two wave processes

n fold darboux transformation and solitonic interactions of a variable coefficient generalized boussinesq system in shallow water abstract under consideration in this paper is a variable coefficient generalized boussinesq system for the long weakly nonlinear and weakly dispersive surface waves in shallow water with the aid of symbolic computation n fold darboux transformation n dt is constructed for that system analytic solutions of the system are obtained via the n dt elastic interactions of three bell shaped and periodic bell shaped solitons are obtained fusion interactions and periodic fusion fission interactions of the solitary waves are graphically analyzed which are inelastic

torus fibrations and localization of index ii local index for acyclic compatible system we give a framework of localization for the index of a dirac type operator on an open manifold suppose the open manifold has a compact subset whose complement is covered by a family of finitely many open subsets each of which has a structure of the total space of a torus bundle under an acyclic condition we define the index of the dirac type operator by using the witten type deformation and show that the index has several properties such as excision property and a product formula in particular we show that the index is localized on the compact set

complexiton solutions of the toda lattice equation a set of coupled conditions consisting of differential difference equations is presented for casorati determinants to solve the toda lattice equation one class of the resulting conditions leads to an approach for constructing complexiton solutions to the toda lattice equation through the casoratian formulation an analysis is made for solving the resulting system of differential difference equations thereby providing the general solution yielding eigenfunctions required for forming complexitons moreover a feasible way is presented to compute the required eigenfunctions along with examples of real complexitons of lower order

traveling wave solutions and conservation laws for nonlinear evolution equation in this work the riccati bernoulli sub ordinary differential equation and modified tanh coth methods are used to reach soliton solutions of the nonlinear evolution equation we acquire new types of traveling wave solutions for the governing equation we show that the equation is nonlinear self adjoint by obtaining suitable substitution therefore we construct conservation laws for the equation using new conservation theorem the obtained solutions in this work may be used to explain and understand the physical nature of the wave spreads in the most dispersive medium the constraint condition for the existence of solitons is stated some three dimensional figures for some of the acquired results are illustrated

positons for the toda lattice and related spectral problems the concept of positons is introduced for the toda lattice equation it is shown that these multiparametric oscillating and slowly decaying solutions when inserted as potentials in the finite difference schrodinger equation of the corresponding lax pair lead to a trivial s matrix the resulting eigenvalues are embedded in the continuous spectrum of this infinite jacobi matrix the singularities connected with the one positon solution are discussed and compared with those of the positons of continuous integrable models the special features of the soliton positon interaction are analysed

an analytic proof of the geometric quantization conjecture of guillemin sternberg we present a direct analytic approach to the guillemin sternberg conjecture gs that geometric quantization commutes with symplectic reduction which was proved recently by meinrenken m1 m2 and vergne v1 v2 et al besides providing a new proof of this conjecture our methods also lead immediately to further extensions in various contexts

asymptotic solutions for singularly perturbed boussinesq equations abstract we consider a family of singularly perturbed boussinesq equations we obtain a rational weak solution to the classical boussinesq equation and demonstrate that this solution can be used to construct perturbation solutions for singularly perturbed high order boussinesq equations these solutions take the form of an algebraic function which behaves similarly to a peakon and which decays as time becomes large we show that approximate solutions obtained via perturbation for the singularly perturbed models are asymptotic to the true solutions as the residual errors rapidly decay away from the origin

a stiffness matrix extrapolation strategy for nonlinear analysis the use of the direct stiffness method of analysis of geometrically nonlinear structural problems was pioneered by the work of turner dill martin and melosh l basically the solution of a nonlinear problem reduces to that of tracing a nonlinear load displacement path by solving a system of nonlinear equations this problem is however normally very involved with multi degrees of freedom and its solution occurs in the multi dimensional space a host of solution strategies for tackling nonlinear structural analysis has since evolved and several reviews on the subject can be found in the literature 2 6 bergan et al 7 found it convenient to classify the solution techniques according to their mathematical formulation from which 3 different categories could be identified for the first category the problem is represented by an energy potential use is then made of the algorithms for unconstrained minimization to search for the stationary value of the energy functional techniques such as the random search method 8 and powell s method of conjugate directions 9 make use of the object function only the method of steepest descent 8 and the conjugate gradient method lo in addition make use of the gradient first derivative while the quasi newton ll 131 secant newton 14 and newton raphson 15 methods require the evaluation of the 2nd derivatives as well the second category of solution techniques is based on the equilibrium conditions which are usually formulated by setting to zero the variation of the total potential energy of the system the method of functional iteration 1161 belongs to this group but it is characterised by poor convergence properties the most powerful of the solution techniques here is the newton raphson algorithm 17 in the third category use is made of the incremental form of the equilibrium equations the easiest technique is the euler cauchy method or incremental stiffness procedure l use of more accurate integration schemes such as the runge kutta 18 or the predictor corrector 18 methods will reduce the drifting effect which is associated with the euler integration the most popular techn ique is to combine both the incremental and iterative methods a simple euler incrementation is first applied and then either the full or modified newton raphson method is evoked to iterate to equilibrium 7 the above techniques as they stand are unsatisfactory for tracing the post limit response of the structure strategies which allow one to follow the post limit behaviour are the method of artificial springs 19 the current stiffness parameter method of bergan 7 the displacement

multiplicities formula for geometric quantization part ii 1 introduction let p be a compact manifold let h be a compact lie group acting on the right on p we assume that the stabilizer of each element y e p is a finite subgroup of h the space m p h is an orbifold and every orbifold can be presented this way if h acts freely then m is a manifold if ae graded space of h invariant solutions of the h horizontal dirac operator on p twisted by the line bundle let m o 9 be the moment map for the g action assume 0 is a regular value of let mred be the reduced orbifold of m that is mred o g consider the reduced orbifold line bundle rd lu lto g on m oa in the ease where both g andh are torus we prove here the formula

new analytical positon negaton and complexiton solutions of a coupled kdv mkdv system new positon negaton and complexiton solutions of a coupled kdv mkdv system are constructed directly from zero seed solution by means of darboux transformation these new exact solutions are all analytical with different spectral parameters and constants selection