quantum monodromy and nonconcentration near a closed semi hyperbolic orbit for a large class of semiclassical operators p h z which includes schrodinger operators on manifolds with boundary we construct the quantum monodromy operator m z associated to a periodic orbit of the classical flow using estimates relating m z and p h z we prove semiclassical estimates for small complex perturbations of p h z in the case is semi hyperbolic as our main application we give logarithmic lower bounds on the mass of eigenfunctions away from semi hyperbolic orbits of the associated classical flow as a second application of the monodromy operator construction we prove if is an elliptic orbit then p h admits quasimodes which are well localized near

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

grammian solutions and pfaffianization of a non isospectral and variable coefficient kadomtsev petviashvili equation abstract in this paper we first present the grammian determinant solutions to the non isospectral and variable coefficient kadomtsev petviashvili vckp equation then by using the pfaffianization procedure of hirota and ohta a new non isospectral and variable coefficient integrable coupled system is generated moreover gramm type pfaffian solutions of the pfaffianized system are proposed

backlund transformation in bilinear form for a higher order nonlinear schrodinger equation backlund transformation in bilinear form is presented for a higher order nonlinear schrodinger equation which describes the propagation of ultrashort light pulses in optical fibers with symbolic computation and starting from the backlund transformation the analytical soliton solution is obtained from a trivial solution and the inverse scattering transform scheme is also derived furthermore the n soliton solution in double wronskian form is given and the value of the arbitrary constant appearing in the backlund transformation is determined for a transformation between the n 1 and n soliton solutions the results obtained from the backlund transformation might be valuable in optical communications

painleve expansions for nonintegrable evolution equations abstract the wtc method is applied to a variety of nonintegrable nonlinear evolution equations and shown to provide a systematic procedure for obtaining special solutions some of the special techniques developed for integrable systems are also shown to be applicable to nonintegrable equations

a combinatorial approach to topological quantum field theories and invariants of graphs the combinatorial state sum of turaev and viro for a compact 3 manifold in terms of quantum 6j symbols is generalized by introducing observables in the form of coloured graphs they satisfy braiding relations and allow for surgeries and a discussion of cobordism theory application of these techniques give the dimension and an explicit basis for the vector space of the topological quantum field theory associated to any riemann surface with arbitrary coloured punctures

the non local delta problem and 2 1 dimensional soliton equations a method of constructing 2 1 dimensional non linear integrable equations and their solutions by means of the non local delta problem is developed a basic set of equations is obtained by using different normalisations of the non local delta problem and the lagrangian of the set is found other integrable equations which are degenerate cases of the basic set are also lagrangian

multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method a new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system compared with most of the existing tanh methods the jacobi elliptic function method or other sophisticated methods the proposed method not only gives new and more general solutions but also provides a guideline to classify the various types of the travelling wave solutions according to the values of some parameters the solutions obtained in this paper include a kink shaped and bell shaped soliton solutions b rational solutions c triangular periodic solutions and d jacobi and weierstrass doubly periodic wave solutions among them the jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition the efficiency of the method can be demonstrated on a large variety of nonlinear evolution equations such as those considered in this paper kdv mkdv ito s fifth mkdv hirota nizhnik novikov veselov broer kaup generalized coupled hirota satsuma coupled schrodinger kdv 2 1 dimensional dispersive long wave 2 1 dimensional davey stewartson equations in addition as an illustrative sample the properties of the soliton solutions and jacobi doubly periodic solutions for the hirota equation are shown by some figures the links among our proposed method the tanh method extended tanh method and the jacobi elliptic function method are clarified generally

gauge transformation between the first order nonisospectral and isospectral heisenberg hierarchies abstract by means of the ablowitz kaup newell segur system with symbolic computation the gauge transformation between the first order nonisospectral and isospectral heisenberg spectral problems is obtained correspondingly the equivalence relation between the first order nonisospectral and isospectral heisenberg hierarchies is given in addition further applications and potential research directions of our method are discussed

novel rule for quantizing chaos numerical tests of a novel semiclassical quantization rule are carried out for three strongly chaotic systems the hyperbolic billiard artin s billiard and the hadamard gutzwiller model the results demonstrate that this novel rule is very effective and capable of generalizing sensible approximations to the quantal energy levels

backlund transformation and multi soliton solutions for a 2 1 dimensional korteweg de vries system via symbolic computation abstract in this paper the 2 1 dimensional korteweg de vries system is symbolically investigated by the bilinear method the n soliton solution is presented then based on the backlund transformation in bilinear form a new backlund transformation is obtained and new representation of the n soliton solution is derived a class of novel multi soliton solutions are obtained by the new backlund transformation and the availability of symbolic computation is demonstrated

an extension of the wronskian technique for the multicomponent wronskian solution to the vector nonlinear schrodinger equation in this paper the wronskian technique is applied to the vector nonlinear schrodinger equation with arbitrary m components which arises from some applications in the multimode fibers photorefractive materials and bose einstein condensates via the iterative algorithm based on the darboux transformation the m 1 component wronskian solution is generated from the zero solution the verification of the solution is finished by using the m 1 component wronskian notation and new determinantal identities with a set of n linearly independent solutions of the zero potential lax pair the m 1 component wronskian solution is found to be the representation of the bright n soliton solution which contains m 1 n parameters for characterizing the asymptotic behavior of the generic bright n soliton solution an algebraic procedure is derived to obtain the explicit expressions of asymptotic solitons as t

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

solitary wave and periodic wave solutions for the thermally forced gravity waves in atmosphere by introducing a new transformation a new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system which extends fan s direct algebraic method to the case when r 4 the solutions of a first order nonlinear ordinary differential equation with a higher degree nonlinear term and fan s direct algebraic method of obtaining exact solutions to nonlinear partial differential equations are applied to the combined kdv mkdv gkdv equation which is derived from a simple incompressible non hydrostatic boussinesq equation with the influence of thermal forcing and is applied to investigate internal gravity waves in the atmosphere as a result by taking advantage of the new first order nonlinear ordinary differential equation with a fifth degree nonlinear term and an eighth degree nonlinear term periodic wave solutions associated with the jacobin elliptic function and the bell and kink profile solitary wave solutions are obtained under the effect of thermal forcing most importantly the mechanism of propagation and generation of the periodic waves and the solitary waves is analysed in detail according to the values of the heating parameter which show that the effect of heating in atmosphere helps to excite westerly or easterly propagating periodic internal gravity waves and internal solitary waves in atmosphere which are affected by the local excitation structures in atmosphere in addition as an illustrative sample the properties of the solitary wave solution and jacobin periodic solution are shown by some figures under the consideration of heating interaction

on the accuracy of the semiclassical trace formula the semiclassical trace formula provides the basic construction from which one derives the semiclassical approximation for the spectrum of quantum systems which are chaotic in the classical limit when the dimensionality of the system increases the mean level spacing decreases as while the semiclassical approximation is commonly believed to provide an accuracy of order independently of d if this was true the semiclassical trace formula would be limited to systems in only in this work we set out to define proper measures of the semiclassical spectral accuracy and to propose theoretical and numerical evidence to the effect that the semiclassical accuracy measured in units of the mean level spacing depends only weakly if at all on the dimensionality detailed and thorough numerical tests were performed for the sinai billiard in two and three dimensions substantiating the theoretical arguments

darboux transformation and hamiltonian structure for the jaulent miodek hierarchy abstract starting from the jaulent miodek spectral problem we derive the associated hierarchy of nonlinear evolution equations in this paper it is shown that this hierarchy is completely integrable in the liouville sense and possesses the hamiltonian structure moreover by virtue of symbolic computation two types of darboux transformations for the whole hierarchy are explicitly constructed which enables us to find the new soliton like shock and anti shock solutions for the jaulent miodek hierarchy figures are presented to discuss the properties of the new soliton like solutions

symbolic dynamics and periodic orbits for the cardioid billiard the periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics the corresponding partition is mapped to a topologically well ordered symbol plane in the symbol plane the pruning front is obtained from orbits running either into or through the cusp we show that all periodic orbits correspond to maxima of the lagrangian and give a complete list up to code length 15 the symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classified using symmetry lines we show that there exists an infinite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned all these orbits are related to finite orbits starting and ending in the cusp we obtain an analytical estimate of the kolmogorov sinai entropy and find a good agreement with the numerically calculated value and the one obtained by averaging periodic orbits furthermore the statistical properties of periodic orbits are investigated

nonisospectral integrable nonlinear equations with external potentials and their gbdt solutions auxiliary systems for matrix nonisospectral equations including coupled nls with external potential and kdv with variable coefficients were introduced explicit up to the inversion of finite matrices solutions of nonisospectral equations were constructed using the gbdt version of the backlund darboux transformation

state sum invariants of compact 3 manifolds with boundary and 6j symbols the authors extend the combinatorial construction of invariants of smooth compact closed 3 manifolds as given by turaev and viro 1991 to obtain invariants of 3 manifolds and with a boundary the techniques uses quantum 6j symbols associated to the quantized universal enveloping algebra uq sl 2 c q a root of unity to give a construction of a state sum for given triangulation this state sum is then invariant under subdivisions and isotopes of the boundary their methods also lead to a simplified proof of the main result of turaev and viro that the state sum is independent of the triangulation and hence gives rise to an invariant of the manifold they use surgery to calculate the state sum for some closed manifolds their results also confirm a finding of turaev 1990 which relates this theory to the topological quantum field theory with a chern simons action in the sense of witten 1989

multi soliton solutions and a backlund transformation for a generalized variable coefficient higher order nonlinear schrodinger equation with symbolic computation abstract in this paper by virtue of symbolic computation the investigation is made on a generalized variable coefficient higher order nonlinear schrodinger equation with varying higher order effects and gain or loss which can describe the femtosecond optical pulse propagation in a monomode dielectric waveguide a modified dependent variable transformation is introduced into the bilinear method to transform such an equation into a variable coefficient bilinear form based on the formal parameter expansion technique the multi soliton solutions of this equation are obtained through the bilinear form under sets of parametric constraints a backlund transformation in bilinear form is also obtained for the first time in this paper finally discussions on the analytic soliton solutions are given and various propagation situations are illustrated

the behaviour of eigenstates of arithmetic hyperbolic manifolds in this paper we study some problems arising from the theory of quantum chaos in the context of arithmetic hyperbolic manifolds we show that there is no strong localization scarring onto totally geodesic submanifolds arithmetic examples are given which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom

gauge transformations and symmetries of integrable systems we analyse several integrable systems in zero curvature form within the framework of invariant gauge theory in the drinfeld sokolov gauge we derive a two parameter family of nonlinear evolution equations which as special cases include the korteweg de vries kdv and harry dym hd equations we find residual gauge transformations which lead to infinitesimal symmetries of this family of equations for kdv and hd equations we find an infinite hierarchy of such symmetry transformations and we investigate their relation with local conservation laws constants of the motion and the bi hamiltonian structure of the equations applying successive gauge transformations of miura type we obtain a sequence of gauge equivalent integrable systems among them the modified kdv and calogero kdv equations

bi hamiltonian aspects of a matrix harry dym hierarchy we study the harry dym hierarchy of nonlinear evolution equations from the bi hamiltonian view point this is done by using the concept of an s hierarchy it allows us to define a matrix harry dym hierarchy of commuting hamiltonian flows in two fields that projects onto the scalar harry dym hierarchy we also show that the conserved densities of the matrix harry dym equation can be found by means of a riccati type equation

derivation and implicit solution of the harry dym equation and its connections with the korteweg de vries equation the harry dym equation which is related to the classical string problem is derived in three different ways an implicit cusp solitary wave solution is constructed via a simple direct method the existing connections between the harry dym and the korteweg de vries equations are uniformised and simplified and transformations between their respective solutions are carried out explicitly whenever possible physical insights are provided

observables in the turaev viro and crane yetter models we define an invariant of graphs embedded in a 3 manifold and a partition function for 2 complexes embedded in a triangulated 4 manifold by specifying the values of variables in the turaev viro and crane yetter state sum models in the case of the three dimensional invariant we prove a duality formula relating its fourier transform to another invariant defined via the colored jones polynomial in the case of the four dimensional partition function we give a formula for it in terms of a regular neighborhood of the 2 complex and the signature of its complement some examples are computed which show that the partition function determines an invariant which can detect non locally flat surfaces in a 4 manifold

soliton solutions for a negative order non isospectral akns equation abstract in this paper the bilinear form of a negative order non isospectral akns equation is given the soliton solutions are obtained through hirota s direct method and the wronskian technique respectively

the exact solutions for the nonisospectral akns hierarchy through the inverse scattering transform abstract exact solutions for the nonisospectral akns hierarchy are obtained through the inverse scattering transform some explicit forms of these solutions are further given reductions are also considered exact solutions of some nonisospectral hierarchies such as the nonisospectral mkdv hierarchy the nonisospectral kdv hierarchy the nonisospectral nonlinear schrodinger hierarchy and the nonisospectral sine gordon hierarchy are obtained

nonpropagating solitons of the variable coefficient and nonisospectral korteweg de vries equation the method of inverse scattering is extended to solve the initial value problem of a variable coefficient and nonisospectral korteweg de vries equation with time varying boundary condition one and two soliton solutions are examined in detail by an appropriate decomposition soliton interactions and asymptotic behaviors are investigated oscillating and asymptotically standing two soliton solutions are discussed

symbolic computation backlund transformation and analytic n soliton like solution for a nonlinear schrodinger equation with nonuniformity term from certain space laboratory plasmas abstract with symbolic computation a bilinear backlund transformation is presented for a nonlinear schrodinger equation with nonuniformity term from certain space laboratory plasmas and correspondingly the one soliton like solution is derived from the backlund transformation simultaneously the n soliton like solution in double wronskian form is also given besides the authors verify that the n 1 and n soliton like solutions satisfy the backlund transformation the results obtained in this paper might be valuable for the study of the nonuniform media

the higher times approach to multisoliton solutions of the harry dym equation an n cusp soliton solution of the harry dym equation hde is constructed explicitly the proposed approach is based on the algebraic effectivization of the reciprocal links between the systems consisting of n higher order analogues of the hde and the corresponding analogues of the kdv equation an important role in the discussed scheme belongs to the reciprocal auto backlund transformation for the dym hierarchy

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

dynamics of a complex interface we study the motion of the interface between two fluids in a pressure field a more viscous fluid surrounds a finite region filled with a less viscous fluid whose pressure is constant the more viscous fluid is incompressible and moves with a velocity proportional to the gradient of its pressure the pressure jump across the interface between the fluids is proportional to the curvature of the interface the proportionality constant is the surface tension in the two dimensional case the interface can be described by a complex function which is analytic in the exterior of the unit circle the dynamics are governed by a nonlocal nonlinear equation for this function if all the singularities of the function are situated near the origin then the nonlocal interactions contribute little to the evolution of the singularities in this case the nonlocal evolution equation can be approximated by a local nonlinear one we prove that this equation has solutions with certain uniform behavior as the surface tension is allowed to become vanishingly small

lax representations and lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations a method of constructing lax representations for isospectral and nonisospectral hierarchies of evolution equations is proposed it is shown that under some simple but essential conditions the corresponding lax operators may constitute infinite dimensional lie algebras with respect to the binary operation given in this paper furthermore the detailed analyses for the kdv couple the akns couple and a new couple of isospectral and nonisospectral hierarchies of integrable equations are presented as examples

on the solutions of the dkp equation the nonlinear riemann hilbert problem longtime behaviour implicit solutions and wave breaking we have recently solved the inverse scattering problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions including the second heavenly equation of plebanski and the dispersionless kadomtsev petviashvili dkp equation we showed in particular that the associated inverse problems can be expressed in terms of nonlinear riemann hilbert problems on the real axis in this paper we make use of the nonlinear riemann hilbert problem of dkp i to construct the longtime behaviour of the solutions of its cauchy problem ii to characterize a class of implicit solutions iii to elucidate the spectral mechanism causing the gradient catastrophe of localized solutions of dkp at finite time as well as in the longtime regime and the corresponding universal behaviours near breaking

trace formulae for three dimensional hyperbolic lattices and application to a strongly chaotic tetrahedral billiard abstract this paper is devoted to the quantum chaology of three dimensional systems a trace formula is derived for compact polyhedral billiards which tessellate the three dimensional hyperbolic space of constant negative curvature the exact trace formula is compared with gutzwiller s semiclassical periodic orbit theory in three dimensions and applied to a tetrahedral billiard being strongly chaotic geometric properties as well as the conjugacy classes of the defining group are discussed the length spectrum and the quantal level spectrum are numerically computed allowing the evaluation of the trace formula as is demonstrated in the case of the spectral staircase n e which in turn is successfully applied in a quantization condition

lie algebraic structure for the akns system a new approach is proposed to construct the master symmetry and to prove its lie algebraic structure for the akns system

the use of backlund transformations in obtaining n soliton solutions in wronskian form the wronskian formulation of the n soliton solutions of various nonlinear evolution equations modified korteweg de vries with both zero and non zero asymptotic conditions sine gordon korteweg de vries kadomtsev petviashvili and boussinesq equations are obtained by inductive use of backlund transformations in hirota s 1980 bilinear notation

semiclassical non concentration near hyperbolic orbits for a large class of semiclassical pseudodifferential operators including schrodinger operators p h h2 g v x on compact riemannian manifolds we give logarithmic lower bounds on the mass of eigenfunctions outside neighbourhoods of generic closed hyperbolic orbits more precisely we show that if a is a pseudodifferential operator which is microlocally equal to the identity near the hyperbolic orbit and microlocally zero away from the orbit then u c log 1 h h p h u clog 1 h i a u this generalizes earlier estimates of colin de verdiere and parisse y colin de verdiere b parisse equilibre instable en regime semi classique i concentration microlocale comm partial differential equations 19 1994 1535 1563 equilibre instable en regime semi classique ii conditions de bohr sommerfeld ann inst h poincare phys theor 61 1994 347 367 obtained for a special case and of burq and zworski n burq m zworski geometric control in the presence of a black box j amer math soc 17 2004 443 471 for real hyperbolic orbits

constructing of new exact solutions to the gkdv mkdv equation with any order nonlinear terms by g g expansion method abstract the g g expansion method is introduced to construct more general exact travelling wave solutions to the generalized kdv mkdv equation with any order nonlinear terms as a result hyperbolic function solution trigonometric function solution and rational solution with parameters are obtained when the parameters take up special values the solitary waves are also derived from the travelling wave solutions the g g expansion method presents a wider applicability for handling nonlinear wave equations

a rational generalization of fan s method and its application to generalized shallow water wave equations in this paper we employ a rational expansion to generalize fan s method for exact travelling wave solutions for nonlinear partial differential equations pdes to verify the reliability of the proposed method the generalized shallow water wave gsww equation has been investigated as an example kinds of new exact travelling wave solutions of a rational form have been obtained this indicates that the proposed method provides a more general result for exact solution of nonlinear equations

delta dressing and exact solutions for the 2 1 dimensional harry dym equation exact implicit solutions with functional parameters of the 2 1 dimensional harry dym hd equation are constructed by the delta dressing method the interrelation between the hd and modified kadomtsev petviashvili mkp equations is established within the framework of the delta dressing method knowledge of the mkp eigenfunctions allows the solutions of the 2 1 dimensional hd equation to be represented in a simple parametric form

topological quantum field theory and invariants of graphs for quantum groups on the basis of generalized 6j symbols we give a formulation of topological quantum field theories for 3 manifolds including observables in the form of coloured graphs it is shown that the 6j symbols associated with deformations of the classical groups at primitive even roots of unity provide examples of this construction calculational methods are developed which in particular yield the dimensions of the state spaces as well as a rather simple proof of the relation previously found by turaev and walker for the case ofu q sl 2 c between these models and corresponding ones based on the ribbon graph construction of reshetikhin and turaev

on the non isospectral kadomtsev petviashvili equation with self consistent sources abstract a new procedure called source generation is applied to construct non isospectral soliton equations with self consistent sources as results the non isospectral kadomtsev petviashvili equation with self consistent sources kpescs and its gramm type determinant solutions are obtained furthermore the non isospectral pfaffianized kp equation with self consistent sources is constructed this coupled system can not only be reduced to the non isospectral pfaffianized kp equation but also reduced to the non isospectral kpescs

arithmetical chaos and violation of universality in energy level statistics a class of strongly chaotic systems revealing a strange arithmetical structure is discussed whose quantal energy levels exhibit level attraction rather than repulsion as an example the nearest neighbor level spacings for artin s billiard have been computed in a large energy range it is shown that the observed violation of universality has its root in the existence of an infinite number of hermitian operators hecke operators which commute with the hamiltonian and generate nongeneric correlations in the eigenfunctions

soliton solutions to the 3rd nonisospectral akns system abstract bilinear form of the nonisospectral akns equation is given the n soliton solutions and double wronskian solution are obtained through hirota s direct method and wronskian technique respectively the nonisospectral mkdv equation and its multi soliton solutions are presented by reducing

backlund transformation lax pair and solutions for the caudrey dodd gibbon equation by using bell polynomials and symbolic computation we investigate the caudrey dodd gibbon equation analytically through a generalization of bells polynomials its bilinear form is derived based on which the periodic wave solution and soliton solutions are presented and the soliton solutions with graphic analysis are also given furthermore backlund transformation and lax pair are derived via the bells exponential polynomials finally the ablowitz kaup newell segur system is constructed

painleve analysis and new analytic solutions for variable coefficient kadomtsev petviashvili equation with symbolic computation abstract the variable coefficient kadomtsev petviashvili kp equation is hereby under investigation painleve analysis is given out and an auto backlund transformation is presented via the truncated painleve expansion based on the auto backlund transformation new analytic solutions are given including the soliton like and periodic solutions it is also reduced to a 1 1 dimensional partial differential equation via classical lie group method and the painleve i equation by ck direct method

on the periodic orbits of a strongly chaotic system abstract a point particle sliding freely on a two dimensional surface of constant negative curvature hadamard gutzwiller model exemplifies the simplest chaotic hamiltonian system exploiting the close connection between hyperbolic geometry and the group su 1 1 1 we construct an algorithm symboliv dynamics which generates the periodic orbits of the system for the simplest compact riemann surface having as its fundamental group the octagon group we present an enumeration of more than 206 million periodic orbits for the length of the n th primitive periodic orbit we find a simple expression in terms of algebraic numbers of the form m 2 n m n n are governed by a particular beatty sequence which reveals a strange arithmetical structure of chaos knowledge of the length spectrum is crucial for quantization via the selberg trace formula periodic orbit theory which in turn is expected to unravel the mystery of quantum chaos

soliton solutions of the korteweg de vries and the kadomtsev petviashvili equations the wronskian technique the well known soliton solutions of the kadomtsev petviashvili equations are written in terms of determinants of wronskian form by using this compact representation together with the hirota bilinear form of the equations it is demonstrated by elementary algebraic methods that the n soliton solution satisfies the evolution equation and the n and n 1 soliton solutions satisfy the associated backlund transformation the relation of these results to the eigensolutions of the inverse scattering method and to the more usual representation of the n soliton solution is also given

the approach to the dispersionless 2 1 harry dym hierarchy a approach is adopted to study the dispersionless harry dym dhd hierarchy moreover this formulism is applied to construct some explicit solutions of the dhd hierarchy

invariants of spin networks embedded in three manifolds we study the invariants of spin networks embedded in a three dimensional manifold which are based on the path integral for su 2 bf theory these invariants appear naturally in loop quantum gravity and have been defined as spin foam state sums by using the chain mail technique we give a more general definition of these invariants and show that the state sum definition is a special case this provides a rigorous proof that the state sum invariants of spin networks are topological invariants we derive various results about the bf theory spin network invariants and we find a relation with the corresponding invariants defined from chern simons theory i e the witten reshetikhin turaev invariants we also prove that the bf theory spin network invariants coincide with v turaev s definition of invariants of coloured graphs embedded in 3 manifolds and thick surfaces constructed by using shadow world evaluations our framework therefore provides a unified view of these invariants

the n soliton solutions for the non isospectral mkdv equation the bilinear form of the non isospectral mkdv equation is given and the exact n soliton solutions are obtained through hirota method and wronskian technique respectively c

solitonic propagation and interaction for a generalized variable coefficient forced korteweg de vries equation in fluids under investigation is a generalized variable coefficient forced korteweg de vries equation in fluids and other fields from the bilinear form of such equation the n soliton solution and a type of analytic solution are constructed with symbolic computation analytic analysis indicates that 1 dispersive and dissipative coefficients affect the solitonic velocity 2 external force term affects the solitonic velocity and background 3 line damping coefficient and some parameters affect the solitonic velocity background and amplitude solitonic propagation and interaction can be regarded as the combination of the effects of various variable coefficients according to a constraint among the nonlinear dispersive and line damping coefficients in this paper the possible applications of our results in the real world are also discussed in three aspects i e solution with the constraint solution without the constraint and approximate solution

the korteweg de vries hierarchy as dynamics of closed curves in the plane the korteweg de vries modified korteweg de vries and harry dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane these purely local dynamics conserve an infinite number of global geometric properties of the curves such as perimeter and enclosed area several techniques used to study these integrable systems are shown to have simple differential geometric interpretations connections with incompressible inviscid fluid flow in two dimensions are suggested

exterior differential expression of the 1 1 dimensional nonlinear evolution equation with lax integrability abstract a 1 1 dimensional nonlinear evolution equation with lax integrability is investigated in this paper with the aid of differential forms and exterior differentials equivalent definition of the lax integrability is given relation between the lax integrability and complete integrability is discussed geometric interpretation of the lax equation is presented gauge transformation and darboux transformation are restated in terms of the differential forms and exterior differentials if i s is a darboux transformation between two lax equations the system that the matrix s should satisfy is given where is the spectral parameter and i is the identity matrix the completely integrable condition of that system is obtained according to the frobenius theorem and it is shown that s h h 1 satisfies that completely integrable condition with h and as two matrices

solitonic interactions and double wronskian type solutions for a variable coefficient variant boussinesq model in the long gravity water waves abstract under investigation in this paper is a variable coefficient variant boussinesq vcvb model for the nonlinear and dispersive long gravity waves in shallow water traveling in two horizontal directions with varying depth connection between the vcvb model and a variable coefficient ablowitz kaup newell segur system is revealed under certain constraints with the help of the symbolic computation multi solitonic solutions in terms of the double wronskian determinant for the vcvb model are derived interactions among the vcvb solitons are discussed a novel dynamic property is observed i e the coexistence of elastic inelastic interactions