the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

application of the exp function method for nonlinear differential difference equations in this paper we established abundant travelling wave solutions for some nonlinear differential difference equations it is shown that the exp function method with the help of symbolic computation provides a very effective and powerful new method for discrete nonlinear evolution equations in mathematical physics

a study on resonant multi soliton solutions to the 2 1 dimensional hirota satsuma ito equations via the linear superposition principle abstract in this paper the existence and non existence of resonant multi soliton solutions to two different 2 1 dimensional hirota satsuma ito hsi equations are explored after applying the linear superposition principle we generate resonant multi soliton solutions to the first hsi equation which appeared in the theory of shallow water wave the conditions of real resonant multi soliton solutions are revealed the presented resonant multi soliton solutions exhibit the inelastic collision phenomenon among the involved solitary waves particularly upon choosing appropriate parameters we demonstrate the characteristics of inelastic interactions among the multi front kink waves both graphically and theoretically moreover non existence of resonant multi soliton solution is considered for the generalized hsi equation via the linear superposition principle

two symmetry constraints for a generalized dirac integrable hierarchy abstract by a two by two matrix spectral problem a generalized dirac integrable hierarchy is presented a hamiltonian structure of the obtained hierarchy is established by trace identity and its liouville integrability is proved then through bargmann symmetry constraint spatial part of the lax pairs and adjoint lax pairs is nonlinearized as a completely integrable finite dimensional hamiltonian system next under an implicit symmetry constraint both spatial part and temporal parts of the lax pairs and adjoint lax pairs are all nonlinearized as completely integrable finite dimensional hamiltonian systems ultimately the involutive representation of solution of the generalized dirac integrable hierarchy is given

infinitely many lax pairs and symmetry constraints of the kp equation starting from a known lax pair one can get some infinitely many coupled lax pairs infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways in this paper taking the well known kadomtsev petviashvili kp equation as a special example we show that infinitely many nonhomogeneous linear lax pairs can be obtained by using infinitely many symmetries differentiating the spectral functions with respect to the inner parameters using a known lax pair and the darboux transformations dt infinitely many nonhomogeneous nonlinear lax pairs can also be obtained by means of the infinitely many lax pairs dt and the conformal invariance of the schwartz form of the kp equation infinitely many new nonlocal symmetries can be obtained naturally infinitely many integrable models in 1 1 dimensions 2 1 dimensions 3 1 dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the kp equation related to the infinitely many lax pairs

non perturbative heat kernel asymptotics on homogeneous abelian bundles we study the heat kernel for a laplace type partial differential operator acting on smooth sections of a complex vector bundle with the structure group g u 1 over a riemannian manifold m without boundary the total connection on the vector bundle naturally splits into a g connection and a u 1 connection which is assumed to have a parallel curvature f we find a new local short time asymptotic expansion of the off diagonal heat kernel u t x x close to the diagonal of m m assuming the curvature f to be of order t 1 the coefficients of this expansion are polynomial functions in the riemann curvature tensor and the curvature of the g connection and its derivatives with universal coefficients depending in a non polynomial but analytic way on the curvature f more precisely on tf these functions generate all terms quadratic and linear in the riemann curvature and of arbitrary order in f in the usual heat kernel coefficients in that sense we effectively sum up the usual short time heat kernel asymptotic expansion to all orders of the curvature f we compute the first three coefficients both diagonal and off diagonal of this new asymptotic expansion

a new algebraic approach for calculating the heat kernel in quantum gravity it is shown that the heat kernel operator for the laplace operator on any covariantly constant curved background i e in symmetric spaces may be presented in the form of an averaging over the lie group of isometries with some nontrivial measure using this representation the heat kernel diagonal i e the heat kernel in coinciding points is obtained related topics concerning the structure of symmetric spaces and the calculation of the effective action are discussed

rogue wave and a pair of resonance stripe solitons to kp equation abstract the rogue wave and a pair of resonance stripe solitons to kp equation are discovered first based on the bilinear method some lump solutions are obtained containing six parameters four of which must cater to the non zero conditions so as to insure the solution analytic and rationally localized second a one stripe soliton lump solution is presented and the interaction shows that the lump soliton can be drowned or swallowed by the stripe soliton conversely the lump soliton is spit out from the stripe soliton finally a new ansatz of combination of positive quadratic functions and hyperbolic functions is introduced and thus a novel nonlinear phenomenon is explored it is interesting that a rogue wave can be excited it is observed that the rogue wave possessing a peak wave profile arises from one of the resonance stripe solitons moves to the other and then disappears therefore a rogue wave can be generated by the interaction between the lump soliton and the pair of resonance stripe solitons however compared with classic rouge wave the dynamics of above nonlinear waves are quite different which are graphically demonstrated

construction of new exact rational form non travelling wave solutions to the 2 1 dimensional generalized broer kaup system abstract with the aid of symbolic computation maple several new families of rational form variable separation solutions with three arbitrary functions to the 2 1 dimensional generalized broer kaup system are derived by using an improved mapping approach and a variable separation approach these solutions include rational solitary wave solutions periodic wave solutions and rational wave solutions the properties of the novel localized excitation are revealed by some figures

construction of solutions for an integrable differential difference equation by darboux backlund transformation abstract we first establish a one fold darboux backlund transformation for the integrable differential difference equation which is presented in ref 6 by means of a proper gauge transformation matrix then as a result of the n times one fold darboux backlund transformation we derive n fold darboux backlund transformation finally using the obtained darboux backlund transformation two exact solutions are worked out

lump solutions to the kadomtsev petviashvili i equation with a self consistent source abstract based on symbolic computations lump solutions to the kadomtsev petviashvili i kpi equation with a self consistent source kpiescs are constructed by using the hirota bilinear method and an ansatz technique in contrast with lower order lump solutions of the kadomtsev petviashvili kp equation the presented lump solutions to the kpiescs exhibit more diverse nonlinear phenomena the method used here is more natural and simpler

multiple soliton solutions and fusion interaction phenomena for the 2 1 dimensional modified dispersive water wave system by means of backlund transformation and the hirota bilinear method the 2 1 dimensional modified dispersive water wave system is investigated with the aid of symbolic computation new multiple soliton solutions with arbitrary functions are derived based on the derived solutions rich coherent structures like dromions peakons and compactons can be derived and moreover the fusion interactions among different types of localized structures are graphically studied which might be helpful to understand the propagation processes for nonlinear and dispersive long gravity waves on shallow waters

twisted alexander norms give lower bounds on the thurston norm we introduce twisted alexander norms of a compact connected orientable 3 manifold with first betti number greater than one generalizing norms of mcmullen and turaev we show that twisted alexander norms give lower bounds on the thurston norm of a 3 manifold using these we completely determine the thurston norm of many 3 manifolds which are not determined by norms of mcmullen and turaev

variable separation solutions to the coupled integrable dispersionless equations abstract in this paper the variable separation solutions of the coupled integrable dispersionless equations are derived these solutions include two arbitrary functions and they are more general than the results presented before furthermore some coherent structures of the physical quantity including decaying bell solitary single dromion soliton type breather peakon type breather and loop type breather are constructed

a new integrable symplectic map associated with lattice soliton equations a method is developed that extends the nonlinearization technique to the hierarchy of lattice soliton equations associated with a discrete 3 3 matrix spectral problem a new integrable symplectic map and its involutive system of conserved integrals are obtained by the nonlinearization of spatial parts and the time parts of lax pairs and their adjoint lax pairs of the hierarchy moreover the solutions of the typical system of lattice equations in the hierarchy are reduced to the solutions of a system of ordinary differential equations and a simple iterative process of the symplectic map

study on the behavior of oscillating solitons using the 2 1 dimensional nonlinear system abstract by means of an extended homogeneous balance method and a variable separation hypothesis a broad general variable separation solution with three specific arbitrary functions of the nonlinear 2 1 dimensional broer kaup bk equations was derived based on the derived solution a number of abundant oscillating solitons such as dromion multi dromion solitoff ring multi lump and so on have been revealed in this study by selecting appropriate functions of the general variable separation solution

solitons and singular solitons for the gardner kp equation abstract we present in this work a reliable treatment for the gardner kp gkp equation the hirota s bilinear method is used to obtain multiple soliton solutions for this completely integrable equation multiple singular soliton solutions are obtained as well the study highlights a variety of multi solitary wave and multi singular solitary solutions of the gardner kp equation

discrete exact solutions of modified volterra and volterra lattice equations via the new discrete sine gordon expansion algorithm abstract recently we have presented a sine gordon expansion method to construct new exact solutions of a wide of continuous nonlinear evolution equations in this paper we further develop the method to be the discrete sine gordon expansion method in nonlinear differential difference equations in particular discrete soliton equations we choose the modified volterra lattice and volterra lattice equation to illustrate the new method such as many types of new exact solutions are obtained moreover some figures display the profiles of the obtained solutions our method can be also applied to other discrete soliton equations

the two singular manifold method i modified korteweg de vries and sine gordon equations we address the problem of finding by painleve analysis only the backlund transformation of partial differential equations pdes having two families of movable singularities with opposite principal parts such as the modified korteweg de vries mkdv sine gordon or nonlinear schrodinger equations this first paper gives an almost algorithmic method which extends the singular manifold method of weiss 1986 that is unable to handle these equations first with only one singular manifold at a time we obtain the darboux transformation second we assume that the ratio of two functions defining the singular manifolds satisfies the most general projective riccati system with undetermined coefficients the darboux transformation then generates a very small number of determining equations admitting a unique solution equivalent to the lax pair of the zakharov shabat ablowitz kaup newell segur scheme by the canonical linearization of the riccati system the method is here applied to the mkdv and sine gordon equations

a trace identity and its applications to the theory of discrete integrable systems a general theory for studying discrete integrable systems is developed based on a trace identity that the author previously proposed a scheme for generating hierarchies of discrete integrable systems is presented under broad assumptions the resulting hierarchies are shown to consist of liouville integrable hamiltonian systems

a new zk ilw equation for algebraic gravity solitary waves in finite depth stratified atmosphere and the research of squall lines formation mechanism abstract the research of algebraic gravity solitary waves is an advanced field which has important practical and theoretical value in physical oceanography aerology and etc by calculation condition and theoretical method limit previous researches mainly focused on the 1 1 dimensional models and 2 1 dimensional models were few considered in this paper from the non static equilibrium equation a new zk ilw equation is derived by using multi scale analysis and perturbation method in finite depth stratified atmosphere which is the first time obtained the model can reduce to zk bo model h and zk model h 0 and is the generalization of the above two models in order to further understand the nature of algebraic gravity solitary waves we get the analytical solution of zk ilw equation by using the trial function method and discuss the conservation laws furthermore the fission process of algebraic gravity solitary waves is studied and we can judge that one of the possible formation mechanism of squall lines is the fission of algebraic gravity solitary waves

solitary wave solutions of 3 1 dimensional extended zakharov kuznetsov equation by lie symmetry approach abstract in this paper the solitary wave solutions of 3 1 dimensional extended zakharov kuznetsov ezk equation are constructed which appear in the magnetized two ion temperature dusty plasma and quantum physics lie group of transformation method is proposed to investigate the solution of 3 1 dimensional ezk equation via lie symmetry method the optimal system of one dimensional lie subalgebra is constructed by using lie point symmetries the three dimensional ezk equation reduced into number of ordinary differential equations odes by applying similarity reductions consequently solutions so extracted are more general than erstwhile known results we have obtained twenty one solutions in the explicit form some of them are likewise general and some are new for the best study of us eventually single soliton quasi periodic soliton multisoliton lump type soliton traveling wave and solitary wave interaction behavior are illustrated graphically through numerical simulation for physical affirmation of the results please check whether the affiliations are correct

new exact solutions for the 2 1 dimensional generalized broer kaup system abstract the 1 1 dimensional broer kaup system which describes the propagation of shallow water waves is extended to a generalized 2 1 dimensional model with painleve property in this paper based on the general variable separation approach and two extended riccati equations we first find several new families of exact soliton like solutions and periodic like wave solutions with arbitrary functions for the 2 1 dimensional simplified generalized broer kaup gbk system b 0 abundant new localized excitations can be found by selecting appropriate functions after that we consider the conditions of b 0 to the gbk and several new results are obtained

the dispersionless lax equations and topological minimal models it is shown that perturbed rings of the primary chiral fields of the topological minimal models coincide with some particular solutions of the dispersionless lax equations the exact formulae for the tree level partition functions of an topological minimal models are found the virasoro constraints for the analogue of the function of the dispersionless lax equation correspond ing to these models are proved

localized waves and interaction solutions to a 3 1 dimensional generalized kp equation abstract based on the hirota bilinear method and long wave limit four kinds of localized waves namely solitons lumps breathers and rogue waves are constructed for the 3 1 dimensional generalized kp equation n soliton solutions are obtained by employing bilinear method then breathers two breathers and interaction breather solutions are obtained by selecting appropriate parameters on two soliton solution and four soliton solution these breathers own different dynamic behaviors in the different planes taking a long wave limit on the two and four soliton solutions under special parameter constraints one order lumps and rogue waves two order lumps and rogue waves and interaction solutions between lumps and rogue waves are derived applying the same method on the three soliton solution interaction solutions between kink solitons with periodic solutions lumps and rogue waves are constructed respectively the influence of parameters on the solution is analyzed the propagation directions phase shifts energies and shapes for these solutions can be affected and controlled by the parameters moreover graphics are presented to demonstrate the properties of the explicit analytical localized wave solutions

asymptotic behaviors of the heat kernel in covariant perturbation theory the trace of the heat kernel is expanded in a basis of nonlocal curvature invariants of nth order the coefficients of this expansion the nonlocal form factors are calculated to third order in the curvature inclusive the early time and late time asymptotic behaviors of the trace of the heat kernel are presented with this accuracy the late time behavior gives the criterion of analyticity of the effective action in quantum field theory the latter point is exemplified by deriving the effective action in two dimensions

analytic solutions for the 2 1 dimensional generalized sine gordon equations in nonlinear optics abstract the 2 1 dimensional generalized sine gordon equations can be used to describe the propagation of femtosecond laser pulse in a systems of two level atoms in this paper bilinear forms of the equations have been constructed through variable transformation and bell polynomial based on the bilinear method and truncated painleve expansion multi soliton and quasi periodic peakon solutions for such a system are derived respectively overtaking elastic interactions between the two solitons and among the three solitons have been found it is presented that the soliton with the smaller amplitude can travel faster than the larger one apart from the soliton other type of nonlinear wave propagating in the systems of two level atoms has been displayed namely the quasi periodic peakon the peakon oscillates in both space and time while its left and right derivatives at peak point are relatively large which means the electric field intensity of the femtosecond laser pulse is rather strong

a new integrable differential difference system and its explicit solutions a new integrable differential difference system is proposed by the dependent variable transformation the system is transformed into multilinear form by introducing an auxiliary variable we further transform it into the bilinear form its corresponding backlund transformation is obtained furthermore nonlinear superposition formulae are presented as an application of the obtained results soliton solutions and rational solutions to the system are derived

heat kernel on homogeneous bundles over symmetric spaces we consider laplace type partial differential operators acting on sections of homogeneous vector bundles over symmetric spaces by using an integral representation of the heat semi group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants the obtained formal solution correctly reproduces the exact heat kernel diagonal after a suitable regularization and analytic continuation this result is used to evaluate the nonperturbative low energy effective action in quantum general relativity

r matrix approach to lattice integrable systems an r matrix formalism is applied to the construction of the integrable lattice systems and their bi hamiltonian structure miura like gauge transformations between the hierarchies are also investigated in the end the ladder of linear maps between generated hierarchies is established and described

a discrete variational identity on semi direct sums of lie algebras the discrete variational identity under general bilinear forms on semi direct sums of lie algebras is established the constant involved in the variational identity is determined through the corresponding solution to the stationary discrete zero curvature equation an application of the resulting variational identity to a class of semi direct sums of lie algebras in the volterra lattice case furnishes hamiltonian structures for the associated integrable couplings of the volterra lattice hierarchy

l2 topological invariants of 3 manifolds we give results on thel2 betti numbers and novikov shubin invariants of compact manifolds especially 3 manifolds we first study the betti numbers and novikov shubin invariants of a chain complex of hilbert modules over a finite von neumann algebra we establish inequalities among the novikov shubin invariants of the terms in a short exact sequence of chain complexes our algebraic results along with some analytic results on geometric 3 manifolds are used to compute thel2 betti numbers of compact 3 manifolds which satisfy a weak form of the geometrization conjecture and to compute or estimate their novikov shubin invariants

covariant algebraic method for calculation of the low energy heat kernel using our recently proposed covariant algebraic approach the heat kernel for a laplace like differential operator in a low energy approximation is studied neglecting all the covariant derivatives of the gauge field strength yang mills curvature and the covariant derivatives of the potential term of third order and higher a closed formula for the heat kernel as well as its diagonal is obtained explicit formulas for the coefficients of the asymptotic expansion of the heat kernel diagonal in terms of the yang mills curvature the potential term and its first two covariant derivatives are obtained

multiple soliton solutions for the generalized 1 1 dimensional and the generalized 2 1 dimensional ito equations abstract the generalized 1 1 dimensional and the generalized 2 1 dimensional ito equations obtained by introducing a new type of bilinearism are studied the tanh coth method is used to obtain single soliton solutions and periodic solutions for the ito equations the hirota bilinear method is used to determine the multiple soliton solutions of sech squared type for these equations new entirely solutions are obtained the analysis emphasizes the power of the two methods and its capability of handling completely integrable equations

a unified approach to painleve expansions abstract the painleve test for partial differential equations developed by weiss tabor and carnevale wtc is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations a simple modification of the wtc procedure used for partial differential equations enables us to determine the lax pairs hirota equations and auto backlund transformations for ordinary differential equations including a new lax pair for an integrable case of the henon heiles system a detailed study of the kdv hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained the role of the singular branches is examined in detail and important new insights obtained in particular we find that each singular branch is simply a re expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce a parallel analysis is carried out for the akns hierarchy

time fractional generalized boussinesq equation for rossby solitary waves with dissipation effect in stratified fluid and conservation laws as well as exact solutions abstract construct fractional order model to describe rossby solitary waves can provide more pronounced effects and deeper insight for comprehending generalization and evolution of rossby solitary waves in stratified fluid in the paper from the quasi geostrophic vorticity equation with dissipation effect and complete coriolis force based on the multi scale analysis and perturbation method a classical generalized boussinesq equation is derived to describe the rossby solitary waves in stratified fluid further by employing the reduction perturbation method the semi inverse method the agrawal method we derive the euler lagrangian equation of classical generalized boussinesq equation and obtain the time fractional generalized boussinesq equation without dissipation effect by using lie group analysis method the conservation laws of time fractional boussinesq equation are given finally with the help of the improved g g expansion method the exact solutions of the above equation are generated meanwhile in order to consider the dissipation effect we have to derive the approximate solutions by adopting the new iterative method we remark that the fractional order model can open up a new window for better understanding the waves in fluid

dressing method for the vector sine gordon equation and its soliton interactions in this paper we develop the dressing method to study the exact solutions for the vector sine gordon equation the explicit formulas for one kink and one breather are derived the method can be used to construct multi soliton solutions two soliton interactions are also studied the formulas for position shift of the kink and position and phase shifts of the breather are given these quantities only depend on the pole positions of the dressing matrices

algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations an algebraic structure related to discrete zero curvature equations is established it is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries the key of the theory is to generate nonisospectral flows t l l 0 from the discrete spectral problem associated with a given system of discrete evolution equations three examples are given

novel interaction phenomena of localized waves in the generalized 3 1 dimensional kp equation abstract based on hirota bilinear method the n soliton solution of the 3 1 dimensional generalized kadomtsev petviashvili equation is derived explicitly from which some localized waves such as soliton breather lump and their interactions are obtained by the approach of long wave limit especially by selecting particular parameter constraints in the n soliton solutions the one breather or one lump can be obtained from two soliton the elastic interaction solutions between one bell shaped soliton and one breather or between one bell shaped soliton and one lump can be obtained from three soliton the elastic interaction solutions among two bell shaped solitons and one breather among two bell shaped solitons and one lump between two breathers or between two lumps can be obtained from four soliton the elastic interactions solutions among one bell shaped soliton and two breathers among one breather and three bell shaped solitons among one lump and three bell shaped solitons among one bell shaped soliton and two breathers or among one breather one lump and one bell shaped soliton can be obtained from five soliton detailed behaviors of such interaction phenomena are illustrated analytically and graphically the results obtained in this paper may be helpful for understanding the evolution of nonlinear localized waves in shallow water

universal l2 torsion polytopes and applications to 3 manifolds given an l2 acyclic connected finite cw complex we define its universal l2 torsion in terms of the chain complex of its universal covering it takes values in the weak whitehead group whw g we study its main properties such as homotopy invariance sum formula product formula and poincare duality under certain assumptions we can specify certain homomorphisms from the weak whitehead group whw g to abelian groups such as the real numbers or the grothendieck group of integral polytopes and the image of the universal l2 torsion can be identified with many invariants such as the l2 torsion the l2 torsion function twisted l2 euler characteristics and in the case of a 3 manifold the dual thurston norm polytope

the n soliton solution and localized wave interaction solutions of the 2 1 dimensional generalized hirota satsuma ito equation abstract in this paper the n soliton solution is constructed for the 2 1 dimensional generalized hirota satsuma ito equation from which some localized waves such as line solitons lumps periodic solitons and their interactions are obtained by choosing special parameters especially by selecting appropriate parameters on the multi soliton solutions the two soliton can reduce to a periodic soliton or a lump soliton the three soliton can reduce to the elastic interaction solution between a line soliton and a periodic soliton or the elastic interaction between a line soliton and a lump soliton while the four soliton can reduce to elastic interaction solutions among two line solitons and a periodic soliton or the elastic interaction ones between two periodic solitons detailed behaviours of such solutions are illustrated analytically and graphically by analysing the influence of parameters finally an inelastic interaction solution between a lump soliton and a line soliton is constructed via the ansatz method and the relevant interaction and propagation characteristics are discussed graphically the results obtained in this paper may be helpful for understanding the interaction phenomena of localized nonlinear waves in two dimensional nonlinear wave equations

a deformed reduced semi discrete kaup newell equation the related integrable family and darboux transformation we propose a deformed reduced semi discrete kaup newell equation we derive the related integrable family through discrete zero curvature equation we establish a darboux transformation of lax pair of the proposed equation three exact solutions are given a deformed reduced semi discrete kaup newell equation and its related integrable family are derived from discrete zero curvature equation a darboux transformation of lax pair of this equation is established with the help of gauge transformation by means of the resulting darboux transformation three exact solutions are given

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

nonlocal symmetry darboux transformation and soliton cnoidal wave interaction solution for the shallow water wave equation abstract in classical shallow water wave sww theory there exist two integrable one dimensional sww equation hirota satsuma hs type and ablowitz kaup newell segur akns type in the boussinesq approximation in this paper we mainly focus on the integrable sww equation of akns type the nonlocal symmetry in form of square spectral function is derived starting from its lax pair infinitely many nonlocal symmetries are presented by introducing the arbitrary spectrum parameter these nonlocal symmetries can be localized and the sww equation is extended to enlarged system with auxiliary dependent variables then darboux transformation for the prolonged system is found by solving the initial value problem similarity reductions related to the nonlocal symmetry and explicit group invariant solutions are obtained it is shown that the soliton cnoidal wave interaction solution which represents soliton lying on a cnoidal periodic wave background can be obtained analytically interesting characteristics of the interaction solution between soliton and cnoidal periodic wave are displayed graphically

localized excitations in 2 1 dimensional systems by means of a special variable separation approach a common formula with some arbitrary functions has been obtained for some suitable physical quantities of various 2 1 dimensional models such as the davey stewartson ds model the nizhnik novikov veselov nnv system asymmetric nnv equation asymmetric ds equation dispersive long wave equation broer kaup kupershmidt system long wave short wave interaction model maccari system and a general n m component ablowitz kaup newell segur akns system selecting the arbitrary functions appropriately one may obtain abundant stable localized interesting excitations such as the multidromions lumps ring soliton solutions breathers instantons etc it is shown that some types of lower dimensional chaotic patterns such as the chaotic chaotic patterns periodic chaotic patterns chaotic line soliton patterns chaotic dromion patterns fractal lump patterns and fractal dromion patterns may be found in higher dimensional soliton systems the interactions between the traveling ring type soliton solutions are completely elastic the traveling ring solitons pass through each other and preserve their shapes velocities and phases some types of localized weak solutions peakons are also discussed especially the interactions between two peakons are not completely elastic after the interactions the traveling peakons also pass through each other and preserve their velocities and phases however they completely exchange their shapes

a modified toda spectral problem and its hierarchy of bi hamiltonian lattice equations starting from a modified toda spectral problem a hierarchy of generalized toda lattice equations with two arbitrary constants is constructed through discrete zero curvature equations it is shown that the hierarchy possesses a bi hamiltonian structure and a hereditary recursion operator which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals two cases of the involved constants present two specific integrable sub hierarchies one of which is exactly the toda lattice hierarchy

degree of l 2 alexander torsion for 3 manifolds for an irreducible orientable compact 3 manifold n with empty or incompressible toral boundary the full l 2 alexander torsion tau 2 n phi t associated to any real first cohomology class phi of n is represented by a function of a positive real variable t the paper shows that tau 2 n phi is continuous everywhere positive and asymptotically monomial in both ends moreover the degree of tau 2 n phi equals the thurston norm of phi the result confirms a conjecture of j dubois s friedl and w luck and addresses a question of w li and w zhang associated to any admissible homomorphism gamma pi 1 n rightarrow g the l 2 alexander torsion tau 2 n gamma phi is shown to be continuous and everywhere positive provided that g is residually finite and n gamma is weakly acyclic in this case a generalized degree can be assigned to tau 2 n gamma phi moreover the generalized degree is bounded by the thurston norm of phi

n solitons breathers lumps and rogue wave solutions to a 3 1 dimensional nonlinear evolution equation abstract based on hirota bilinear method n solitons breathers lumps and rogue waves as exact solutions of the 3 1 dimensional nonlinear evolution equation are obtained the impacts of the parameters on these solutions are analyzed the parameters can influence and control the phase shifts propagation directions shapes and energies for these solutions the single kink soliton solution and interactions of two and three kink soliton overtaking collisions of the hirota bilinear equation are investigated in different planes the breathers in three dimensions possess different dynamics in different planes via a long wave limit of breathers with indefinitely large periods rogue waves are obtained and localized in time it is shown that the rogue wave possess a growing and decaying line profile that arises from a nonconstant background and then retreat back to the same nonconstant background again the results can be used to illustrate the interactions of water waves in shallow water moreover figures are given out to show the properties of the explicit analytic solutions

wave fronts and cascades of soliton interactions in the periodic two dimensional volterra system in the paper we develop the dressing method for the solution of the two dimensional periodic volterra system with a period n we derive soliton solutions of arbitrary rank k and give a full classification of rank 1 solutions we have found a new class of exact solutions corresponding to wave fronts which represent smooth interfaces between two nonlinear periodic waves or a periodic wave and a trivial zero solution the wave fronts are non stationary and they propagate with a constant average velocity the system also has soliton solutions similar to breathers which resembles soliton webs in the kp theory we associate the classification of soliton solutions with the schubert decomposition of the grassmannians grr k n and grc k n

integrable discretizations of derivative nonlinear schrodinger equations we propose integrable discretizations of derivative nonlinear schrodinger dnls equations such as the kaup newell equation the chen lee liu equation and the gerdjikov ivanov equation by constructing lax pairs the discrete dnls systems admit the reduction of complex conjugation between two dependent variables and possess bi hamiltonian structure through transformations of variables and reductions we obtain novel integrable discretizations of the nonlinear schrodinger nls modified kdv mkdv mixed nls matrix nls matrix kdv matrix mkdv coupled nls coupled hirota coupled sasa satsuma and burgers equations we also discuss integrable discretizations of the sine gordon equation the massive thirring model and their generalizations

interaction properties of complex modified korteweg de vries mkdv solitons abstract interaction properties of complex solitons are studied for the two u 1 invariant integrable generalizations of the modified korteweg de vries mkdv equation given by the hirota equation and the sasa satsuma equation which share the same traveling wave single soliton solution having a sech profile characterized by a constant speed and a constant phase angle for both equations nonlinear interactions in which a fast soliton collides with a slow soliton are shown to be described by 2 soliton solutions that can have three different types of interaction profile depending on the speed ratio and the relative phase angle of the individual solitons in all cases the shapes and speeds of the solitons are found to be preserved apart from a shift in position such that their center of momentum moves at a constant speed moreover for the hirota equation the phase angles of the fast and slow solitons are found to remain unchanged while for the sasa satsuma equation the phase angles are shown to undergo a shift such that the relative phase between the fast and slow solitons changes sign

heat kernel coefficients for the matrix schrodinger operator the heat kernel coefficients hk to the schrodinger operator with a matrix potential are investigated we present algorithms and explicit expressions for the taylor coefficients of the hk special terms are discussed and for the one dimensional case some improved algorithms are derived

lump solutions to a 2 1 dimensional extended kp equation abstract with the aid of a computer algebra system we present lump solutions to a 2 1 dimensional extended kadomtsev petviashvili equation ekp and give the sufficient and necessary conditions to guarantee analyticity and rational localization of the solutions we plot a few solutions for some specific values of the free parameters involved and finally derive one of the lump solutions of the kadomtsev petviashvili kp equations from the lump solutions of the ekp equation

integrable couplings of relativistic toda lattice systems in polynomial form and rational form their hierarchies and bi hamiltonian structures integrable couplings of relativistic toda lattice systems in polynomial form and rational form and their hierarchies are derived from a four by four discrete matrix eigenvalue problem the bi hamiltonian structure for every integrable coupling in the two hierarchies obtained is established by means of the discrete variational identity ultimately liouvolle integrability of the obtained integrable couplings is demonstrated

hamiltonian structure of discrete soliton systems we describe an approach for investigating the hamiltonian structures of the lattice isospectral evolution equations associated with a general discrete spectral problem by using the so called implicit representations of the isospectral flows we demonstrate the existence of the recursion operator l which is a strong and hereditary symmetry of the flows it is then proven that every equation in the isospectral hierarchy possesses the hamiltonian structure if l has a skew symmetric factorization and the first equation ut k 0 in the hierarchy satisfies some simple condition we obtain related properties such as the implectic symplectic factorization of l the liouville complete integrability and the multi hamiltonian structures of the isospectral hierarchy four examples are given

lump solutions of a generalized calogero bogoyavlenskii schiff equation abstract a generalized calogero bogoyavlenskii schiff equation is considered and based on its hirota bilinear form a class of lump solutions is explicitly generated via symbolic computations with maple together with plots of a specific lump solution the result enlarges the category of nonlinear partial differential equations which possess lump solutions

a generating scheme for conservation laws of discrete zero curvature equations and its application abstract a generating scheme for conservation laws of discrete integrable equations arising as discrete zero curvature equations is presented from pairs of associated matrix spectral problems an illustrative application to the volterra lattice equation is made through a pair of 2 2 matrix spectral problems the general theory is not limited to 2 2 matrix spectral problems but for arbitrary order matrix spectral problems

semidirect sums of lie algebras and discrete integrable couplings a relation between semidirect sums of lie algebras and integrable couplings of lattice equations is established and a practicable way to construct integrable couplings is further proposed an application of the resulting general theory to the generalized toda spectral problem yields two classes of integrable couplings for the generalized toda hierarchy of lattice equations the construction of integrable couplings using semidirect sums of lie algebras provides a good source of information on complete classification of integrable lattice equations

nonlinear schrodinger equations and simple lie algebras we associate a system of integrable generalised nonlinear schrodinger nls equations with each hermitian symmetric space these nls equations are considered as reductions of more general systems this time associated with a reductive homogeneous space the nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space the hamiltonian structure is investigated using r matrix techniques and shown to be canonical for all these equations throughout the reduction procedure this hamiltonian structure does not degenerate each of the above systems of equations is gauge equivalent to a generalised ferromagnet reductions of the latter are discussed in terms of the corresponding nls type equations

the two singular manifold method ii classical boussinesq system for pt i see musette et al ibid vol 24 p 3895 1994 the two singular manifold method a generalization of the singular manifold method of weiss 1983 is applied to the classical boussinesq system also known as the broer kaup system from the point of view of its singularity analysis the important feature of this system is the existence of two principal families with opposite principal parts the usual singular manifold method takes into account only one of these families at a time our generalization takes into account both families and in this way we are able to derive the lax pair and darboux transformation and hence the auto backlund transformation for the classical boussinesq system from its painleve analysis

diversity of interaction solutions to the 2 1 dimensional ito equation abstract we aim to show the diversity of interaction solutions to the 2 1 dimensional ito equation based on its hirota bilinear form the proof is given through maple symbolic computations an interesting characteristic in the resulting interaction solutions is the involvement of an arbitrary function special cases lead to lump solutions lump soliton solutions and lump kink solutions two illustrative examples of the resulting solutions are displayed by three dimensional plots and contour plots

dynamic behaviors of interaction solutions of 3 1 dimensional shallow water wave equation abstract rational solutions and interaction solutions of the 3 1 dimensional shallow water wave equation are obtained depending on n soliton solution by the hirota bilinear method and long wave limit method and breather lump solutions and line rogue waves are obtained from two soliton solution as well as their interaction solutions are derived from four soliton solution after the collision breathers and lumps remain unchanged but the line rogue wave has a difference furthermore three soliton solution is discussed the interactions between some hybrid solutions such as hybrid solution between the stripe soliton and breather hybrid solution between stripe soliton and lump solution and so on are demonstrated in detail more interestingly stripe soliton remain unchanged but the amplitudes of the periodic line waves lump solutions and rogue waves have a great change after the collision