complexiton and resonant multiple wave solutions to the 2 1 dimensional konopelchenko dubrovsky equation abstract in this paper the 2 1 dimensional konopelchenko dubrovsky kd equation is investigated using the hirota direct method and linear superposition principle the complexiton and resonant multiple wave solutions are successfully constructed for the hirota direct method it is essential that finding the bilinear form to the kd equation and applying its 2 n soliton formulation in real field to build the n complexiton solutions under the action of pairs of conjugate wave variables the linear superposition in real field of exponential traveling waves can generate the resonant multiple wave solutions and then generalize linear superposition principle to complex field we obtain some new resonant multiple wave type solutions the phenomena of complexiton resonant multiple wave and new resonant multiple wave type solutions are presented by figures

rate and power allocation under the pairwise distributed source coding constraint we consider the problem of rate and power allocation for a sensor network under the pairwise distributed source coding constraint for noiseless source terminal channels we show that the minimum sum rate assignment can be found by finding a minimum weight arborescence in an appropriately defined directed graph for orthogonal noisy source terminal channels the minimum sum power allocation can be found by finding a minimum weight matching forest in a mixed graph numerical results are presented for both cases showing that our solutions always outperform previously proposed solutions the gains are considerable when source correlations are high

lax pair and first integrals of the traveling wave reduction for the kdv hierarchy abstract the traveling wave reduction of the korteweg de vries hierarchy is considered the linear system of equations associated with this hierarchy is found the lax pair is used to obtain the first integrals of the traveling wave reduction for the korteweg de vries hierarchy exact formulas for the first integrals of the hierarchy are given the first three members of the hierarchy are considered in more detail these first integrals are the examples of the integrable nonlinear differential equations with the painleve property exact solutions in the form of the soliton for the traveling wave reduction of the korteweg de vries hierarchy and its first integrals are presented

solving the linear matroid parity problem as a sequence of matroid intersection problems in this paper we present an o r4n algorithm for the linear matroid parity problem our solution technique is to introduce a modest generalization the non simple parity problem and identify an important subclass of non simple parity problems called easy parity problems which can be solved as matroid intersection problems we then show how to solve any linear matroid parity problem parametrically as a sequence of easy parity problems

localized finite amplitude disturbances and selection of solitary waves it turns out that evolution of localized finite amplitude disturbances in perturbed kdv equation is qualitatively different compared with conventional small amplitude initial conditions namely relatively fast solitary waves with one and the same amplitude and velocity are formed ahead of conventional chaotic like irregular structures the amplitude and velocity of the waves obtained from the asymptotic theory are in excellent agreement with numerics

novel solitary wave solutions for the 3 1 dimensional extended jimbo miwa equations abstract this paper retrieves new periodic solitary wave solutions for the 3 1 dimensional extended jimbo miwa equations based on the hirota bilinear method by utilizing maple software as a result the hirota bilinear method is successfully employed and acquired several classes of solitary wave solutions in terms of a new combination of exponential function trigonometric function and hyperbolic functions all solutions have been verified back into its corresponding equation by maple we depicted the physical explanation of the extracted solutions with the free choice of the different parameters by plotting some 3d and 2d illustrations finally it has been illustrated that the executed method is robust and more efficient than other methods and the obtained solutions are trustworthy in the applied sciences

orthogonal laurent polynomials a new algebraic approach abstract in this paper some important algebraic aspects in the theory of orthogonal laurent polynomials such as the three term recurrence relation the christoffel darboux identity or the liouville ostrogradski formula are revisited from the riemann hilbert window these topics are considered for general ordered laurent polynomial sequences and not only for the usual balanced cases in addition a connection with szego polynomials orthogonal polynomials in the unit circle is explored

computing orthogonal rational functions with poles near the boundary computing orthogonal rational functions is a far from trivial problem especially for poles close to the boundary of the support of the orthogonality measure in this paper we analyze some of the difficulties involved and present two different approaches for solving this problem

a note on the g g expansion method abstract we demonstrate that the g g expansion method which is often used in finding exact solutions of nonlinear differential equation is equivalent to the well known tanh method and application of these methods gives the same exact solutions of nonlinear differential equations

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

another quadrature rule of highest algebraic degree of precision we consider certain quadrature rules of highest algebraic degree of precision that involve strong stieltjes distributions i e strong distributions on the positive real axis the behavior of the parameters of these quadrature rules when the distributions are strong c inversive stieltjes distributions is given a quadrature rule whose parameters have explicit expressions for their determination is presented an application of this quadrature rule for the evaluation of a certain type of integrals is also given

interaction solutions of a 2 1 dimensional dispersive long wave system abstract with the help of the consistent tanh expansion this paper obtains the interaction solutions between solitons and potential burgers waves of a 2 1 dimensional dispersive long wave system based on some known solutions of the potential burgers equation the multiple resonant soliton wave solutions soliton error function wave solutions soliton rational function wave solutions and soliton periodic wave solutions are obtained directly

orthogonal laurent polynomials and strong moment theory a survey these topics are found in many parts of 20th century mathematics and its applications in mathematical physics chemistry statistics and engineering historically the analytic theory of continued fractions has played a central role in both the origin and the development of the other topics continued fractions are intimately related to pad e approximants and special functions emphasis is given to the development of strong moment theory and orthogonal laurent polynomials and to the related continued fractions quadrature formulas integral transforms and linear functionals by a strong moment problem we mean the following for a given bisequence n n of real numbers does there exist a distribution function such that

companion orthogonal polynomials we give some properties relating the recurrence relations of orthogonal polynomials associated with any two symmetric distributions email protected 1 x and d 2 x such that email protected 2 x 1 kx 2 d 1 x as applications of properties recurrence relations for many interesting systems of orthogonal polynomials are obtained

orthogonal laurent polynomials corresponding to certain strong stieltjes distributions with applications to numerical quadratures in this paper we shall be mainly concerned with sequences of orthogonal laurent polynomials associated with a class of strong stieltjes distributions introduced by a s ranga algebraic properties of certain quadratures formulae exactly integrating laurent polynomials along with an application to estimate weighted integrals on 1 1 with nearby singularities are given finally numerical examples involving interpolatory rules whose nodes are zeros of orthogonal laurent polynomials are also presented

pade approximants and quadratures related to certain strong distributions letbe a c inversive strong distribution as de5ned in ranga numer math 68 1994 283 in this paper two point pad e approximants and quadrature formulas related to the distributionare analyzed c 2001 elsevier science b v all rights reserved msc 41a21 30e05 34e05

optimum matching forests i special weights we introduce the concept of matching forests as a generalization of branchings in a directed graph and matchings in an undirected graph given special weights on the edges of a mixed graph we present an efficient algorithm for finding an optimum weight sum matching forest the algorithm is a careful application of known branching and matching algorithms the maximum cardinality matching forest problem is solved as a special case

optimum matching forests iii facets of matching forest polyhedra in 3 we presented a linear system which definesp g the convex hull of incidence vectors of matching forests of a mixed graphg however many of the inequalities of this system may be redundant here we describe the dimension of the facets ofp g obtained by setting one inequality of the defining system forp g to an equation this leads to a presentation of a minimal defining linear system forp g i e to a presentation of the facets ofp g this generalizes earlier characterizations of facets of 1 matching polyhedra and of branching polyhedra

gaussian quadrature rules and numerical examples for strong extensions of mass distribution functions abstract the theory of strong moment problems has provided gaussian quadrature rules for approximate integration with respect to strong distributions in hagler ph d thesis university of colorado boulder 1997 and hagler et al lecture notes in pure and applied mathematics marcel dekker new york in press a transformation of the form v x 1 x x 0 is used to obtain strong mass distribution functions from mass distribution functions this transformation also links the systems of orthogonal polynomials and laurent polynomials and their zeros in this paper we show how the transformation method can be used to obtain the gaussian quadrature rules for strong extensions of mass distribution functions we then provide numerical examples of strong gaussian quadrature approximations to the integrals of elementary functions with respect to selected strong distributions

m lump solutions to a 3 1 dimensional nonlinear evolution equation abstract this paper aims at computing the m lump solutions which decay to a uniform state in all directions for a 3 1 dimensional nonlinear evolution equation these solutions are constructed by taking a long wave limit of the corresponding n soliton solutions obtained by direct methods the dynamic properties of m lump solutions describing multiple collisions of lumps are presented in addition we investigate the interaction between stripe solitons and lumps which is further discussed implying that lumps will be drowned or swallowed by the stripe solitons finally the dynamic properties of interactive wave solutions are graphically depicted by choosing the values of parameters

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

optimum matching forests ii general weights matching forests generalize branchings in a directed graph and matchings in an undirected graph we present an efficient algorithm the pmf algorithm for the problem given a mixed graphg and a real weight on each of its edges find a perfect matching forest of maximum weight sum the pmf algorithm proves the sufficiency of a linear system which definesp g andp g the convex hull of incidence vectors of perfect matching forests and matching forests respectively ofg the algorithm also provides a generalization of tutte s theorem on the existence of perfect matchings in an undirected graph

some lump solutions for a generalized 3 1 dimensional kadomtsev petviashvili equation abstract in this paper a generalized 3 1 dimensional kadomtsev petviashvili equation which can be reduced to the classical equation is investigated based on the hirota bilinear method the lump and lump strip solutions for this equation are obtained with the help of symbolic computation those solutions are derived from polynomial solutions and can be simply classified into some classes analysis for the obtained solutions are presented and their dynamic properties are discussed results are helpful for the study of soliton interactions in nonlinear mathematical physics

orthogonal laurent polynomials and continued fractions associated with log normal distributions abstract this paper describes properties and computational procedures related to orthogonal laurent polynomials continued fractions two point pade approximants strong moment problems and l gaussian quadrature associated with log normal distribution functions t defined by t q1 2 2 e lnt 2 2 0

on computing rational gauss chebyshev quadrature formulas we provide an algorithm to compute the nodes and weights for gauss chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary real poles outside 1 1 contrary to existing rational quadrature formulas the computational effort is very low even for extremely high degrees and under certain conditions on the poles it can be shown that the complexity is of order o n this method is based on the derivation of explicit expressions for chebyshev orthogonal rational functions which are thus far the only examples of explicitly known orthogonal rational functions on 1 1 with arbitrary real poles outside this interval

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

strong stieltjes distributions and orthogonal laurent polynomials with applications to quadratures and pade approximation starting from a strong stieltjes distribution general sequences of orthogonal laurent polynomials are introduced and sonic of their most relevant algebraic properties are studied from this perspective the connection between certain quadrature formulas associated with the distribution and two point pade approximants to the stieltjes transform of is revisited finally illustrative numerical examples are discussed

two point pade expansions for a family of analytic functions abstract each member g z of a family of analytic functions defined by stieltjes transforms is shown to be represented by a positive t fraction the approximants of which form the main diagonal in the two point pade table of g z the positive t fraction is shown to converge to g z throughout a domain d a b z z b a uniformly on compact subsets in addition truncation error bounds are given for the approximants of the continued function these bounds supplement previously known bounds and apply in part of the domain of g z not covered by other bounds the proofs of our results employ properties of orthogonal l polynomials laurent polynomials and l gaussian quadrature which are of some interest in themselves a number of examples are considered

two point pade type approximation to the ccuchy transform of certain strong distributions abstract in this paper we are mainly concerned with the cauchy transform of certain strong distributions satisfying a type of symmetric property introduced by a s ranga algebraic properties of the corresponding two point pade type approximants are given along with results about convergence for sequences of such approximants

stability of wave equation with locally viscoelastic damping and nonlinear boundary source abstract in this paper we consider a nonlinear wave equation with nonlinear boundary damping and source terms subject to a localized viscoelastic damping and localized frictional damping we obtain the global existence of solution via potential well method furthermore by introducing suitable lyapunov functionals using multiplier method and constructing convex differential inequality we compare the effect of locally viscoelastic term and frictional damping and establish an explicit and general decay rate result this stability result allows the localized viscoelastic damping and localized frictional damping acting simultaneously or complementarily in the domain which gives the competing relationship of the viscoelastic and frictional damping

on a symmetry in strong distributions a strong stieltjes distribution d t is called symmetric if it satises the property t d 2 t 2 t d t for t 2 a b 0 1 2 2 z and 0 in this article some consequences of symmetry on the moments the orthogonal l polynomials and the quadrature formulae associated with the distribution are given c 1999 elsevier science b v all rights reserved msc 30e05 30b70 41a55

families of quasi rational solutions of the nls equation and multi rogue waves we construct a multi parametric family of the solutions of the focusing nonlinear schr dinger equation nls from the known results describing the multi phase almost periodic elementary solutions given in terms of riemann theta functions we give a new representation of their solutions in terms of wronskians determinants of order 2n composed of elementary trigonometric functions when we perform a special passage to the limit when all the periods tend to infinity we obtain a family of quasi rational solutions this leads to efficient representations for the peregrine breathers of orders n 1 2 3 first constructed by akhmediev and his co workers and also allows us to obtain a simpler derivation of the generic formulas corresponding the three or six rogue wave formation in the frame of the nls model first explained by v b matveev in 2010 our formulation allows us to isolate easily the second or third order peregrine breathers from generic solutions and also to compute the peregrine breathers of orders 2 and 3 easily with respect to other approaches in the cases n 2 3 we obtain the comfortable formulas to study the deformation of a higher peregrine breather of order 2 to the three rogue wave or order 3 to the six rogue wave solutions via the variation of the free parameters of our construction

a chebyshev type quadrature rule with some interesting properties we give here an n point chebyshev type rule of algebraic degree of precision n 1 but having nodes that can be given explicitly this quadrature rule also turns out to be one with an almost highest algebraic degree of precision

breather lump and x soliton solutions to nonlocal kp equation abstract breather lump and x soliton solutions are presented via the hirota bilinear method to the nonlocal 2 1 dimensional kp equation derived from the alice bob system the resulting breather contains two cases one is the line breather and another is the normal breather both of which are different from the solutions of the classical 2 1 dimensional kp equation the x soliton is found with the long wave limit by some constraints to the parameters the lump solution is obtained in virtue of two methods one is as the long wave limit of breather theoretically the other is with the quadratic function method which can be guaranteed rationally localized in all directions in the space under some constraints of the parameters by choosing specific values of the involved parameters the dynamic properties of some breather lump solutions to nonlocal kp equation are plotted as illustrative examples

rational solutions and lump solutions to the generalized 3 1 dimensional shallow water like equation abstract through symbolic computation with maple the rational solutions and the lump solutions of the generalized 3 1 dimensional shallow water like equation are presented by using the generalized bilinear operator when the parameter p 3 it is pointed that these rational solutions are classified into five classes because they are obtained mainly by depending on the polynomial solutions the resulting lump solutions which are rationally localized in all directions in the space are acquired by making use of the quadratic function however not every nonlinear partial differential equation has the lump solutions if any the quantity of the lump solutions is fewer than the rational solutions only one class of lump solutions of the generalized 3 1 dimensional shallow water like equation is gotten and three 3d plots with specific values of the involved parameters are plotted

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

new periodic wave cross kink wave and the interaction phenomenon for the jimbo miwa like equation abstract by means of symbolic computation with the help of maple diverse of exact solutions for the jimbo miwa like equation are successfully derived based on the generalized bilinear method these new solutions named the periodic wave and cross kink wave have greatly enriched the existing literature on the jimbo miwa like equation particularly we have obtained the interaction solutions between the lump solution and an arbitrary function for the jimbo miwa like equation as long as the researcher needs it he can choose an arbitrary function to observe the collision phenomenon which he wants between the lump and the solitary waves via the three dimensional images and density images with the help of maple the physical characteristics of these waves are described very well that will be widely used to describe many interesting physical phenomena in the fields of gas plasma optics acoustics heat transfer fluid dynamics classical mechanics and so on

on the convergence of certain gauss type quadrature formulas for unbounded intervals we consider the convergence of gauss type quadrature formulas for the integral f 0 f x x dx where w is a weight function on the half line 0 the n point gauss type quadrature formulas are constructed such that they are exact in the set of laurent polynomials p q 1 k p q 1 p a k x k where p p n is a sequence of integers satisfying 0 p n 2n and q q n 2n p n it is proved that under certain carleman type conditions for the weight and when p n or q n goes to oo then convergence holds for all functions f for which fw is integrable on 0 some numerical experiments compare the convergence of these quadrature formulas with the convergence of the classical gauss quadrature formulas for the half line

the sagitta method for solving linear programs abstract a new non simplex active set method for solving linear programs is presented this method uses a global viewpoint of the problem it starts without any iteration point and its basic characteristic is that the first feasible point determined by the method is generally an optimal solution the computational results obtained for a range space implementation are highly encouraging

the lump solutions and the backlund transformation for the three dimensional three wave resonant interaction a backlund transformation is found for the three dimensional three wave resonant interaction and from it n lump exact solutions may be constructed the one lump solution is analyzed in detail and it is shown that it describes such effects as pulse decay upconversion and explosive instabilities all in three dimensions

algebro geometric solutions for the complex sharma tasso olver hierarchy this paper is dedicated to provide theta function representation of algebro geometric solutions and related crucial quantities for the complex sharma tasso olver csto hierarchy our main tools include the polynomial recursive formalism to derive the csto hierarchy the hyper elliptic curve with finite number of genus the baker akhiezer functions the meromorphic function the dubrovin type equations for auxiliary divisors and the associated trace formulas by use of these tools the explicit representations of the baker ahhiezer functions the meromorphic function and the algebro geometric solutions are obtained for the entire csto hierarchy

exact solutions of the equation for surface waves in a convecting fluid abstract a method for finding exact solutions and the first integrals is presented the basic idea of the method is to use the value of the fuchs index that appears in the painleve test to construct the auxiliary equation for finding the first integrals and exact solutions of nonlinear differential equations it allows us to obtain the first integrals and new exact solutions of some nonlinear ordinary differential equations the main feature of the method is that we do not assign a solution function at the beginning we find this function during calculations this approach is conceptually equivalent to the third step of the painleve test and sometimes allows us to change this step our approach generalizes a number of other methods for finding exact solutions of nonlinear differential equations we demonstrate a method for finding the traveling wave solutions and the first integrals of the well known nonlinear evolution equation for description of surface waves in a convecting liquid the general solution of this equation at some conditions on parameters and new traveling wave solutions of the fourth order equation are found

the use of rational functions in numerical quadrature quadrature problems involving functions that have poles outside the interval of integration can profitably be solved by methods that are exact not only for polynomials of appropriate degree but also for rational functions having the same or the most important poles as the function to be integrated constructive and computational tools for accomplishing this are described and illustrated in a number of quadrature contexts the superiority of such rational polynomial methods is shown by an analysis of the remainder term and documented by numerical examples

interaction solutions for a dimensionally reduced hirota bilinear equation abstract with the help of hirota direct method and symbolic computation two types of interaction solutions to the dimensionally reduced equations in 2 1 dimensions are derived respectively by searching for the solutions to the associated bilinear equations as a combination of positive quadratic function and an exponential hyperbolic cosine function it is interesting that the interaction solutions between lump soliton and one stripe soliton generate the fission and fusion phenomena while the interaction solutions between lump soliton and twin stripe solitons generate the rogue wave

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

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

global analysis and numerical simulations of a novel stochastic eco epidemiological model with time delay abstract in this paper a new stochastic eco epidemiological model is proposed and investigated and the high dimensional model includes time delay and a general incidence rate which make the study more complex first the existence and uniqueness of the global positive solution with any positive initial value are verified second using stochastic analysis methods and some differential inequality techniques we explore the long time asymptotic properties of the stochastic delayed system furthermore we prove that the solution of the stochastic delayed system has a unique stationary distribution and it has ergodic property third the sufficient conditions of permanence in mean or extinction for the ecological populations are obtained finally we introduce a series of computer simulations to show the performance of the theoretical results

a riemann hilbert approach for a new type coupled nonlinear schrodinger equations abstract we develop a riemann hilbert approach to the cauchy problem on the line for a new type of coupled nonlinear schrodinger cnls equations i q 1 t q 1 x x 2 q 1 2 2 q 2 2 q 1 2 q 2 2 q 1 0 i q 2 t q 2 x x 2 2 q 1 2 q 2 2 q 2 2 q 1 2 q 2 0 this approach allows us to give a representation of the solution to the cauchy problem of the cnls equations in terms of the solution of a 4 4 riemann hilbert problem formulated in the complex k plane due to the energy conservation law of above system is q 1 2 q 2 2 d x it is difficult to obtain a solution for this system by using the energy estimate method of pde s therefore this approach efficiently provides a new way in studying the nonlinear problems that pde s theory can t solve furthermore this representation is then used for retrieving the soliton solutions

on linear combinations of l orthogonal polynomials associated with distributions belonging to symmetric classes this paper deals with the classes s 3 w b b of strong distribution functions defined on the interval b 2 b b 0 b

proving total dual integrality with cross free families a general framework we present a theorem stating that certain classes of linear programming problems have integer optimal primal and dual solutions the theorem includes as special cases earlier results of johnson edmonds and giles frank hoffman and schwartz groflin and hoffman and lawler and martel the proof method consists of deriving total dual integrality for the corresponding system of linear inequalities from the total unimodularity of certain cross free subsystems the scheme presented here differs from the one proposed earlier by grishuhin in that grishuhin requires the total unimodularity of cross free subsystems in the axioms whereas here this follows from easier verifiable axioms

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