spectra of digraphs in this primarily expository paper we survey classical and some more recent results on the spectra of digraphs equivalently the spectra of 0 1 matrices with emphasis on the spectral radius

new exact periodic solitary wave solutions for the 3 1 dimensional generalized kp and bkp equations the extended homoclinic test approach ehta is a straightforward and efficient method to solve nonlinear partial differential equations in this paper new exact solutions including kinky periodic solitary wave solutions periodic soliton solutions and periodic kink solutions for the 3 1 dimensional generalized kadomtsev petviashvili kp and b type kadomtsev petviashvili bkp equations are obtained moreover the strangely mechanical features of these wave solutions are investigated these solutions enrich the variety of the dynamics of higher dimensional nonlinear wave field

a search for bilinear equations passing hirota s three soliton condition ii mkdv type bilinear equations in this paper the results of a search for complex bilinear equations with two soliton solutions are presented the following basic types are discussed a the nonlinear schrodinger equation b dx g f 0 a dx dt f f gg and b the benjamin ono equation p dx f f 0 it is found that the existence of two soliton solutions is not automatic but introduces conditions that are like the usual three and four soliton conditions the search was limited by the degree of a 2 and by degree of p 4 the main results are the following 1 iad3x dxdt idy b g f 0 d2xf f gg 2 d2x ad2y idt b g f 0 dxdy f f gg 3 iad3x d2x idt f f 0 and 4 dxdt i adx bdt f f 0

a new proof for decell s finite algorithm for generalized inverses a new proof for decell s finite algorithm for determining the moore penrose generalized inverse of a rectangular matrix is given in this proof we develop additional properties of the algorithm and point out how it can be used for further development of the algebraic properties of the moore penrose generalized inverse

2 1 dimensional mkdv n equations by the mkdv recursion operator multiple soliton and multiple singular soliton solutions abstract in this work we present a new variety of 2 1 dimensional mkdv n equations with distinct even orders we use the recursion operator of the mkdv equation to develop achieve this goal we show that these equations give multiple soliton solutions and multiple singular soliton solutions the same as the multiple soliton and multiple singular soliton solutions of the mkdv hierarchy except for the dispersion relations we also develop a generalization for the dispersion relations and the order of these equations

bounds for levinger s function of nonnegative almost skew symmetric matrices the analysis of the perron eigenspace of a nonnegative matrix a whose symmetric part has rank one is continued improved bounds for the perron root of levinger s transformation 1 a a t 0 1 and its derivative are obtained the relative geometry of the corresponding left and right perron vectors is examined the results are applied to tournament matrices to obtain a comparison result for their spectral radii

the extended tanh method for new solitons solutions for many forms of the fifth order kdv equations abstract the extended tanh method is used to derive new solitons solutions for several forms of the fifth order nonlinear kdv equation the forms include the lax sawada kotera sk sawada kotera parker dye skpd kaup kupershmidt kk kaup kupershmidt parker dye kkpd and the ito equations the criteria established in a m wazwaz abundant solitons solutions for several forms of the fifth order kdv equation by using the tanh method appl math comput in press doi 10 1016 j amc 2006 02 047 to build up reliable relations between the parameters of the equation are confirmed by using this new approach entirely new bell shaped solitons are determined

four 2 1 dimensional integrable extensions of the kadomtsev petviashvili equation in this work four 2 1 dimensional nonlinear extensions of the kadomtsev petviashvili kp equation are developed the complete integrability of these models are investigated multiple soliton solutions and multiple singular soliton solutions are determined to demonstrate the compatibility of these models the resonance phenomenon does not exist for any of the derived models

n soliton solutions for shallow water waves equations in 1 1 and 2 1 dimensions abstract a variety of shallow water waves equations in 1 1 and 2 1 dimensions are investigated we first show that these models are completely integrable we next determine multiple soliton solutions for each equation the simplified hirota s bilinear method developed by hereman will be employed to achieve this goal a comparison between dispersion relations and the phase shifts will be conducted but possess the same coefficients for the polynomials of exponentials

an upper bound on the perron value of an almost regular tournament matrix we provide an asymptotic upper bound on the perron value of an almost regular tournament matrix improving upon an existing bound due to friedland our approach employs techniques from contrained optimization

the generalized 1 1 dimensional and 2 1 dimensional ito equations the multiple exp function algorithm as a generalization of hirota s perturbation scheme is used to construct multiple wave solutions to the generalized 1 1 dimensional and 2 1 dimensional ito equations some of the resulting solutions involve generic phase shifts and wave frequencies

a variety of exact travelling wave solutions for the 2 1 dimensional boiti leon pempinelli equation abstract in this paper the 2 1 dimensional boiti leon pempinelli blp equation will be studied the tanh coth method will be used to obtain exact travelling wave solutions for this equation the exp function method will also be applied to the blp equation to derive a new variety of travelling wave solutions with distinct physical structures

quasi periodic wave solution and asymptotic behavior for the 2 1 dimensional toda lattice equation abstract one and two periodic wave solutions for the 2 1 dimensional toda lattice equation are presented based on the hirota bilinear method and the riemann theta function the asymptotic behaviors of these two solutions are considered and the rigorous proof is given that the periodic wave solutions tend to the soliton solutions in an appropriate limiting procedure

non integrable variants of boussinesq equation with two solitons abstract three variants of the boussinesq equation namely the 2 1 dimensional boussinesq equation the 3 1 dimensional boussinesq equation and the sixth order boussinesq equation are studied the hirota bilinear method is used to construct two soliton solutions for each equation the study highlights the fact that these equations are non integrable and do not admit n soliton solutions although these equations can be put in bilinear forms

regular soliton solutions and singular soliton solutions for the modified kadomtsev petviashvili equations abstract in this paper we study the positive and the negative modified kadomtsev petviashvili mkp equations the mkp equations are completely integrable equations that admit n soliton solutions and an infinite number of conserved densities multiple soliton solutions are obtained for the positive model whereas multiple singular soliton solutions are established for the negative model the analysis depends mainly on the hirota s bilinear method

on almost regular tournament matrices abstract spectral and determinantal properties of a special class m n of 2n 2n almost regular tournament matrices are studied in particular the maximum perron value of the matrices in this class is determined and shown to be achieved by the brualdi li matrix which has been conjectured to have the largest perron value among all tournament matrices of even order we also establish some determinantal inequalities for matrices in m n and describe the structure of their associated walk spaces

breather type multi solitary waves to the kadomtsev petviashvili equation with positive dispersion abstract in this work kp equation with positive dispersion is studied by virtue of hirota bilinear method and generalized three wave type of ansatz approach we obtain the breather type multi solitary waves with fission and fusion behaviors of kp equation based on the structures of those 2 1 dimensional solutions and show via computer simulation the fission and fusion interactions characteristics of breather type multi solitary waves solutions are discussed which might provide us with the useful information on the dynamics of the relevant physical fields

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

solitons and rouge waves for a generalized 3 1 dimensional variable coefficient kadomtsev petviashvili equation in fluid mechanics evolution of the long water waves and small amplitude surface waves with the weak nonlinearity weak dispersion and weak perturbation in fluid mechanics in three spatial dimensions can be described by a generalized 3 1 dimensional variable coefficient kadomtsev petviashvili equation which is studied in this paper with symbolic computation via the truncated painleve expansion an auto backlund transformation is derived based on which under certain variable coefficient constraints one soliton two soliton homoclinic breather wave and rouge wave solutions are respectively obtained via the hirota method graphic analysis shows that the soliton propagates with the varying soliton direction change of the value of any one of g t m t n t h t q t and l t in the equation can cause the change of the soliton shape while the soliton amplitude cannot be affected by that change where g t represents the dispersion m t and n t respectively stand for the disturbed wave velocities along the y and z directions h t q t and l t are the perturbed effects y and z are the scaled spatial coordinates and t is the temporal coordinate soliton direction and type of the interaction between the two solitons can vary with the change of the value of g t while they cannot be affected by m t n t h t q t and l t homoclinic breather wave and rouge wave are respectively displayed where the rouge wave comes from the extreme behaviour of the homoclinic breather wave

a 3 1 dimensional nonlinear evolution equation with multiple soliton solutions and multiple singular soliton solutions in this work a 3 1 dimensional nonlinear evolution equation is investigated the hirota s bilinear method is applied to determine the necessary conditions for the complete integrability of this equation multiple soliton solutions are established to confirm the compatibility structure multiple singular soliton solutions are also derived the resonance phenomenon does not exist for this model

orthogonal laurent polynomials on the unit circle extended cmv ordering and 2d toda type integrable hierarchies abstract we connect the theory of orthogonal laurent polynomials on the unit circle and the theory of toda like integrable systems using the gauss borel factorization of a cantero moral velazquez moment matrix that we construct in terms of a complex quasi definite measure supported on the unit circle the factorization of the moment matrix leads to orthogonal laurent polynomials on the unit circle and the corresponding second kind functions we obtain jacobi operators 5 term recursion relations christoffel darboux kernels and corresponding christoffel darboux formulas from this point of view in a completely algebraic way we generalize the cantero moral velazquez sequence of laurent monomials recursion relations christoffel darboux kernels and corresponding christoffel darboux formulas in this extended context we introduce continuous deformations of the moment matrix and we show how they induce a time dependent orthogonality problem related to a toda type integrable system which is connected with the well known toeplitz lattice we obtain the lax and zakharov shabat equations using the classical integrability theory tools we explicitly derive the dynamical system associated with the coefficients of the orthogonal laurent polynomials and we compare it with the classical toeplitz lattice dynamical system for the verblunsky coefficients of szeg polynomials for a positive measure discrete flows are introduced and related to darboux transformations finally we obtain the representation of the orthogonal laurent polynomials and their second kind functions using the formalism of miwa shifts in terms of functions and the subsequent bilinear equations

the n component kp hierarchy and representation theory it is the aim of the present article to give all formulations of the n component kp hierarchy and clarify connections between them the generalization to the n component kp hierarchy is important because it contains many of the most popular systems of soliton equations like the davey stewartson system for n 2 the two dimensional toda lattice for n 2 the n wave system for n 3 and the darboux egoroff system it also allows us to construct natural generalizations to the davey stewartson and toda lattice systems of course the inclusion of all these systems in the n component kp hierarchy allows us to construct their solutions by making use of vertex operators

bounds for the perron root using max eigenvalues abstract using the techniques of max algebra a new proof of al pin s lower and upper bounds for the perron root of a nonnegative matrix is given the bounds depend on the row sums of the matrix and its directed graph if the matrix has zero main diagonal entries then these bounds may improve the classical row sum bounds this is illustrated by a generalized tournament matrix

a note on the sequence of brualdi li matrices abstract we consider the sequence 2n n 1 2 2n where 2n is the perron value of the brualdi li matrix of order 2n we prove that the sequence converges find its limit and show that asymptotically the sequence is monotonically decreasing in particular this addresses some problems raised by friedland and katz

complete integrability of the kadomtsev petviashvili equation abstract the total hierarchy of the kadomtsev petviashvili kp equation is transformed to the system of linear partial differential equations with constant coefficients the complete integrability of the kp equation is proved by using this linear system the existence and uniqueness theorem of the cauchy problem of the kp hierarchy is obtained

a finite algorithm for the drazin inverse of a polynomial matrix based on greville s finite algorithm for drazin inverse of a constant matrix we propose a finite numerical algorithm for the drazin inverse of polynomial matrices we also present a new proof for decell s finite algorithm through greville s finite algorithm

n soliton solutions for the integrable bidirectional sixth order sawada kotera equation abstract in this work the integrable bidirectional sixth order sawada kotera equation is examined the equation considered is a kdv6 equation that was derived from the fifth order sawada kotera equation multiple soliton solutions and multiple singular soliton solutions are formally derived for this equation the cole hopf transformation method combined with the hirota s bilinear method are used to determine the two sets of solutions where each set has a distinct structure

the tanh coth method for solitons and kink solutions for nonlinear parabolic equations abstract the tanh coth method is used to derive solitons and kink solutions for some of the well known nonlinear parabolic partial differential equations the equations include the fisher equation newell whithead equation allen cahn equation fitzhugh nagumo equation fisher s equation and the burgers fisher equation the new tanh coth approach provides abundant solitons and kink solutions in addition to the existing ones the power of this manageable method is confirmed

exact three wave solution for higher dimensional kdv type equation a new periodic type of three wave solutions including periodic two solitary solution doubly periodic solitary solution and breather type of two solitary solution for the 1 2 dimensional and 1 3 dimensional kdv type equations are obtained using hirota s bilinear form and generalized three wave type of ansatz approach

the interaction processes of the n soliton solutions for an extended generalization of vakhnenko equation the n soliton solutions for an extended generalization of vakhnenko equation are calculated by applying the improved hirota method and the variable transformations the n soliton solutions can be expressed explicitly furthermore the interaction processes are discussed for the n soliton solutions

multiple soliton solutions for the sixth order ramani equation and a coupled ramani equation in this work the completely integrable sixth order nonlinear ramani equation and a coupled ramani equation are studied multiple soliton solutions and multiple singular soliton solutions are formally derived for these two equations the hirota s bilinear method is used to determine the two distinct structures of solutions the resonance relations for the three cases are investigated

resonant soliton and complexiton solutions for 3 1 dimensional boiti leon manna pempinelli equation abstract in this letter the linear superposition principle is used to discuss the 3 1 dimensional boiti leon manna pempinelli equation with bilinear derivatives as a result we obtain new resonant soliton and complexiton solutions by discussing two different cases involved the parameters these solutions are a class of n wave solutions of linear combinations of exponential traveling waves

the hirota s direct method for multiple soliton solutions for three model equations of shallow water waves abstract multiple soliton solutions for three model equations for shallow water waves are determined the three models are completely integrable the hirota bilinear method is used to determine multiple soliton solutions of sech squared type for these equations the tanh coth method is used to obtain single soliton solutions and other solutions for these three models the three models have different linear dispersion relations but possess the same coefficients for the polynomials of exponentials

two systems of two component integrable equations multiple soliton solutions and multiple singular soliton solutions abstract in this work we study two systems of two component integrable equations the cole hopf transformation and hirota s bilinear method are applied to emphasize the integrability of each system multiple soliton solutions and multiple singular soliton solutions are formally derived

multiple soliton solutions for the 2 1 dimensional asymmetric nizhnik novikov veselov equation abstract the asymmetric nizhnik novikov veselov equation will be investigated the hirota bilinear method will be used to derive n soliton solutions of this completely integrable equation multiple singular soliton solutions will be developed as well the asymmetric nizhnik novikov veselov equation does not show any resonant phenomenon

toda lattice hierarchy and generalized string equations string equations of thepth generalized kontsevich model and the compactifiedc 1 string theory are re examined in the language of the toda lattice hierarchy as opposed to a hypothesis postulated in the literature the generalized kontsevich model atp 1 does not coincide with thec 1 string theory at selfdual radius a broader family of solutions of the toda lattice hierarchy including these models is constructed and shown to satisfy generalized string equations the status of a variety ofc 1 string models is discussed in this new framework

multiple soliton solutions for the lax kadomtsev petviashvili lax kp equation a new completely integrable dispersive equation is derived the new equation is obtained by extending the lax fifth order equation using the sense of the kadomtsev petviashvili kp equation in extending the kdv equation the newly derived lax kadomtsev petviashvili lax kp equation is investigated by using the tanh coth method and the hirota bilinear method to derive single soliton solution and n soliton solutions respectively the study highlights the multiple soliton solutions of the derived completely integrable equation

multiple orthogonal polynomials of mixed type gauss borel factorization and the multi component 2d toda hierarchy multiple orthogonality is considered in the realm of a gauss borel factorization problem for a semi infinite moment matrix perfect combinations of weights and a finite borel measure are constructed in terms of m nikishin systems these perfect combinations ensure that the problem of mixed multiple orthogonality has a unique solution that can be obtained from the solution of a gauss borel factorization problem for a semi infinite matrix which plays the role of a moment matrix this leads to sequences of multiple orthogonal polynomials their duals and second kind functions it also gives the corresponding linear forms that are bi orthogonal to the dual linear forms expressions for these objects in terms of determinants from the moment matrix are given recursion relations are found which imply a multi diagonal jacobi type matrix with snake shape and results like the abc theorem or the christoffel darboux formula are re derived in this context using the factorization problem and the generalized hankel symmetry of the moment matrix the connection between this description of multiple orthogonality and the multi component 2d toda hierarchy which can be also understood and studied through a gauss borel factorization problem is discussed deformations of the weights natural for m nikishin systems are considered and the correspondence with solutions to the integrable hierarchy represented as a collection of lax equations is explored corresponding lax and zakharov shabat matrices as well as wave functions and their adjoints are determined the construction of discrete flows is discussed in terms of miwa transformations which involve darboux transformations for the multiple orthogonality conditions the bilinear equations are derived and the tau function representation of the multiple orthogonality is given

2 1 dimensional burgers equations be m n 1 using the recursion operator in this work we develop a variety of 2 1 dimensional burgers equations be n of distinct orders we use the recursion operator of the burgers equation to derive these equations the obtained equations possess multiple kink solutions the same as the multiple kink solutions of the burgers hierarchy but differ only in the dispersion relations we finally establish a generalization for the dispersion relations

a finite algorithm for computing the weighted moore penrose inverse a mn this paper presents a finite algorithm for computing the weighted moore penrose inverse of a rectangular matrix this algorithm may be viewed as an extension of decell s algorithm

exact solutions of nonlinear evolution equations with variable coefficients using exp function method abstract in this paper we apply the exp function method to construct generalized solitary and periodic solutions of nonlinear evolution equations with variable coefficients the proposed technique is tested on the zakharov kuznetsov and 2 1 dimensional broer kaup equations with variable coefficients these equations play a very important role in mathematical physics and engineering sciences the suggested algorithm is quite efficient and is practically well suited for use in these problems obtained results clearly indicate the reliability and efficiency of the proposed exp function method

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

soliton solutions backlund transformation and wronskian solutions for the extended 2 1 dimensional konopelchenko dubrovsky equations in fluid mechanics abstract this paper is to investigate the extended 2 1 dimensional konopelchenko dubrovsky equations which can be applied to describing some phenomena in the stratified shear flow the internal and shallow water waves and plasmas bilinear form equations are transformed from the original equations and n soliton solutions are derived via symbolic computation bilinear form backlund transformation and single soliton solution are obtained and illustrated wronskian solutions are constructed from the backlund transformation and single soliton solution

a new approach to the self dual yang mills equations inspired by sato s new theory for soliton equations we find a new approach to the self dual yang mills equations we first establish a correspondence of solutions between the self dual yang mills equations and a new system of equations with infinitely many unknown functions it then turns out that the latter equations can be easily solved by a simple explicit procedure this leads to an explicit description of a very broad class of solutions to the self dual yang mills equations and also to a construction of transformations acting on these solutions

solvability of the super kp equation and a generalization of the birkhoff decomposition the unique solvability of the initial value problem for the total hierarchy of the super kadomtsev petviashvili system is established to prove the existence we use a generalization of the birkhoff decomposition which is obtained by replacing the loop variable and loop groups in the original setting by a super derivation operator and groups of infinite order super micro i e pseudo differential operators to show the uniqueness we generalize the fact that every flat connection admits horizontal sections to the case of an infinite dimensional super algebra bundle defined over an infinite dimensional super space the usual kp system with non commutative coefficients is also studied the kp system is obtained from the super kp system by reduction modulo odd variables on the other hand the first modified kp equation can be obtained from the super kp system by elimination of odd variables thus the super kp system is a natural unification of the kp system and the modified kp systems

an alternative limit expression of drazin inverse and its application abstract we present an alternative limit expression of drazin inverse of a square matrix and a new proof for a finite algorithm for determining the drazin inverse is given using our new limit expression

multiple soliton solutions and multiple singular soliton solutions for the 3 1 dimensional burgers equations abstract in this work the 3 1 dimensional integrable burgers equation is studied multiple kink solutions and multiple singular kink solutions are established for this completely integrable equation the backlund transformation method combined with the hirota s bilinear sense are used to determine two distinct structures of regular soliton solutions and singular soliton solutions the tanh coth method will also be implemented to obtain new single soliton solutions

an efficient algorithm to construct multi soliton rational solutions of the 2 1 dimensional kdv equation with variable coefficients abstract herein we present an efficient algorithm to construct multi soliton rational solutions of the 2 1 dimensional korteweg de vries equation with time dependent coefficients we employ the generalized unified method which presents a wider applicability for handling many other nonlinear evolution equations in different branches of science to find these solutions the dynamical behavior of the traveling wave solutions and their structures are discussed by different choices for the arbitrary functions in the obtained solutions

the integrable kdv6 equations multiple soliton solutions and multiple singular soliton solutions abstract in this work the completely integrable sixth order nonlinear equations kdv6 are studied three distinct cases of these equations are selected for a reliable treatment multiple soliton solutions and multiple singular soliton solutions are formally derived for these equations the cole hopf transformation method combined with the hirota s bilinear sense are used to determine the two distinct structures of solutions the resonance relations for the three cases are investigated

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

extensions of faddeev s algorithms to polynomial matrices starting from algorithms introduced in ky m vu an extension of the faddeev s algorithms in proceedings of the ieee multi conference on systems and control on september 3 5th 2008 san antonio tx which are applicable to one variable regular polynomial matrices we introduce two dual extensions of the faddeev s algorithm to one variable rectangular or singular matrices corresponding algorithms for symbolic computing the drazin and the moore penrose inverse are introduced these algorithms are alternative with respect to previous representations of the moore penrose and the drazin inverse of one variable polynomial matrices based on the leverrier faddeev s algorithm complexity analysis is performed algorithms are implemented in the symbolic computational package mathematica and illustrative test examples are presented

normal matrices and their principal submatrices of co order one abstract let a be a normal matrix v be any of its indices a v be the matrix obtained from a by deleting the v th row and column and be an eigenvalue of a v in our paper we construct the eigenspace of a associated with from that of a v in particular it is shown that if there is a unique jordan block of size strictly greater than one in the part of the jordan form of a v corresponding to then the geometric multiplicity of decreases by one under the transition from a v to a in other words the typical change of the spectral properties holds for the results obtained are applied to circulant matrices moreover in appendix to our paper we consider almost regular tournament matrices as principal submatrices of co order one of regular tournament matrices in particular it is observed that the brualdi li tournament matrix b 2 n of order 2n is permutationally similar to a principal submatrix of co order one of the circulant matrix of order 2n 1 with the first row 0 1 1 n 0 0 n as a consequence of this fact the weak brualdi li conjecture is formulated for principal submatrices of co order one of the adjacency matrices of cayley tournaments

n soliton solutions for the combined kdv cdg equation and the kdv lax equation abstract the hirota ito approach presented in r hirota m ito resonance of solitons in one dimension j phys soc jpn 52 3 1983 744 748 for extending fifth order integrable equations with a nonvanishing boundary conditions to combined equations is used in this work the generalized fifth order caudrey dodd gibbon cdg and lax equations are extended to combined integrable equations the hirota s bilinear method is used to derive multiple soliton solutions for the extended kdv cdg and kdv lax equations

multiple kink solutions for m component burgers equations in 1 1 dimensions and 2 1 dimensions abstract m component burgers equations in 1 1 dimensions and 2 1 dimensions are examined for complete integrability the cole hopf transformation method and the simplified form of hereman s method are used to achieve this goal multiple kink solutions and multiple singular kink solutions are formally derived for each vector equation

bilinear backlund transformations kink periodic solitary wave and lump wave solutions of the bogoyavlenskii kadomtsev petviashvili equation abstract this paper investigates the bogoyavlenskii kadomtsev petviashvili equation by using the binary bell polynomials method and differential constraint technique several kinds of more general bilinear representations and baklund transformations are presented at the same time multiple wave solutions including the kink periodic solitary wave and bright dark lump wave solutions are constructed the resonant interactions between two kink solitary waves are investigated and exhibited mathematically and graphically lump wave solutions a kind of rational function localized in all directions of the space are obtained by the taylor expansion of the kink periodic solitary wave also we discuss the dynamical behavior of the lump wave solutions and their structures

the hirota s bilinear method and the tanh coth method for multiple soliton solutions of the sawada kotera kadomtsev petviashvili equation abstract in this work we derive a new completely integrable dispersive equation the equation is obtained by combining the sawada kotera sk equation with the sense of the kadomtsev petviashvili kp equation the newly derived sawada kotera kadomtsev petviashvili sk kp equation is studied by using the tanh coth method to obtain single soliton solution and by the hirota bilinear method to determine the n soliton solutions the study highlights the significant features of the employed methods and its capability of handling completely integrable equations

young diagrams and n soliton solutions of the kp equation we consider n soliton solutions of the kp equation an n soliton solution is a solution u x y t which has the same set of n line soliton solutions in both asymptotics y and y the n soliton solutions include all possible resonant interactions among those line solitons we then classify those n soliton solutions by defining a pair of n numbers n n with n n 1 n n n j 1 2n which labels n line solitons in the solution the classification is related to the schubert decomposition of the grassmann manifolds gr n 2n where the solution of the kp equation is defined as a torus orbit then the interaction pattern of n soliton solution can be described by the pair of young diagrams associated with n n we also show that n soliton solutions of the kdv equation obtained by the constraint u y 0 cannot have resonant interaction

four 2 1 dimensional integrable extensions of the kdv equation multiple soliton and multiple singular soliton solutions in this work four 2 1 dimensional nonlinear completely integrable equations generated by extending the kdv equation are developed the necessary condition for the complete integrability of these equation are formally derived multiple soliton solutions and multiple singular soliton solutions are determined to emphasize the compatability of these models the dispersion relations of these models are characterized by distinct physical structures the resonance phenomenon for these equations does not exist for any model

multiple soliton solutions and multiple singular soliton solutions for two higher dimensional shallow water wave equations two 3 1 dimensional shallow water wave equations are studied for complete integrability the hirota s bilinear method is used to determine the multiple soliton solutions for these equations moreover multiple singular soliton solutions will also be determined for each model the analysis highlights the capability of the direct method in handling completely integrable equations

multiple soliton solutions of two extended model equations for shallow water waves abstract multiple soliton solutions for two extensions of model equations for shallow water waves presented in m j ablowitz d j kaup a c newell h segur the inverse scattering transform fourier analysis for nonlinear problems stud appl math 53 1974 249 315 r hirota j satsuma n soliton solutions of model equations for shallow water waves j phys soc jpn 40 2 1976 611 612 are obtained the two extensions possess identical dispersion relations but with different structures for the n soliton solutions for n 1 the hirota bilinear method is used to determine multiple soliton solutions of sech squared type for these extended equations the tanh coth method is used to obtain single soliton solutions

a note on perron vectors for almost regular tournament matrices abstract let t be an almost regular tournament matrix of order n with right perron vector w we show that if the i th row sum of t is n 2 2 and the j th row sum is n 2 then w i w j thus in the round robin competition corresponding to t the ranking schemes of kendall and wei and of ramanujacharyula agree with the ranking generated by the row sums of t

n soliton solutions of the kp equation by exp function method abstract in this paper the exp function method is generalized to determine the 1 soliton 2 soliton and 3 soliton solutions of the kadomtsev petviashvili kp equation then the uniform formula of n soliton solutions is derived the method is simple and straightforward in constructing the n soliton of the nonlinear evolution equation in mathematical physics

perron vector bounds for a tournament matrix with applications to a conjecture of brualdi and li abstract for a primitive generalized tournament matrix we present upper and lower bounds on an entry in its perron vector in terms of the corresponding row sum of the matrix the bounds are then used to help prove that if n is even and sufficiently large any tournament matrix of order n which maximizes the perron value must be almost regular throughout we use both analytic and combinatorial techniques

new exact solutions for solving the initial value problem of the kdv kp equation via the lie group method the traveling wave solutions of the two dimensional kdv kp equation are studied via the lie group method the kdv kp equation is a nonlinear partial differential equation in two spatial and one temporal coordinate which describes the evolution of nonlinear long waves of small amplitude with slow dependence on the transverse coordinate the investigation of exact solutions of the kdv kp via the application of lie group and the extended f expansion methods plays the main role in understanding the nonlinear physical phenomena

multiple kink solutions and multiple singular kink solutions for the 2 1 dimensional burgers equations abstract in this work we study the 2 1 dimensional painleve integrable burgers equation multiple kink solutions and multiple singular kink solutions are formally derived for this equation the analysis depends mainly on the hirota s bilinear method

on the application of the exp function method to the kp equation for n soliton solutions abstract we observe that the form of the kadomstev petviashvili equation studied by yu 2011 s yu n soliton solutions of the kp equation by exp function method appl math comput 2011 doi 10 1016 j amc 2010 12 095 is incorrect we claim that the n soliton solutions obtained by means of the basic exp function method and some of its known generalizations do not satisfy the equation considered we emphasize that yu s results except only one cannot be solutions of the correct form of the kadomstev petviashvili equation in addition we provide some correct results using the same approach

single and multiple soliton solutions for the 2 1 dimensional kdv equation abstract the 2 1 dimensional kdv equation is studied by four distinct methods the hirota s bilinear method is used to derive multiple soliton solutions for this completely integrable equation the tanh coth method and the cosh ansatz are used to obtain single soliton solutions and travelling wave solutions the exp function method gives travelling wave solutions with some free parameters the work highlights the reliability of the four methods

solution of the conjecture of brualdi and li abstract we establish the conjecture of brualdi and li on the maximal perron root of a tournament matrix of even order

multiple soliton solutions for the kp equation by hirota s bilinear method and by the tanh coth method abstract in this work we use the hirota s direct method combined with the simplified version of this method to determine the n soliton solutions n 1 2 3 4 5 for the kadomtsev petviashvili kp equation the one soliton solutions will be handled by the tanh coth method the work emphasizes the power of the two methods that will be used

solitary wave solutions of the generalized shallow water wave gsww equation by hirota s method tanh coth method and exp function method abstract the generalized short wave equation gsww is studied by three distinct methods the hirota s bilinear method is used to derive multiple soliton solutions for the completely integrable forms of the gsww equation the tanh coth method is used to obtain single soliton solutions for this equation the exp function method is also applied to derive a variety of travelling wave solutions with distinct physical structures for this non linear evolution equation

1 soliton solution of the generalized kp equation with generalized evolution this paper obtains the 1 soliton solution of three variants of the generalized kp equation with generalized evolution the solitary wave ansatz is used to carry out the integration of such equations the parameter domain is also identified in the process the numerical simulations are also obtained in this paper

a sub ode method for generalized gardner and bbm equation with nonlinear terms of any order abstract in this paper a method with the aid of a sub ode and its solutions is used for constructing new periodic wave solutions for nonlinear gardner equation and bbm equation with nonlinear terms of any order arising in mathematical physics as a result many exact traveling wave solutions are successfully obtained the method in the paper is very direct and it can also be applied to other nonlinear evolution equations

the cole hopf transformation and multiple soliton solutions for the integrable sixth order drinfeld sokolov satsuma hirota equation abstract the completely integrable sixth order nonlinear drinfeld sokolov satsuma hirota equation is studied three distinct methods namely the cole hopf transformation method the tanh coth method and the exp function method are used for a reliable treatment of this equation solitons multiple soliton solutions multiple singular soliton solutions and plane periodic solutions are obtained for this equations the study highlights the power of each method

multiple soliton solutions for the ninth order kdv equation and sixth order boussinesq equation abstract two nonlinear dispersive equations namely the ninth order kdv equation and the sixth order boussinesq equation are formally derived by generalizing the bilinear forms of the kdv and boussinesq equations respectively the two equations are approached by using the tanh coth method to obtain single soliton solutions and by the hirota bilinear method to determine the multiple soliton solutions the study highlights the fact that both equations are completely integrable and admits n soliton solutions

analytic solutions of the 2 1 dimensional nonlinear evolution equations using the sine cosine method in this paper we establish exact solutions for 2 1 dimensional nonlinear evolution equations the sine cosine method is used to construct exact periodic and soliton solutions of 2 1 dimensional nonlinear evolution equations many new families of exact traveling wave solutions of the 2 1 dimensional boussinesq breaking soliton and bkp equations are successfully obtained these solutions may be important of significance for the explanation of some practical physical problems it is shown that the sine cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics