symmetry preserving difference schemes for some heat transfer equations lie group analysis of differential equations is a generally recognized method which provides invariant solutions integrability conservation laws etc in this paper we present three characteristic examples of the construction of invariant difference equations and meshes where the original continuous symmetries are preserved in discrete models conservation of symmetries in difference modelling helps to retain qualitative properties of the differential equations in their difference counterparts

new periodic and soliton solutions by application of exp function method for nonlinear evolution equations in this letter the kaup kupershmidt 2 1 dimensional potential kadomtsev petviashvili shortly pkp equations are presented and the exp function method is employed to compute an approximation to the solution of nonlinear differential equations governing the problem it has been attempted to show the capabilities and wide range applications of the exp function method this method can be used as an alternative to obtain analytic and approximate solution of different types of differential equations applied in engineering mathematics

distinct variants of the kdv equation with compact and noncompact structures in this paper we examine three distinct variants of the generalized kdv equation both the compact and the noncompact structures are treated for these variants we also emphasize the different physical structures of the two classes the focusing branch and the defocusing branch

new exact solutions for the kdv equation with higher order nonlinearity by using the variational method the korteweg de vries kdv equation with higher order nonlinearity models the wave propagation in one dimensional nonlinear lattice a higher order extension of the familiar kdv equation is produced for internal solitary waves in a density and current stratified shear flow with a free surface the variational approximation method is applied to obtain the solutions for the well known kdv equation explicit solutions are presented and compared with the exact solutions very good agreement is achieved demonstrating the high efficiency of variational approximation method the existence of a lagrangian and the invariant variational principle for the higher order kdv equation are discussed the simplest version of the variational approximation based on trial functions with two free parameters is demonstrated the jost functions by quadratic cubic and fourth order polynomials are approximated also we choose the trial jost functions in the form of exponential and sinh solutions all solutions are exact and stable and have applications in physics

symmetries as integrability criteria for differential difference equations in this paper we review the results obtained by the generalized symmetry method in the case of differential difference equations during the last 20 years together with general theory of the method classification results are discussed for classes of equations which include the volterra toda and relativistic toda lattice equations

patterns on liquid surfaces cnoidal waves compactons and scaling abstract localized patterns and nonlinear oscillation formations on the bounded free surface of an ideal incompressible liquid are investigated cnoidal modes solitons and compactons as traveling non axially symmetric shapes are discussed a finite difference differential generalized korteweg de vries kdv equation is shown to describe the three dimensional motion of the fluid surface and in the limit of long and shallow channels one recovers the well known kdv equation a tentative expansion formula for the representation of the general solution of a nonlinear equation for given initial conditions is introduced the model is useful in multilayer fluid dynamics cluster formation and nuclear physics since up to an overall scale these systems display a free liquid surface behavior

bifurcations of travelling wave solutions for 2 1 dimensional boussinesq type equation variants of the two dimensional boussinesq type equations with positive and negative exponents are studied the bifurcation theory of dynamic systems is fruitfully used to carry out the analysis the dynamical behavior of different physical structure solitary patterns solitons kink breaking and periodic wave solutions is obtained the quantitative change in the physical structure of the solutions is shown to depend on the systemic parameters under different parametric conditions various sufficient conditions to guarantee the existence of the above solutions are given

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

new solitary wave solutions to the modified forms of degasperis procesi and camassa holm equations abstract in this work we establish new solitary wave solutions to the modified forms of degasperis procesi and camassa holm equations unlike the standard degasperis procesi and camassa holm equations where multi peakon solutions arise the modified forms cause a change in the characteristic of these solutions and change it to bell shaped solitons periodic and complex solutions the extended tanh method the rational hyperbolic functions method and the rational exponential functions method are employed to reveal these new solutions

some physical structures for the 2 1 dimensional boussinesq water equation with positive and negative exponents in this paper a mathematical method is constructed to study two variants of the two dimensional boussinesq water equation with positive and negative exponents in terms of travelling wave solutions the partial differential equations are transformed to nonlinear ordinary differential equations exact solutions are then derived for various cases to describe the different physical structures such as compactons solitons solitary patterns and periodic solutions the exponent of the wave function u and the ratio of the two coefficients a and b in the boussinesq equation are shown to qualitatively determine the physical structures of the solutions

an extended simplest equation method and its application to several forms of the fifth order kdv equation abstract in this paper an extended simplest equation method is proposed to seek exact travelling wave solutions of nonlinear evolution equations as applications many new exact travelling wave solutions for several forms of the fifth order kdv equation are obtained by using our method the forms include the lax sawada kotera sawada kotera parker dye caudrey dodd gibbon kaup kupershmidt kaup kupershmidt parker dye and the ito forms

algebraic gauss manin systems and brieskorn modules we study the algebraic gauss manin system and the algebraic brieskorn module associated to a polynomial mapping with isolated singularities since the algebraic gauss manin system does not contain any information on the cohomology of singular fibers we first construct a non quasi coherent sheaf which gives the cohomology of every fiber then we study the algebraic brieskorn module and show that its position in the the algebraic gauss manin system is determined by a natural map to quotients of local analytic gauss manin systems and its pole part by the vanishing cycles at infinity comparing it with the deligne extension this implies for example a formula for the determinant of periods in the two dimensional case we can describe the global structure of the algebraic gauss manin system rather explicitly

center conditions rigidity of logarithmic differential equations abstract in this paper we prove that any degree d deformation of a generic logarithmic polynomial differential equation with a persistent center must be logarithmic again this is a generalization of ilyashenko s result on hamiltonian differential equations the main tools are picard lefschetz theory of a polynomial with complex coefficients in two variables specially the gusein zade a campo s theorem on calculating the dynkin diagram of the polynomial and the action of gauss manin connection on the so called brieskorn lattice petrov module of the polynomial we will also generalize j p francoise recursion formula and condition for a polynomial which is a product of lines in a general position some applications on the cyclicity of cycles and the bautin ideals will be given

nonlinear schrodinger type equations from multiscale reduction of pdes ii necessary conditions of integrability for real pdes the multiscale method has been applied in j math phys 41 6399 2000 to evolution pdes to obtain various nonlinear schrodinger type equations here these results are exploited to derive necessary conditions of integrability for nonlinear real evolution pdes in 1 1 dimensions these conditions are given by explicit formulas and several examples are detailed

periodic wave solutions and solitary wave solutions of generalized modified boussinesq equation and evolution relationship between both solutions abstract in this paper we focus on studying the exact solitary wave solutions and periodic wave solutions of the generalized modified boussinesq equation u t t u t t x x a 1 u a 2 u 2 a 3 u 3 x x 0 as well as the evolution relationship between these solutions detailed qualitative analysis is conducted on traveling wave solutions of this equation and global phase portraits in various parameter conditions are proposed various significant results about the existence of both solutions including three forms of solitary wave solutions and four exact bounded periodic wave solutions in different conditions are obtained then we further discuss the relationship between energy of hamiltonian system corresponding to this equation and the periodic wave solutions and solitary wave solutions it is concluded that the essential reason of periodic wave solutions and solitary wave solutions is the different values for the energy of hamiltonian system corresponding to this equation in addition the limited relations of periodic wave solutions and solitary wave solutions with the energy of hamiltonian system are proposed and the schematic diagram of evolution from periodic wave solutions to solitary wave solutions is drawn

the extended tanh method for abundant solitary wave solutions of nonlinear wave equations the extended tanh method is used to establish abundant solitary wave solutions of nonlinear wave equations the obtained solutions include solitons kinks and plane periodic solutions the study is an extension to the remarkable development by malfliet 1 the extended tanh method presents a wider applicability for handling nonlinear wave equations

multiscale reduction of discrete korteweg de vries equations we show how through a multiscale reduction technique performing the analysis at orders beyond the nonlinear schrodinger equation one can effectively prove if some nonlinear partial difference equation is not integrable the example is carried out on a symmetric discretization of the kdv equation and is compared to a similar reduction performed on the integrable lattice potential kdv equation

compactons and solitary patterns structures for variants of the kdv and the kp equations in this paper we discuss mathematical variants in higher dimensions of the kdv and the kp equations it is shown that the focusing branches of these variants exhibit compactons solitons with finite wavelength whereas the defocusing branches support solitary patterns solutions with infinite slopes or cusps the study presents a fairly complete understanding of the compact and noncompact dispersive structures

difference schemes with point symmetries and their numerical tests symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented they have the same three or four dimensional symmetry groups as the original differential equations the new difference schemes are tested as numerical methods the obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost for an example involving a solution with a singularity in the integration region the symmetry preserving scheme contrary to standard ones provides solutions valid beyond the singular point

a computational approach to soliton solutions of the kadomtsev petviashvili equation in this paper we present a computational approach to develop soliton solutions of the nonlinear kadomtsev petviashvili equation our approach rests mainly on the adomian decomposition method to include few components of the decomposition series the proposed framework is presented in a general way so that it can be used in nonlinear evolution equations of the same type numerical examples are tested to illustrate the proposed scheme

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

darboux transformation and explicit solutions for the integrable sixth order kdv equation for nonlinear waves abstract korteweg de vries kdv type equations can describe some physical phenomena in fluids nonlinear optics quantum mechanics plasmas etc in this paper with the aid of symbolic computation the integrable sixth order kdv equation is investigated darboux transformation dt with an arbitrary parameter is presented explicit solutions are derived with the dt relevant properties are graphically illustrated which might be helpful to understand some physical processes in fluids plasmas optics and quantum mechanics

solitary wave solutions of nonlinear evolution and wave equations using a direct method and macsyma the direct algebraic method for constructing travelling wave solutions on nonlinear evolution and wave equations has been generalized and systematized the class of solitary wave solutions is extended to analytic rather than rational functions of the real exponential solutions of the linearized equation expanding the solutions in an infinite series in these real exponentials an exact solution of the nonlinear pde is obtained whenever the series can be summed methods for solving the nonlinear recursion relation for the coefficients of the series and for summing the series in closed form are discussed the algorithm is now suited to solving nonlinear equations by any symbolic manipulation program this direct method is illustrated by constructing exact solutions of a generalized kdv equation the kuramoto sivashinski equation and a generalized fisher equation

compactons and solitary wave solutions for the boussinesq wave equation and its generalized form in this paper we examine the boussinesq wave equation and its generalized form compactons solitons solitary patterns and periodic solutions are obtained the tanh method is effectively employed in this work

the korteweg de vries kawahara equation in a bounded domain and some numerical results abstract we are concerned with the initial boundary problem associated to the korteweg de vries kawahara perturbed by a dispersive term which appears in several fluids dynamics problems we obtain local smoothing effects that are uniform with respect to the size of the interval we also propose a simple finite different scheme for the problem and prove its unconditional stability finally we give some numerical examples

new exact solutions of the double sine gordon equation using symbolic computations the double sine gordon equation is studied some complex hyperbolic functions are proposed to derive travelling wave solutions based on the symbolic computation program maple many new exact solutions are obtained

nonlinear variants of the improved boussinesq equation with compact and noncompact structures variants of the improved boussinesq equation with positive and negative exponents are investigated it is formally shown that these nonlinear variants give rise to compact and non compact physical structures where compactons solitons solitary patterns and periodic solutions are developed the presented sine cosine ansatz is reliable

the improved fan sub equation method and its application to the sk equation in this paper many types of exact solutions of a first order nonlinear ordinary differential equation called fan sub equation is further investigated by using bifurcation method of dynamical systems as a result more types of exact solutions to sawada kotera sk equation are obtained which include more general soliton solutions kink solutions triangular function solutions jacobian elliptic function solutions with double periods and so on

linear b spline finite element method for the improved boussinesq equation in this paper we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved boussinesq equation the finite element method with linear b spline basis functions is used to discretize the nonlinear partial differential equation in space and derive a second order system involving only ordinary derivatives it is shown that the coefficient matrix for the second order term in this system is invertible consequently for the first time the initial boundary value problem can be reduced to an explicit initial value problem to which many accurate numerical methods are readily applicable various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting wave interaction and blow up behavior

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

analytical and numerical aspects of certain nonlinear evolution equations i analytical abstract nonlinear partial difference equations are obtained which have as limiting forms the nonlinear schrodinger korteweg devries and modified korteweg devries equations these difference equations have a number of special properties they are constructed by methods related to the inverse scattering transform they can be used as a basis for numerical schemes to the associated nonlinear evolution equations experiments have shown that they compare very favorably with other known numerical methods papers ii iii in paper ii the ablowitz ladik scheme for the nonlinear schrodinger equation is compared to other known numerical schemes and generally proved to be faster than all utilized finite difference schemes but somewhat slower than the finite fourier pseudospectral methods in paper iii a proposed scheme for the korteweg de vries equation proved to be faster than both the finite difference and finite fourier methods already considered

on the mixed hodge structure on the cohomology of the milnor fibre in 18 the second author introduced a mixed hodge structure on the cohomology of the milnor fibre of an isolated hypersurface singularity for the definition of a mixed hodge structure el 17 sect 3 4 the weight filtration is essentially the monodromy weight filtration 18 sect 4 and thus simplifies the jordan normal form of the unipotent part of the monodromy the significance of the hodge filtration however is not so dear and its description seems difficult to use this paper gives another description of the hodge filtration which we hope is easier to understand and apply its definition does not use resolution of singularities instead it relies on the theory of holonomic modules in one variable with regular singularities this paper arose from conversations at the 1980 arbeitstagung in bonn where varchenko s conjectured hodge filtration 22 had been discussed in brieskorn s seminar we have since learned that varchenko obtained similar results to those in this paper in the summer and autumn of 1980 23 25 in sect 1 the milnor fibration is embedded in a family of smooth projective hypersurfaces the hodge theory of a smooth projective hypersurface is explained in the fashion of brylinski 2 in sect 2 in sect 3 we recall the description of the gauss manin system from 8 our new formula for the hodge filtration is explained in sects 4 5 in sect 6 we prove that it gives the same result as 18 we prove a thom sebastiani formula for the hodge filtration in sect 7 which leads to a proof of conjecture 5 4 of 18 in sect 8 we show how to prove varchenko s result about the jordan normal forms of multiplication by f in the jacobian ring of f and the logarithm of the unipotent monodormy of f with our method in the last chapter the mixed hodge structure is calculated for two examples

nonlinear schrodinger type equations from multiscale reduction of pdes i systematic derivation in this article we begin a systematic investigation via multiscale expansions of nonlinear evolution pdes partial differential equations in this first article we restrict consideration to a single autonomous but otherwise generic pde in 1 1 variables space time of first order in time whose linear part is dispersive and to solutions dominated by a single plane wave satisfying the linear part of the pde the expansion parameter is an assumedly small coefficient multiplying this plane wave the main indeed asymptotically exact effect of the weak nonlinearity is then to cause a modulation of the amplitude of the plane wave and of its harmonics which is generally described in appropriately defined coarse grained time and space variables by evolution equations of nonlinear schrodinger type a systematic analysis of such equations is presented corresponding to various assumptions on the resonances occurring for the first few harmonics

general solutions with solitary patterns for the defocusing branch of the nonlinear dispersive k n n equations in higher dimensional spaces in this work the defocusing branch of the genuinely nonlinear dispersive k n n equation that exhibits solutions with solitary patterns is investigated the equation with constant coefficients to the derivatives is examined in one two and three dimensional spaces general formulas for solutions with solitary patterns are developed for all positive integers n n 1 two sets of entirely new formulas one set produces solutions for even integers n whereas the second set produces solutions in opposite directions for odd integers n n 1 are established

solitons solutions for some nonlinear evolution equations abstract in this paper we establish new solitary wave solutions to the modified kawahara equation by the sine cosine method moreover the periodic solutions and bell shaped solitons solutions to the generalized fifth order kdv equation are obtained the tanh method is used to handle the double sine gordon equation and the double sinh gordon equation families of exact travelling wave solutions are formally derived the rational triangle sine cosine method is introduced and to be constructed complex solutions to the modified degasperis procesi dp equation and the modified camassa holm ch equation

lie group classification of second order ordinary difference equations a group classification of invariant difference models i e difference equations and meshes is presented in the continuous limit the results go over into lie s classification of second order ordinary differential equations the discrete model is a three point one and we show that it can be invariant under lie groups of dimension 0 n 6

symbolic analysis and exact travelling wave solutions to a new modified novikov equation in this paper a new modified novikov equation is introduced multiple exact travelling wave solutions are obtained by using the extended tanh function method furthermore based on the homogeneous balance method an auto backlund transformation is derived and some solitary wave solutions to this new equation are established

exact soliton solutions for the fifth order sawada kotera equation exact soliton solutions for the fifth order sawada kotera equation are obtained by using the hirota bilinear method these solutions include one soliton solutions periodic two soliton solutions and singular periodic soliton solutions the results show that there exist periodic two soliton solutions and singular periodic soliton solutions for the fifth order sawada kotera equation

new periodic and soliton solutions of nonlinear evolution equations abstract in this paper the tanh and sine cosine methods are used to construct exact periodic and soliton solutions of nonlinear evolution equations arising in mathematical physics many new families of exact travelling wave solutions of the generalized hirota satsuma coupled kdv system generalized zakharov equations and 2 1 dimensional broer kaup kupershmidt system are successfully obtained the obtained solutions include solitons kinks and plane periodic solutions these solutions may be important of significance for the explanation of some practical physical problems

abundant solitons solutions for several forms of the fifth order kdv equation by using the tanh method in this work we use the tanh method for solving several forms of the fifth order nonlinear kdv equation the forms include the lax sawada kotera sk kaup kupershmidt kk and the ito forms and other related special cases abundant solitons solutions are derived two necessary criteria are established to build up reliable strategies that govern the relation between the parameters of the equation previously known solutions are recovered and entirely new bell shaped solitons are determined

variants of the two dimensional boussinesq equation with compactons solitons and periodic solutions variants of the two dimensional boussinesq equation with positive and negative exponents are studied the sine cosine ansatz is fruitfully used to carry out the analysis exact solutions of different physical structures compactons solitary patterns solitons and periodic solutions are obtained the quantitative change in the physical structure of the solutions is shown to depend mainly on the exponent of the wave function u x t and on the ratio a b of the derivatives of u x t

the tanh method exact solutions of the sine gordon and the sinh gordon equations in this work we use the tanh method for traveling wave solutions of the sine gordon and the sinh gordon equations several exact solutions of distinct physical structures are obtained the method is powerful with minimal algebra work and is demonstrated for four models

some remarks on the fifth order kdv equations abstract in this paper we present the differential operators for the generalized fifth order kdv equation we give formal proofs on the hamiltonian properties including the skew adjointness and jacobi identity by using the prolongation method our results show that there are three third order hamiltonian operators which can be used to construct the hamiltonians however no fifth order operators are shown to pass the hamiltonian test although there are an infinite number of them and they are skew adjoint

exact travelling wave solutions for a generalized zakharov kuznetsov equation in this paper by means of a proper transformation and symbolic computation we study the exact travelling wave solutions for a generalized zakharov kuznetsov gzk equation by using the extended tanh method and direct assumption method as a result rich exact travelling wave solutions which contain new kink shaped solitons bell shaped solitons periodic solutions combined formal solitons rational solutions and singular solitons for gzk equation are obtained

variants of the generalized kdv equation with compact and noncompact structures abstract in this paper nonlinear dispersive variants of the generalized kdv equation are investigated the compact and the noncompact physical structures are studied two general formulas for the focusing branch and the defocusing branch that are of substantial interest are formally derived

general compactons solutions for the focusing branch of the nonlinear dispersive k n n equations in higher dimensional spaces in this paper we will study the focusing branch of the genuinely nonlinear dispersive k n n equation that exhibits compactons solitons with finite wavelengths the equation will be studied in one two and three dimensional spaces general formulas for compactons solutions are developed for all positive integers n n 1 two sets of entirely new formulas one set produces compactons for even integers n whereas the second set produces compactons and anticompactons for odd integers n n 1 are established

relative cohomology with respect to a lefschetz pencil let m be a complex projective manifold of dimension n 1 and f a meromorphic function on m obtained by a generic pencil of hyperplane sections of m the n th cohomology vector bundle of f0 f m r where r is the set of indeterminacy points of f is defined on the set of regular values of f0 and we have the usual gauss manin connection on it following brieskorn s methods in br we extend the n th cohomology vector bundle of f0 and the associated gauss manin connection to p1 by means of differential forms the new connection turns out to be meromorphic on the critical values of f0 we prove that the meromorphic global sections of the vector bundle with poles of arbitrary order at p1 is isomorphic to the brieskorn module of f in a natural way and so the brieskorn module in this case is a free c t module of rank n where c t is the ring of polynomials in t and n is the dimension of n th cohomology group of a regular fiber of f0

discretization of partial differential equations preserving their physical symmetries a procedure for obtaining a minimal discretization of a partial differential equation preserving all of its lie point symmetries is presented minimal in this case means that the differential equation is replaced by a partial difference scheme involving n difference equations where n is the number of independent and dependent variables we restrict ourselves to one scalar function of two independent variables as examples invariant discretizations of the heat burgers and korteweg de vries equations are presented some exact solutions of the discrete schemes are obtained

the multiscale expansions of difference equations in the small lattice spacing regime and a vicinity and integrability test i we propose an algorithmic procedure i to study the distance between an integrable pde and any discretization of it in the small lattice spacing regime and at the same time ii to test the asymptotic integrability properties of such discretization this method should provide in particular useful and concrete information on how good is any numerical scheme used to integrate a given integrable pde the procedure illustrated on a fairly general ten parameter family of discretizations of the nonlinear schr dinger equation consists of the following three steps i the construction of the continuous multiscale expansion of a generic solution of the discrete system at all orders in following degasperis et al 1997 physica d 100 187 211 ii the application to such an expansion of the degasperis procesi dp integrability test degasperis a and procesi m 1999 asymptotic integrability symmetry and perturbation theory spt98 ed a degasperis and g gaeta singapore world scientific pp 23 37 degasperis a 2001 multiscale expansion and integrability of dispersive wave equations lectures given at the euro summer school what is integrability isaac newton institute cambridge uk 13 24 august integrability lecture notes in physics vol 767 ed a mikhailov berlin springer to test the asymptotic integrability properties of the discrete system and its distance from its continuous limit iii the use of the main output of the dp test to construct infinitely many approximate symmetries and constants of motion of the discrete system through novel and simple formulas

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

multiple scale perturbation beyond the nonlinear schroedinger equation i abstract we consider the effect of weak nonlinearity on the propagation of one dimensional strongly dispersive waves in the standard quasi monochromatic approximation it is well known that the first order amplitude modulation satisfies the nonlinear schroedinger equation in slow variables with the assumption that this amplitude modulation has no discrete spectrum component i e no soliton we explore the higher order effects and provide a well defined procedure to compute all coefficients of the perturbative expansion as relatively bounded functions no secularity effects of slow variables the main effect of this procedure is that the leading amplitude modulation satisfies at higher orders the nonlinear schroedinger hierarchy of evolution equations in slow variables key ingredient is the integrability of the nonlinear schroedinger equation and its commuting symmetries

recursion operators and bi hamiltonian structures in multidimensions ii the algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied in particular the factorization of certain operators called recursion operators establishes the bi hamiltonian nature of all these equations recently we have presented the recursion operator and the bi hamiltonian formulation of the kadomtsev petviashvili equation a two spatial dimensional analogue of the korteweg devries equation here we present the general theory associated with recursion operators for bi hamiltonian equations in two spatial and one temporal dimensions as an application we show that general classes of equations which include the kadomtsev petviashvili and the davey stewartson equations possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct poisson brackets furthermore we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems

cylindrical solitary pulses in a two dimensional stabilized kuramoto sivashinsky system abstract by linearly coupling a generalized zakharov kuznetsov equation alias the two dimensional 2d benney equation to an extra linear dissipative equation a 2d extension of a recently proposed stabilized kuramoto sivashinsky system is developed the model applies to the description of surface waves on 2d liquid layers in various physical settings the extra equation provides for the stability of the zero state in the system thus paving a way to the existence of stable 2d localized solitary pulses sps a perturbation theory based on a family of cylindrical solitons existing in the conservative counterpart of the system is developed by treating dissipation and gain in the model as small perturbations it is shown that the system may select two steady state solitons from the continuous family provided by the conservative counterpart of which the one with larger amplitude is expected to be stable numerical simulations support the analytical predictions quite well additionally it is found that a shallow quasi one dimensional 1d trough is attached to the stable sp if the integration domain is not very large and an explanation to this feature is proposed stable double humped bound states of two pulses are found too

a new approach on numerical solutions of the improved boussinesq type equation using quadratic b spline galerkin finite element method in the present manuscript some numerical solutions of an improved boussinesq type equation are obtained by means of quadratic b spline galerkin finite element method then error norms l2 and l have been calculated to test the accuracy of the current method in the manuscript solitary wave movement and interaction of solitary antisolitary waves are considered as test problems

the extended tanh method and its applications for solving nonlinear physical models the extended tanh method with a computerized symbolic computation is used for constructing the travelling wave solutions of coupled nonlinear equations arising in physics the obtained solutions include solitons kinks and plane periodic solutions the applied method will be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics

on exact travelling wave solutions for two types of nonlinear k n n equations and a generalized kp equation in this paper we study two types of genuinely nonlinear k n n equations and a generalized kp equation by developing a mathematical method based on the reduction of order of nonlinear differential equations we derive general formulas for the travelling wave solutions of the three equations the compactons solitary patterns solitons and periodic solutions obtained are expressed analytically it is shown that the y and z components of the wave number vectors in the travelling wave solutions of the generalized kp equation remain free and arbitrary constants the work generalizes the known results of travelling wave solutions for the three equations

multi scale expansions in the theory of systems integrable by the inverse scattering transform abstract it is shown that using multi scale expansions conventionally employed in the theory of nonlinear waves one can transform systems integrable by the ist method into other systems of this type

a class of nonlinear fourth order variant of a generalized camassa holm equation with compact and noncompact solutions abstract in this paper we study a class of nonlinear fourth order analogue of a generalized camassa holm equation it is shown that this class gives compactons conventional solitons solitary patterns and periodic solutions it is also found that the qualitative change in the physical structure of solutions depends mainly on the exponent of the wave function u x t positive or negative and on the coefficient of u n as well

a study on nonlinear dispersive partial differential equations of compact and noncompact solutions in this paper we study two genuinely nonlinear dispersive partial differential equations in one and higher dimensional spaces we show that the focusing branch and the defocusing branch of these equations each leads to a different physical structure in addition we develop general solutions of compact and noncompact support which are not obtainable through classic techniques

the tanh method for traveling wave solutions of nonlinear equations in this work we employ the tanh method for traveling wave solutions of nonlinear equations the study is extended to equations that do not have tanh polynomial solutions the efficiency of the method is demonstrated by applying it for a variety of selected equations

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