a method for constructing nontraveling wave solutions for 1 1 dimensional evolution equations a method is proposed to construct nontraveling wave solution for 1 1 dimensional evolution equations by extending the linear traveling wave transformation of the g g expansion method into a nonlinear transformation owing to a built in arbitrary function included in the solution abundant solutions can be excited taking the vakhnenko equation as an example a series of nontraveling wave solutions with variable separation is obtained some new solutions are excited and the known solitary wave solutions are special cases of the nontraveling wave solutions

global existence and blow up for the generalized sixth order boussinesq equation abstract in this paper we prove local well posedness in l 2 r and h 1 r for the generalized sixth order boussinesq equation u t t u x x u x x x x u x x x x x x u u x x our proof relies in the oscillatory integrals estimates introduced by kenig et al 1991 14 we also show that under suitable conditions a global solution for the initial value problem exists in addition we derive the sufficient conditions for the blow up of the solution to the problem

boundary behavior in the loewner nirenberg problem abstract let r n be a bounded domain of class c 2 0 1 we show that if n 3 and u is the maximal solution of equation u n n 2 u n 2 n 2 in then the hyperbolic radius v u 2 n 2 is of class c 2 up to the boundary the argument rests on a reduction to a nonlinear fuchsian elliptic pde

analytic solutions to nonlinear differential difference equations by means of the extended g g expansion method in this paper a discrete extension of the g g expansion method is applied to a relativistic toda lattice system and a discrete nonlinear schrodinger equation in order to obtain discrete traveling wave solutions closed form solutions with more arbitrary parameters which reduce to solitary and periodic waves are exhibited new rational solutions are also obtained the method is straightforward and concise and its applications in physical sciences are promising

discrete exact solutions to some nonlinear differential difference equations via the g g expansion method we extended the g g expansion method to two well known nonlinear differential difference equations the discrete nonlinear schrodinger equation and the toda lattice equation for constructing traveling wave solutions discrete soliton and periodic wave solutions with more arbitrary parameters as well as discrete rational wave solutions are revealed it seems that the utilized method can provide highly accurate discrete exact solutions to nddes arising in applied mathematical and physical sciences

application of the g g expansion method to 3 1 dimensional nonlinear evolution equations in this paper the g g expansion method is employed to solve the 3 1 dimensional kadomtsev petviashvili kp equation the 3 1 dimensional potential ytsf equation and the 3 1 dimensional jimbo miwa jm equation exact traveling wave solutions are obtained the traveling wave solutions are expressed in terms of hyperbolic functions the trigonometric functions and the rational functions

a direct method for constructing the traveling wave solutions of a modified generalized vakhnenko equation abstract proposing a direct method namely the traveling wave reduction combined with a smart variable transformation a series of traveling wave solutions are constructed for a modified generalized vakhnenko equation these solutions include the jacobi elliptic function solution the weierstrass elliptic function solution and the trigonometric function solution

the structure of wtc expansions and applications we construct generalized painleve expansions with logarithmic terms for a general class of non integrable scalar equations and describe their structure in detail these expansions were introduced without logarithms by weiss tabor carnevale wtc 1983 the construction of the formal solutions is shown to involve semi invariants of binary forms and tools from invariant theory are applied to the determination of the type of logarithmic terms that are required for the most general singular series the structure of the series depends strongly on whether 1 is or is not a resonance the convergence of these series is obtained as a consequence of the general results of littman and kichenassamy 1993 the results are illustrated on a family of fifth order models for water waves and other examples we also give necessary and sufficient conditions for 1 to be a resonance

application of the g g expansion to travelling wave solutions of the broer kaup and the approximate long water wave equations by using the g g expansion proposed recently the travelling wave solutions involving parameters of the broer kaup equations and the approximate long water wave equations are found out the travelling wave solutions are expressed by three types of functions which are the hyperbolic functions the trigonometric functions and the rational functions when the parameters are taken as special values the solitary wave solutions are obtained c 2008 elsevier inc all rights reserved

the g g expansion method for some nonlinear evolution equations abstract in this paper the g g expansion method is applied to the liouville sine gordon and new coupled mkdv equations to obtain their some generalized exact travelling wave solutions

a local existence result for poincare einstein metrics abstract given a closed riemannian manifold m g m of dimension n 3 we prove the existence of a conformally compact einstein metric g defined on a collar neighborhood m 0 1 whose conformal infinity is g m

exact and explicit solutions to some nonlinear evolution equations by utilizing the g g expansion method in this paper we demonstrate the effectiveness of the so called g g expansion method by examining some nonlinear evolution equations with physical interest our work is motivated by the fact that the g g expansion method provides not only more general forms of solutions but also periodic and solitary waves if we set the parameters in the obtained wider set of solutions as special values then some previously known solutions can be recovered the method appears to be easier and faster by means of a symbolic computation system

observations on the tanh coth expansion method for finding solutions to nonlinear evolution equations abstract the tanh coth expansion method for finding solitary travelling wave solutions to nonlinear evolution equations has been used extensively in the literature it is a natural extension to the basic tanh function expansion method which was developed in the 1990s it usually delivers three types of solution namely a tanh function expansion a coth function expansion and a tanh coth expansion it is known that for every tanh function expansion solution there is a corresponding coth function expansion solution it is shown that there is a tanh coth expansion solution that is merely a disguised version of the coth solution in many papers such tanh coth solutions are erroneously claimed to be new however other tanh coth solutions may be delivered that are genuinely new in the sense that they would not be delivered via the basic tanh function method similar remarks apply to tan cot and tan cot expansion solutions

on a conjecture of fefferman and graham abstract in 1985 fefferman and graham have constructed a local embedding of an arbitrary real analytic manifold of odd dimension n into a ricci flat manifold of dimension n 2 admitting a homothety they conjectured that their result remains valid in even dimensions if logarithms are allowed in the expansion of the metric in this paper we i prove that such expansions exist and converge and ii establish the degree of non uniqueness of the solution in terms of the coefficients in the expansion

boundary blow up and degenerate equations let r2 be a bounded domain of class c2 0 1 we show that if u is the solution of u 4exp 2u which tends to as x y then the hyperbolic radius v exp u is also of class c2 up to the boundary the proof relies on new schauder estimates for degenerate elliptic equations of fuchsian type

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

application of the g g expansion method to kawahara type equations using symbolic computation abstract in this paper kawahara type equations are selected to illustrate the effectiveness and simplicity of the g g expansion method with the aid of a symbolic computation system three types of more general traveling wave solutions including hyperbolic functions trigonometric functions and rational functions with free parameters are constructed solutions concerning solitary and periodic waves are also given by setting the two arbitrary parameters involved in the traveling waves as special values

the ablowitz ladik lattice system by means of the extended g g expansion method abstract we analyzed the ablowitz ladik lattice system by using the extended g g expansion method further discrete soliton and periodic wave solutions with more arbitrary parameters are obtained we observed that some previously known results can be recovered by assigning special values to the arbitrary parameters

application of g g expansion method to two nonlinear evolution equations abstract two nonlinear evolution equations are separately reduced to nonlinear ordinary differential equations ode by using a simple transformation when the parameters are taken as special values the traveling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions the proposed method can be used for many other nonlinear evolution equations

ambient metrics for n dimensional pp waves we provide an explicit formula for the fefferman graham ambient metric of an n dimensional conformal pp wave in those cases where it exists in even dimensions we calculate the obstruction explicitly furthermore we describe all 4 dimensional pp waves that are bach flat and give a large class of bach flat examples which are conformally cotton flat but not conformally einstein finally as an application we use the obtained ambient metric to show that even dimensional pp waves have vanishing critical q curvature

towards a unified method for exact solutions of evolution equations an application to reaction diffusion equations with finite memory transport we present a brief report on the different methods for finding exact solutions of nonlinear evolution equations explicit exact traveling wave solutions are the most amenable besides implicit and parametric ones it is shown that most of methods that exist in the literature are equivalent to the generalized mapping method that unifies them by using this method a class of formal exact solutions for reaction diffusion equations with finite memory transport is obtained attention is focused to the finite memory transport fisher and nagumo equations

travelling wave solutions of two nonlinear evolution equations by using the g g expansion method abstract by using the g g expansion method proposed recently we give the exact travelling wave solutions of two different types of nonlinear evolution equations in this paper the travelling wave solutions are expressed by three types of functions the computation for the method appears to be easier and faster by general mathematical software

the improved g g expansion method and its applications to the broer kaup equations and approximate long water wave equations by introducing a new general ansatze the improved g g expansion method is proposed to construct exact solutions of both broer kaup equations and approximate long water wave equations as a result some new travelling wave solutions involving parameters expressed by three types of functions which are the hyperbolic functions the trigonometric functions and the rational functions are obtained when the parameters are taken as special values the solitary wave solutions are derived from the hyperbolic function solutions the proposed method is straightforward concise and effective and can be applied to other nonlinear evolution equations in mathematical physics

application of g g expansion method to regularized long wave rlw equation in this paper the g g expansion method with the aid of maple is used to obtain a generalized soliton solution for the generalized regularized long wave rlw equation each of the obtained solutions namely hyperbolic function solutions and trigonometric function solutions contain an explicit linear function of the variables in the considered equation it is shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering problems

new application of the g g expansion method to higher order nonlinear equations in this paper we extend the g g expansion method to higher order nonlinear equations as a result a series of new travelling wave solutions of the modified kawahara equation and the fifth order kdv equation have been obtained the travelling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions it is shown that the g g expansion method presents a wider applicability for handling nonlinear evolution equations

on the validity and reliability of the g g expansion method by using higher order nonlinear equations in this study we demonstrate the validity and reliability of the so called g g expansion method via symbolic computation for illustrative examples we choose the sixth order boussinesq equation and the ninth order korteweg de vries equation as a result the power of the employed method is confirmed

the compound g g expansion method and double non traveling wave solutions of 2 1 dimensional nonlinear partial differential equations abstract to seek the exact double non traveling wave solutions of nonlinear partial differential equations the compound g g expansion method is firstly proposed in this paper with the aid of symbolic computation this new method is applied to construct double non traveling wave solutions of 2 1 dimensional painleve integrable burgers equation and 2 1 dimensional breaking soliton equation as a result abundant double non traveling wave solutions including double hyperbolic function solutions double trigonometric function solutions double rational solutions and a series of complexiton solutions of these two equations are obtained via the proposed method these exact solutions contain arbitrary functions which may be helpful to explain some complex phenomena when the parameters are taken as special values the double solitary like wave solutions can be derived from double hyperbolic function solutions furthermore the time evolutions of double solitary like wave solutions are discussed in detail

exact solutions for a system of two coupled discrete nonlinear schrodinger equations with a saturable nonlinearity in this paper we are going to solve a system of two coupled discrete nonlinear schrodinger equations with a saturable nonlinearity using the g g expansion method the traveling wave solutions found include hyperbolic functions trigonometric functions and rational functions from the hyperbolic functions solutions obtained in this study the dark bright and multiple dark solitons solutions are derived simultaneously using the variation of the parameter of saturation

a construction of compact and noncompact solutions for nonlinear dispersive equations of even order a construction of compact and noncompact solutions for nonlinear dispersive partial differential equations of even order is considered two model equations of even orders k 4 are adopted to carry out this study the analysis reveals the presence of distinct roots of different indices that play a significant role in the solutions

exact traveling solutions of some nonlinear evolution equation by g g expansion method in this paper we establish exact solutions for some nonlinear evolution equations the g g expansion method was used to construct traveling wave solutions of ramani s equations in this method we take the advantages of the general solutions of the second order lode to solve effectively many nonlinear evolution equations the g g expansion method presents a wider applicability for handling nonlinear wave equations

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