a series of new analytical solutions to the nonlinear equation yt yxxx 6y2yx 6λyx 0 i following the backlund transformation and using the theorems proved by the same authors previously we obtained sets of new solutions to the kdv equation and the nonlinear equation yt yxxx 6y2yx 6 yx 0 which transforms into the modified kdv equation when 0 in this paper we present another new series of solutions to the above nonlinear equation one of our analytical solutions under a certain special condition vacuum parameter b 0 and 0 is found to be identical to the one soliton solution obtained via the inverse scattering method

a straightforward method for finding implicit solitary wave solutions of nonlinear evolution and wave equations the authors present a straightforward method for finding implicit solutions for nonlinear evolution and wave equations the method is illustrated by finding implicit single solitary wave solutions for the harry dym korteweg de vries modified korteweg de vries boussinesq and the generalised korteweg de vries equations

regularized long wave equation and inverse scattering transform the sech2 solitary wave solution of the regularized long wave equation is reobtained via the inverse scattering transform the wave function of the eigenvalue problem of the relevant schrodinger equation is proved reflectionless with sech2 potential of arbitrary amplitude moreover the nonexistence of n solitary wave solution sech2 form and n 2 is confirmed for the regularized long wave rlw equation and the method provides the possibility of solving some incompletely integrable equations via the inverse scattering transform

reciprocal transformation for one dimensional conservation equations one dimensional conservation equations oce of the form n t f x 0 with n n x t 0 and f f n n x 2n x2 admit a symmetric reciprocal transformation x x x t n n x t n 1 f f n 1f which produces an equivalent oce for n in x space certain oce of contemporary interest are reciprocal invariant in the sense that f f n n x 2n x 2 there also exists a class of essentially nonlinear oce for which the reciprocal transformation produces a linear oce and thus equations in this class are solvable analytically

on the integrability of 2 1 dimensional quasilinear systems a 2 1 dimensional quasilinear system is said to be integrable if it can be decoupled in infinitely many ways into a pair of compatible n component one dimensional systems in riemann invariants exact solutions described by these reductions known as nonlinear interactions of planar simple waves can be viewed as natural dispersionless analogues of n gap solutions it is demonstrated that the requirement of the existence of sufficiently many n component reductions provides the effective classification criterion as an example of this approach we classify integrable 2 1 dimensional systems of conservation laws possessing a convex quadratic entropy

a note on travelling wave solutions to lax s seventh order kdv equation ganji and abdollahzadeh d d ganji m abdollahzadeh appl math comput 206 2008 438 444 derived three supposedly new travelling wave solutions to lax s seventh order kdv equation each solution was obtained by a different method it is shown that any two of the solutions may be obtained trivially from the remaining solution furthermore it is noted that one of the solutions has been known for many years

painleve test and exact similarity solutions of a class of nonlinear diffusion equations for 1 1 dimensional diffusion equations where the diffusion coefficient is an arbitrary power of the dependent variable a painleve test is performed on both the partial differential equation and the similarity reductions the invariance of the painleve property under a large class of similarity reductions is shown for the reductions with the painleve property the explicit similarity solutions are given it is demonstrated how results of a painleve analysis can be used to obtain further exact similarity solutions even in cases without the painleve property

generalized lorentz invariance and solitary waves for coupled nonlinear klein gordon equations a generalized lorentz transformation that preserves the invariance of a nonlinear klein gordon equation is presented and is used to extend the set of solitary wave solutions for a special set of n coupled nonlinear schrodinger equations to become solutions of the corresponding set of coupled nonlinear klein gordon equations

exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method the authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear equations the physical concept is one of the mixing of these elementary solutions through the nonlinearities in the system the emphasis is however on the mathematical aspects i e the formal procedure necessary to find single solitary wave solutions by means of examples it is shown how various cases of pulse type and kink type solutions are to be obtained by this method an exhaustive list of equations so treated is presented in tabular form together with the particular intermediate relations necessary for deriving their solutions the extension of the technique to construct n soliton solutions and indicate connections with other existing methods is outlined

asymptotic behavior for a nonlinear degenerate diffusion equation in population dynamics we consider a spatially aggregating population model which provides the homogenizing process due to density dependent diffusion and the dehomogenizing one due to a certain long range transport the result asserts that by a balance between two processes an initial distribution of populations forms itself into a traveling solitary wave pattern for large time which exhibits phenomenologically a kind of aggregation of a species

existence and asymptotic behavior of pade approximants to the korteweg de vries multisoliton solutions the summation procedure of the pade type is applied to the perturbation expansion of the solution of the potential korteweg de vries equation k d v introduced by rosales for the n soliton solution without background the n 1 n pade approximants are shown to exist for n n their asymptotic behavior is investigated and it is found that it corresponds to a system of n solitons with the leading velocity parameters the analogous results for the k d v then follow in agreement with some previous numerical observations

link between solitary waves and projective riccati equations many solitary wave solutions of nonlinear partial differential equations can be written as a polynomial in two elementary functions which satisfy a projective hence linearizable riccati system from that property the authors deduce a method for building these solutions by determining only a finite number of coefficients this method is much shorter and obtains more solutions than the one which consists of summing a perturbation series built from exponential solutions of the linearized equation they handle several examples for the henon heiles hamiltonian system they obtain several exact solutions one of them defines a new solitary wave solution for a coupled system of boussinesq and nonlinear schrodinger equations for a third order dispersive equation with two monomial nonlinearities they isolate all cases where the general solution is single valued

backlund transformations and abundant exact explicit solutions of the sharma tasso olver equation abstract by using an extension of the homogeneous balance method and maple the backlund transformations for the sharma tasso olver equation are derived the connections between the sharma tasso olver equation and some linear partial differential equations are found with the aid of the transformations given here and the computer program maple 12 abundant exact explicit special solutions to the sharma tasso olver equation are constructed in addition to all known solutions re deriving in a systematic way several entirely new and more general exact explicit solitary wave solutions can also be obtained these solutions include a the algebraic solitary wave solution of rational function b single soliton solutions c double soliton solutions d n soliton solutions e singular traveling solutions f the periodic wave solutions of trigonometric function type and g many non traveling solutions by using the airy s function and the backlund transformations obtained here the exact explicit solution of the initial value problem for the sto equation is presented the variety of the structure of the solutions for the sharma tasso olver equation is illustrated

interval computing periodic orbits of maps using a piecewise approach abstract interval arithmetic applied to simulation of dynamical systems has attracted a great deal of interest in recent years much of this research has been carried out in the calculation of fixed points or low period windows for nonlinear discrete maps this study proposes a novel interval computation based on a piecewise method to calculate periodic orbits for the logistic map using the cobweb plot three rounding situations have been applied to a correct outward rounding as required by interval arithmetic the proposed method is compared with results in the literature and with the results obtained by means of the matlab toolbox intlab the comparison is accomplished for nine case studies using the logistic map numerical results explicitly indicate that the proposed method produces intervals that are substantially narrower than those obtained with the traditional techniques

on series expansions giving closed form solutions of korteweg de vries like equations korteweg devries kdv like equations with higher degree nonlinearity are solved by a direct algebraic technique due to hereman et al j phys a 19 1986 pp 607 628 for two kdv like equations one with fifth degree nonlinearity the other a combined kdv and mkdv equation for particular choices of the coefficients of the nonlinear terms the kink and antikink solutions found by dey are recovered furthermore soliton solutions of the combined kdv and mkdv equation are found for all values of the coefficients closed form solutions for the calogero degasperis fokas modified mkdv equation are also developed applications of the solutions of these equations in quantum field theory plasma physics and solid state physics are mentioned the hereman et al method is illustrated and slightly extended and this direct series method is briefly compared to hirota s method

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

exact solutions of some nonlinear partial differential equations abstract we implemented a generalized tanh function method for approximating the solution of the combined korteweg de vries ckdv equation and sharma tasso olver equation by using this scheme we found some exact solutions of the ckdv equation and sharma tasso olver equation for the further investigation of the method some exact solutions of the boussinesq equation and the complexly coupled kdv equations have been obtained

derivation and implicit solution of the harry dym equation and its connections with the korteweg de vries equation the harry dym equation which is related to the classical string problem is derived in three different ways an implicit cusp solitary wave solution is constructed via a simple direct method the existing connections between the harry dym and the korteweg de vries equations are uniformised and simplified and transformations between their respective solutions are carried out explicitly whenever possible physical insights are provided

special set and solutions of coupled nonlinear schrodinger equations a special set of n coupled nonlinear schrodinger equations which consists of 2n interaction types each of which is characterized by a specific array of interaction parameters is presented it is shown that every lame function of order n n is a solution for one or more components for one or more interaction types of this special set simple rules that relate interaction and solution types and interesting features of these relationships are presented

direct and inverse scattering problems of the nonlinear intermediate long wave equation the inverse scattering transformation method associated with a nonlinear singular integrodifferential equation is discussed the equation describes long internal gravity waves in a stratified fluid of finite depth and reduces to the korteweg de vries equation as shallow water limit and the benjamin ono equation as deep water limit both limits of the method and novel aspects of the theory are also discussed

the density wave in a car following model according to the optimal velocity model the condition for stable traffic flow is deduced nonlinear analysis shows that the density fluctuation in traffic flow itself induces two types of local density waves a weak fluctuation occurring near the stability state in a wide range of headways forms a soliton determined by the korteweg de vries kdv equation the appearance of such a soliton shows that drivers tend to reach the safety distance when they are away from it this density wave degenerates to a travelling wave at the critical point a strong fluctuation occurring around the critical point forms a kink or a soliton determined by the modified korteweg de vries mkdv equation

benjamin bona mahony bbm equation with variable coefficients similarity reductions and painleve analysis abstract the lie group formalism is applied to investigate the symmetries of the benjamin bona mahony bbm equation with variable coefficients we derive the infinitesimals and the admissible forms of the coefficients that admit the classical symmetry group the ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained

analytic study of the generalized burger s huxley equation by hyperbolic tangent method abstract in this research exact traveling wave solutions of the generalized burger s huxley equation are obtained which occurs in the description of many non linear wave phenomena the analysis rests mainly on the standard hyperbolic tangent tanh method this method is a powerful technique for the computation of closed form tanh solutions for nonlinear partial differential equations the types of exact solutions are analyzed depending on the values of the equation parameters

auto backlund transformation and similarity reductions for coupled burger s equation in this work we applied backlund transformation and similarity reduction for coupled burger s equation clarkson and kruskal developed a direct and simple method to obtain more similarity solutions of nonlinear partial differential equation we received our inspiration from fan s article as far as we think that our work is an extended one from the fan s has done in his paper as a result of this study we obtained solitary wave solutions and traveling wave solutions of coupled burger s equation

group transformations and the nonlinear heat diffusion equation a nonlinear diffusion equation with a power law diffusion coefficient is studied self similar and partially invariant solutions are shown to be identical the theory of homology is introduced and used to generate classes of equations the solutions of which are related through backlund transformations an example of an application is given which generalizes the solution of storm j appl phys 22 1951 pp 940 951 on heat conduction in metals

on exact solutions of the regularized long wave equation a direct approach to partially integrable equations i solitary wave and solitons hitherto the bilinear transformation method has been used to obtain analytic solutions of completely integrable or nonlinear evolution equations in this first of two studies designated i and ii it is shown that the method may apply equally well to those nonlinear wave equations dubbed partially integrable here the regularized long wave rlw and rlw boussinesq rlwb equations are considered as examples in each case the bilinear form is found to have an extra equation which would appear to signal their nonintegrability their solitary wave solutions are reconstructed and the nonexistence of multisoliton solutions is demonstrated the present work provides the basis for the construction of periodic solutions in ii

hamiltonian formulation of odd burgers hierarchy the derivation of the odd burgers hierarchy is revisited a first hamiltonian formulation of the basic equation of the hierarchy is presented in relation with its linear counterpart the generalized poisson bracket is given explicitly it contains exponentials of integrals of the dynamic variable it verifies jacobi identity by construction and through direct calculations a second hamiltonian formulation is also presented it means that the equation as expected is bi hamiltonian this property permits as usual the construction of all the hierarchy extension to matrix burgers systems is suggested

phase space analysis for the direct algebraic method for nonlinear evolution and wave equations abstract the basis of the direct algebraic method employing a series in real exponential functions for solving nonlinear evolution and wave equations is examined in terms of the general theory of autonomous ordinary differential equations odes and phase space analysis the closed form solutions obtained by the direct algebraic method are identified with special trajectories in phase space that connect a critical point to itself or other critical points the phase portrait is investigated for odes resulting from a combined kdv and mkdv equation a kdv like equation with fifth degree nonlinearity a special case of the calogero degasperis fokas modified mkdv equation and a generalized fisher equation the phase space analysis serves to clarify and partially justify the direct algebraic method

homogeneous diffusion in ℝ with power like nonlinear diffusivity we study the nonnegative solutions of the initial value problem ut ur ux p 1ux x u x 0 l1 where p 0 r p 0 the local velocity of propagation of the solutions is identified as v vx vx p 1 where v cu with r p 1 p and c r p r p 1 is the nonlinear potential our main result is the a priori estimate vx vx p 1 x frac 1 2p r t

lattice solitary wave solutions of discrete nonlinear wave equations using a direct method it is shown that the real exponential approach to the solutions of nonlinear evolution and wave equations can also he applied to discrete systems as examples the lattice solitary wave and kink solutions of the discrete modified kdv dmkdv equation and some modified versions of this equation are derived using this approach

hodograph transformations of linearizable partial differential equations in this paper an algorithmic method is developed for transforming quasilinear partial differential equations of the form u t g u u nx f u u x cdots u n 1 x u mx equiv partial m u partial x m where dg du not equiv 0 into semilinear equations i e equations of the above form with g u 1 this crucially involves the use of hodograph transformations i e transformations involving the interchange of dependent and independent variables furthermore the most general quasilinear equation of the above form is found that can be mapped via a hodograph transformation to a semilinear form this algorithm provides a method for establishing whether a given quasilinear equation is linearizable i e is solvable in terms of either a linear partial differential equation or of a linear integral equation in particular this method is used to show how the painleve tests may be applied to quasilinear equations this appears to resolve the problem that solutions of lineariz

isogroup and general similarity solution of a nonlinear diffusion equation for a class of one dimensional nonlinear diffusion equations where the diffusion coefficient varies as some power of the dependent variable the invariance group and its lie algebra are given the isovector fields which generate the isogroup are then used to derive a general similarity solution for a particular case the solution can be reduced to a one parameter group solution which is in full agreement with previously published results

noisy integer maps abstract the combined effects of discretization and noise in iterated maps are studied with simulations and a transfer matrix technique the resulting changes in a typical map s invariant distribution are illustrated

painleve analysis and integrability of coupled non linear schrodinger equations considering a system of coupled non linear schrodinger nls equations the authors discuss the integrability properties through painleve p analysis for the two coupled nls equations they show that there exists a pair of parametric values possessing the p property for which the associated backlund transformation bt and the hirota bilinearisation are constructed these parametric choices are identical to those of zakharov and schulman 1982 who established the integrability in terms of motion invariants finally they extend the p analysis to the n coupled nls system and identify two parametric choices possessing the p property which are natural generalisations of the two coupled nls cases

the direct linearization of a class of nonlinear evolution equations the paper deals with the direct linearization an approach used to generate particular solutions of the partial differential equations that can be solved through the inverse scattering transform linear integral equations are presented which enable one to find broad classes of solutions to certain nonlinear evolution equations in 1 1 and 2 1 dimensions

exact solutions of equations for the burgers hierarchy some classes of the rational periodic and solitary wave solutions for the burgers hierarchy are presented the solutions for this hierarchy are obtained by using the generalized cole hopf transformation

construction of solutions of the sine gordon equation by means of fredholm determinants abstract a new straightforward and rigorous approach to the theory of the sine gordon equation is presented it provides a simple and unified view of the three known methods for solution namely hirota s method of generating n soliton solutions the inverse scattering transform and backlund transformations in addition new types of solutions corresponding to multiple poles in the scattering data can be easily obtained the principal tool used is the theory of fredholm determinants generalisations to other soliton equations seem to be possible

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

the fredholm determinant method for the kdv equations abstract the fredholm determinant method is a new and rigorous way to investigate soliton equations and to construct their solutions it consists essentially of establishing a comparatively simple relation between the soliton equation and its linearized form the inverse scattering method hirota s method of constructing n soliton solutions and backlund transformations are given a new and unified interpretation moreover some new results can be obtained in this paper the fredholm determinant method is applied to the korteweg de vries kdv modified kdv and the hierarchy of generalized kdv equations

new exact solutions for some coupled nonlinear partial differential equations using extended coupled sub equations expansion method abstract an extended coupled sub equations expansion method is proposed to seek new traveling wave solutions of the coupled nonlinear partial differential equations the generalized schrodinger boussinesq and coupled nonlinear klein gordon schrodinger equations are used to illustrate the validity and the advantages of this method

travelling wave solutions to nonlinear evolution and wave equations we have studied a series of ansatze ordinary differential equations of the first order which correspond to the travelling and or solitary wave solutions of some nonlinear partial differential equations we have investigated the conditions under which the nonlinear partial differential equations have certain kinds of travelling and or solitary wave solutions as a consequence of applications we can take the trial procedures to obtain the travelling wave solutions which is a very efficient method for solving several classes of nonlinear partial differential equations

chaos number theory and computers abstract a case study is presented in which the logistic map is shown to have anomalous precision dependent behavior it is also demonstrated that using greater precision does not necessarily lead to greater accuracy and in fact the contrary can be true though the presention deals only with this particular map it illustrates that the actual math package being used should be examined and cannot be treated as a black box

new applications of variational iteration method abstract the variational iteration method is used for solving three types of nonlinear partial differential equations such as coupled schrodinger kdv generalized kdv and shallow water equations the exact and numerical solutions obtained by variational iteration method are compared with that obtained using adomian decomposition method the comparison shows that the two solutions obtained are excellent agreement the method is made showing that the former is more effective than the latter in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

some non local transformations between nonlinear diffusion equations generalized backlund transformations are applied to derive links between a large number of different types of nonlinear diffusion equations including many which are of physical significance some new exactly linearizable forms are determined

reciprocal auto backlund transformations invariant backlund transformations of reciprocal type are derived for a class of n 1 th order conservation laws a new auto backlund transformation for the harry dym equation is presented as a special case of the analysis

an application of the adm to seven order sawada kotara equations we implemented the adomian decomposition method for short adm for approximating the solution of the seventh order sawada kotera for short ssk and a lax s seventh order kdv for short lskdv equations by using this scheme explicit exact solution is calculated we obtain the exact solitary wave solutions and numerical solutions of the lskdv and ssk equations for the initial conditions the numerical solutions are compared with the known analytical solutions their remarkable accuracy are finally demonstrated for the both seven order equations

on the exactly solvable equation s_t beta s gamma 2 s_x _x alpha beta s gamma 2 s_x occurring in two phase flow in porous media the symmetry approach of fokas j math phys 21 1980 pp 1318 1325 is used to show that the most general equation of the form s t g s s xx f s s x dg ds ne 0 which is exactly solvable is equivalent to the equation s t beta s gamma 2 s x x alpha beta s gamma 2 s x this equation is then mapped to the linear heat equation through a series of backlund transformations the above result is used to solve an initial boundary value problem modeling the process of waterflooding i e the displacement of oil initially in a porous reservoir by continuously injected water

nonlinear mode decoupling for classes of evolution equations the method of third order mode coupling is applied to a general evolution equation which includes as special cases many of the modified and higher order koreteweg de vries equations which have recently appeared in the literature the equations for the slow changes in the amplitudes and phases are derived and specialised to two different classes of evolution equations the first class exhibits the property of nonresonant mode decoupling the evolution of each wave being governed only by its own parameters included in this class are the kdv equation some of the mkdv equations and the sharma tasso olver equation equations of the second class leave the amplitudes and hence the energetics of the waves constant as long as the mode coupling is non resonant here one finds the fifth order kdv equations besides the kdv and sto equations

approximations to large amplitude solitary waves on nonlinear electrical lattices abstract in this paper we describe an approximate method to characterise solitary wave solutions of nonlinear lattice equations it is based upon one and two point pade approximations to a series of the real exponential travelling wave solutions of the underlying dispersive system the theory is applied to an example of a lattice system which models an experimental nonlinear transmission line and the results obtained are consistent with numerical simulations even for relatively large amplitude solitary waves the speed amplitude relation is investigated and compared with the derived using quasi continuum methods