results on infinite dimensional topology and applications to the structure of the critical set of nonlinear sturm liouville operators we consider the nonlinear sturm liouville dieren tial operator f u u 00 f u for u 2 h 2 d 0 a sobolev space of functions satisfying dirichlet boundary conditions for a generic nonlinearity f r r we show that there is a dieomorphism in the domain of f converting the critical set c of f into a union of isolated parallel hyperplanes for the proof we show that the homotopy groups of connected components of c are trivial and prove results which permit to replace homotopy equivalences of systems of innite dimensional hilbert manifolds by dieomorphisms

the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

on stabilized finite element methods for the reissner mindlin plate model stabilized finite element formulation for the reissner mindlin plate model is considered physical interpretation for the stabilization procedure for low order elements is established explicit interpolation functions for linear and bilinear stabilized mitc elements are derived some numerical examples including buckling and frequency analyses are presented copyright 2007 john wiley sons ltd

einstein maxwell solitons the application of the inverse scattering method to the einstein maxwell equations for stationary axisymmetric exterior fields leads the authors to an explicit formula for the ernst and electromagnetic potentials of new exact solutions generated from an arbitrary seed solution

painleve expansions for nonintegrable evolution equations abstract the wtc method is applied to a variety of nonintegrable nonlinear evolution equations and shown to provide a systematic procedure for obtaining special solutions some of the special techniques developed for integrable systems are also shown to be applicable to nonintegrable equations

the calculation of novel class of solutions of a non linear fourth order evolution equation by the weierstras transform method abstract in this paper the weierstras transform method is applied to a non linear partial differential equation npde of fourth order to obtain new exact class of solutions in terms of the weierstras elliptic function g 2 g 3 the major aspect of this paper however is the fact that we are able to calculate distinctive class of solutions of a less studied non linear evolution equation explicitly given by u tt u xxxt 3 u xx u xt 0 the derived class of solutions cannot be found in the current literature more precisely the mathematical structure of these class of solutions are quite different from those derived by classical ansatz methods in other words using this special algebraic procedure the solution manifold is augmented to further class of solutions including special functions they are connected in a complicate way and allow a more general representation as found in previous time simultaneously we would like to stress the necessity of such sophisticated methods since a general theory of npdes does not exist in addition this paper is a logical completion of our recent studies where we have applied successfully the weierstras transform method to the important kdv mkdv equation

a generalized weierstrass elliptic function expansion method for solving some nonlinear partial differential equations the present paper deals with families of non trivial solutions of the equation email protected 2 pw 4 x qw 2 x r on the basis of these solutions a direct and generalized algebraic algorithm is described for constructing the new solutions of some nonlinear partial differential equations nlpdes subsequently many new and more general exact solutions in terms of the weierstrass elliptic function x g 2 g 3 are obtained the method can be applied to other nlpdes in mathematical physics

gauge transformation between the first order nonisospectral and isospectral heisenberg hierarchies abstract by means of the ablowitz kaup newell segur system with symbolic computation the gauge transformation between the first order nonisospectral and isospectral heisenberg spectral problems is obtained correspondingly the equivalence relation between the first order nonisospectral and isospectral heisenberg hierarchies is given in addition further applications and potential research directions of our method are discussed

a method of linearization for painleve equations painleve iv v abstract a method to linearize the initial value problem of the painleve equations iv v is given the procedure involves formulating a riemann hilbert boundary value problem on intersecting lines for the inverse monodromy problem this boundary value problem is reduced to a sequence of standard problems on single lines in a certain range of parameter space schlesinger transformations allow one to completely cover the parameter space special solutions are constructed from special cases of the riemann problem as well

exact solutions of the bogoyavlenskii equation using the multiple g g expansion method in this work exact traveling wave solutions of the bogoyavlenskii equation are studied by the g g expansion method and its variants the solutions obtained are general solutions which are in the form of hyperbolic trigonometric and rational functions and a variety of special solutions like kink shaped antikink shaped bell type soliton solutions etc can easily be derived from the general results under certain domain these methods are more effective and simple than other methods and a number of solutions can be obtained at the same time in order to visualize the underlying dynamics of the obtained solutions two and three dimensional plots are drawn

non generic connections corresponding to front solutions a classification of special nonlinear front solutions for certain one time and one space reaction diffusion equations is presented using the method of weiss tabor and carnevale wtc these results are related to known stability criteria in particular the steepness criterion of van saarloos 1989 the wtc method is shown to be equivalent to a special first order reduction and both of these methods are shown to work for reaction diffusion equations with special nonlinearities of particular interest is the fact that the special first order reduction is shown to give separatrices in appropriate phase spaces an example of reaction diffusion equation is presented without these special nonlinearities while this equation is shown to have a special nonlinear connection and resulting stability properties it is intractable for either a singular manifold expansion of a first order reduction a lie symmetry analysis is carried out and it is shown that equations with continuous groups other than translational invariance are only a subclass of equations which are amenable to the special solution techniques however the rescaling ansatz of cariello and tabor 1991 suggests that some symmetries are present

travelling wave solutions to zufiria s higher order boussinesq type equations abstract zufiria s higher order boussinesq type equations are studied by transforming them into solvable ordinary differential equations various families of their travelling wave solutions are generated which include periodic wave solitary wave periodic like wave soliton like wave jacobi elliptic function periodic wave combined non degenerative jacobi elliptic function like wave weierstrass elliptic function periodic and rational solutions the presented approach can be also applied to nonlinear wave equations with variable coefficients in mathematical physics and mechanics

some notes on the singular manifold method several manifolds and constraints some questions arising when various modifications are made to the singular manifold method are considered the solutions are shown to lie outside the framework of the singular analysis the approach is illustrated by a number of examples with ordinary differential equations further perspectives and the question of optimization of the calculations are also discussed

exact soliton solutions of the one dimensional complex swift hohenberg equation abstract using painleve analysis the hirota multi linear method and a direct ansatz technique we study analytic solutions of the 1 1 dimensional complex cubic and quintic swift hohenberg equations we consider both standard and generalized versions of these equations we have found that a number of exact solutions exist to each of these equations provided that the coefficients are constrained by certain relations the set of solutions include particular types of solitary wave solutions hole dark soliton solutions and periodic solutions in terms of elliptic jacobi functions and the weierstrass function although these solutions represent only a small subset of the large variety of possible solutions admitted by the complex cubic and quintic swift hohenberg equations those presented here are the first examples of exact analytic solutions found thus far

painleve analysis and integrability properties of a 2 1 nonrelativistic field theory a model for planar phenomena introduced by jackiw and pi and described by a lagrangian including a chern simons term is considered the associated equations of motion among which there is a 2 1 gauged nonlinear schrodinger equation are rewritten into a gauge independent form involving the modulus of the matter field application of a painleve analysis as adapted to partial differential equations by weiss tabor and carnevale shows up resonance values that are all integer however compatibility conditions need be considered which cannot be satisfied consistently in general such a result suggests that the examined equations are not integrable but provides tools for the investigation of the integrability of different reductions this in particular puts forward the familiar integrable liouville and 1 1 nonlinear schrodinger equations

wronskian and grammian solutions for a 2 1 dimensional date jimbo kashiwara miwa equation abstract in this paper a 2 1 dimensional date jimbo kashiwara miwa equation is investigated based on the hirota method and auxiliary variable backlund transformation is obtained under certain conditions the hirota bilinear forms can be reduced to the plucker and jacobi identities via the wronskian and grammian solutions through the wronskian technique and pfaffian derivative formulae n soliton solutions in the wronskian and grammian are given graphically presented soliton collisions are elastic both in the wronskian and grammian and after each collision the solitons keep their amplitudes unchanged except for the phase shifts

symbolic computation of some new nonlinear partial differential equations of nanobiosciences using modified extended tanh function method abstract by means of computerized symbolic computation and a modified extended tanh function method the multiple travelling wave solutions of nonlinear partial differential equations is presented and implemented in a computer algebraic system applying this method we consider some of nonlinear partial differential equations of special interest in nanobiosciences and biophysics namely the transmission line models of microtubules for nano ionic currents the nonlinear equations elaborated here are quite original and first proposed in the context of important nanosciences problems related with cell signaling it could be even of basic importance for explanation of cognitive processes in neurons as results we can successfully recover the previously known solitary wave solutions that had been found by other sophisticated methods the method is straightforward and concise and it can also be applied to other nonlinear equations in physics

linearity inside nonlinearity exact solutions to the complex ginzburg landau equation abstract systems of two linear partial differential equations pdes with constant coefficients define a natural basis of elementary functions to build some exact solutions to nonlinear pdes for the one dimensional complex ginzburg landau equation the usual representations of the complex field re a or a are multivalued which makes difficult the search for exact solutions in this manner fortunately the painleve analysis naturally introduces an elementary single valued like representation of a by two fields z grad respectively complex abd real uniquely defined by an explicit expression for and a ze i another important feature is the invariance by parity on a which increases the class of expected solutions this allows to retrieve quite easily the four famous solutions of nozaki and bekki represented by the constant coefficients of two linear partial differential equations and a finite set of constants

new variable separation approach application to nonlinear diffusion equations the concept of the derivative dependent functional separable solution ddfss as a generalization to the functional separable solution is proposed as an application it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach as a consequence a complete list of canonical forms for such equations which admit the ddfss is obtained and some exact solutions to the resulting equations are described

explicit solutions of the four wave mixing model dynamical degenerate four wave mixing is studied analytically in detail by removing the unessential freedom we first characterize this system by a lower dimensional closed subsystem of a deformed maxwell bloch type involving only three physical variables the intensity pattern the dynamical grating amplitude the relative net gain we then classify by the painlev test all the cases when single valued solutions may exist according to the two essential parameters of the system the real relaxation time and the complex response constant in addition to the stationary case the only two integrable cases occur for a purely nonlocal response re 0 these are the complex unpumped maxwell bloch system and another one which is explicitly integrated with elliptic functions for a generic response re 0 we display strong similarities with the cubic complex ginzburg landau equation

the sine gordon equations complete and partial integrability the sine gordon equation in one space one time dimension is known to possess the painleve property and to be completely integrable it is shown how the method of singular manifold analysis obtains the backlund transform and the lax pair for this equation a connection with the sequence of higher order kdv equations is found the modified sine gordon equations are defined in terms of the singular manifold these equations are shown to be identically painleve also certain rational solutions are constructed iteratively the double sine gordon equation is shown not to possess the painleve property however if the singular manifold defines an affine minimal surface then the equation has integrable solutions this restriction is termed partial integrability the sine gordon equation in n 1 variables n space 1 time where n is greater than one is shown not to possess the painleve property the condition of partial integrability requires the singular manifold to be an einstein space with null scalar curvature the known integrable solutions satisfy this constraint in a trivial manner finally the coupled kdv or hirota satsuma equations possess the painleve property the associated modified equations are derived and from these the lax pair is found

exact solutions of the multidimensional derivative nonlinear schrodinger equation for many body systems of criticality the authors investigate the problem of strongly interacting many body systems near criticality as recently described in terms of nonlinear dynamics by dixon and tuszynski 1989 they show that in the first order of approximation the equation of motion for the order parameter can be mapped on the derivative nonlinear schrodinger equation and thus can be solved exactly they solve the equations both in the absence and presence of current densities

symmetry reductions and exact solutions of a class of nonlinear heat equations abstract classical and nonclassical symmetries of the nonlinear heat equation ut uxx f u are considered the method of differential grobner bases is used both to find the conditions on f u under which symmetries other than the trivial spatial and temporal translational symmetries exist and to solve the determining equations for the infinitesimals a catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of the nonlinear heat equation for cubic f u in terms of the roots of f u 0

comment on on the extended applications of homogeneous balance method in this letter we show that all results presented in senthilvelan s paper can be actually obtained by a direct method

non classical symmetries and the singular manifold method revisited the connection between the singular manifold method painleve expansions truncated at the constant term and symmetry reductions of two members of a family of cahn hilliard equations is considered the conjecture that similarity information for a nonlinear partial differential equation may always be fully recovered from the singular manifold method is violated for these equations and is thus shown to be invalid in general given that several earlier examples demonstrate the connection between the two techniques in some cases it now becomes necessary to establish when such a relationship exists a question related to a deeper understanding of painleve analysis this issue is also briefly discussed

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

solutions of a 2 1 dimensional dispersive long wave equation a special type of multisoliton solution with a particular dispersion relation is obtained for wu zhang equation which describes 2 1 dimensional dispersive long waves by the standard weiss tabor carnvale painleve truncation expansion using a nonstandard truncation of a modified conte s invariant painleve expansion two different types of soliton solutions without any dispersive relation is found two types of periodic wave solutions expressed by jacobi elliptic functions are found by the truncations of a special extended painleve expansion the soliton solutions are special cases of the corresponding periodic solutions

the structures and interactions of solitary waves in a 2 1 dimensional coupled nonlinear extension of the reaction diffusion equation the complete integrability of a 2 1 dimensional coupled nonlinear extension of the reaction diffusion cnlerd equation using the technique of painlev p analysis is investigated using the formalism of weiss et al 1983 j math phys 24 522 the arbitrariness of the expansion coefficients are proved besides following the hirota s formalism hirota r 1980 direct methods in soliton theory soliton berlin springer combined to weiss et al s methodology weiss et al 1984 j math phys 25 13 the consistency of the truncation is shown thus without the use of kruskal s simplification the bcklund transformation bt of the equations is obtained via the truncation procedure taking into account the arbitrariness of the expansion coefficients in the foregoing truncation a typical spectrum of localized coherent structures may be unearthed the scattering behavior of such structures is also investigated

exact periodic solutions of the complex ginzburg landau equation three new exact periodic solutions of the complex ginzburg landau equation are obtained in terms of the weierstrass elliptic function furthermore the new periodic solutions and other shock solutions appear as their bounded limits along the real axis for particular relationships between the coefficients in the equation in the general case bounded limits are nothing but the already known pulse hole and shock solutions it is also shown that the shapes of the solutions are quite different from the shape of the usual envelope wave solution in particular the spatial structure of the new bounded periodic solutions varies in time while the pulse solution may exhibit breather like behavior

a new truncation in painleve analysis the author observes that when the expansion variable in painleve analysis satisfies a system of riccati equations truncation at a level higher than constant level is allowed this extends the range of exact solutions to nonlinear partial differential equations that one is able to obtain using truncated painleve expansions

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

generalized knizhnik zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform maxwell bloch system with pumping canonical quantization of the isomonodromy solutions of equations integrable via the inverse scattering transform leads to generalized knizhnik zamolodchikov equations one can solve these equations by the off shell bethe ansatz method provided the knizhnik zamolodchikov equations are related with the highest weight representations of the corresponding lie algebras these solutions can be written in terms of multi variable generalizations of special functions of the hypergeometric type in this work we consider a realization of the above scheme for the maxwell bloch system with pumping quantum states for this system are found in terms of the multi variable confluent hypergeometric function

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

conditional lie backlund symmetries and functionally generalized separable solutions to the generalized porous medium equations with source abstract the functionally generalized separable solutions of the generalized porous medium equations with power law and exponential diffusivity are studied by using the conditional lie backlund symmetry method the variant forms of the considered equations which admit the linear conditional lie backlund symmetries are identified a number of examples are considered and some exact solutions defined on the polynomial trigonometric and exponential invariant subspaces determined by the linear conditional lie backlund symmetries are constructed

similarity solutions of the deformed maxwell bloch system the authors have found similarity reductions for the deformed maxwell bloch system to the fifth and second painleve equations asymptotics of the solutions of these equations which are relevant to two level atomic systems with pumping have also been derived

painleve test generalized symmetries backlund transformations and exact solutions to the third order burgers equations in this paper the painleve analysis is performed on the physical form of the third order burgers equation the painleve property and integrability c integrable of the equation is verified then the generalized symmetries of the equation are presented and the generalized symmetries of the other equation are given by the symmetry transformation method the backlund transformations of the equations are constructed based on the symmetries respectively furthermore the exact explicit solutions to the equations are investigated in terms of the symmetries backlund transformations and transformations of the equations

application of the darboux transformation to inverse problems with variable spectral parameters it is shown that darboux transformations can be used to find explicit solutions of deformed equations which have recently been found through the inverse method with a variable spectral parameter the author constructs in particular the soliton solution of the deformed maxwell bloch equations and analyses its asymptotic behaviour

a series of explicit and exact travelling wave solutions of the b m n equations abstract in this paper explicit and exact travelling wave solutions of b m n equations are investigated by using a generalized algebraic method a series of explicit exact solutions such as solitary wave triangular periodic rational jacobi and weierstrass elliptic function periodic solutions are obtained many new results are presented

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

painleve analysis of the generalized burgers huxley equation a complete painleve test 1900 is applied to the generalized burgers huxley equation using the version of the painleve analysis recently developed by weiss tabor and carnevale 1983 for partial nonlinear differential equations in so doing the authors are able to find a complete set of new solutions as well as recovering some previous particular solutions already found by using ad hoc methods which have been recently published

exact travelling wave solutions of some nonlinear equations by g g expansion method abstract here we find exact solutions of some nonlinear evolution equations within the framework of the g g expansion method exact solutions of five nonlinear equations of physical importance viz the coupled schrodinger kdv equation the coupled nonlinear reaction diffusion equation the foam drainage equation the phi four equation and the dodd bullough mikhailov equation are obtained these general solutions can be reduced in some standard results derived by some other methods two and three dimensional plots of some of the results are also presented

separation of variables of a generalized porous medium equation with nonlinear source this paper considers a general form of the porous medium equation with nonlinear source term ut d u u n x f u n 1 the functional separation of variables of this equation is studied by using the generalized conditional symmetry approach we obtain a complete list of canonical forms for such equations which admit the functional separable solutions as a consequence some exact solutions to the resulting equations are constructed and their behavior are also investigated 2002 elsevier science usa all rights reserved

an improved generalized f expansion method and its applications an improved generalized f expansion method is proposed to seek exact solutions of nonlinear evolution equations with the aid of symbolic computation we choose the generalized nonlinear schrodinger equation with a source to illustrate the validity and advantages of the proposed method many new and more general non travelling wave solutions are obtained including single and combined non degenerate jacobi elliptic function solutions solitons like solutions trigonometric function solutions and rational solutions the method is straightforward and concise and it can also be applied to other nonlinear evolution equations in physics

comparison of three semi analytical methods for solving 1 1 dimensional dispersive long wave equations in this work we consider how adomian s decomposition method adm the homotopy analysis method ham and the homotopy perturbation method hpm can be used to investigate wave solutions of 1 1 dimensional dispersive long wave equations it is also worth noting that the advantage of the approximation of the series methodologies is a fast convergence of the solutions

analytic solutions of a microstructure pde and the kdv and kadomtsev petviashvili equations by invariant painleve analysis and generalized hirota techniques abstract truncated painleve expansions invariant painleve analysis and generalized hirota expansions are employed in combination to solve partially reduce to quadrature the integrable kdv and kp equations and a nonintegrable generalized microstructure gms equation although the multisolitons of the kdv and kp equations are very well known the solutions obtained here for all the three nlpdes are novel and non trivial the solutions obtained via invariant painleve analysis are all complicated rational functions with arguments which themselves are confluent hypergeometric kdv or trigonometric gms functions of various distinct non traveling kdv and traveling wave variables in some cases this is slightly reminiscent of doubly periodic elliptic function solutions when nonlinear ode systems are reduced to quadratures the solutions obtained by the use of recently generalized hirota type expansions in the truncated painleve expansions are closer in functional form to conventional hyperbolic secant solutions although with non trivial traveling wave arguments which are distinct for the three nlpdes considered here

symmetry reductions of partial differential equations related to singular manifold expansions it is shown that all the information yielded by the reduction methods of bluman and cole 1974 and clarkson and kruskal 1989 can be obtained using the singular manifold expansion the burgers equation modified korteweg de vries equation caudrey dodd gibbon equation and the fitzhugh nagumo equation are used as illustrative examples several new exact solutions are presented

a unified approach to painleve expansions abstract the painleve test for partial differential equations developed by weiss tabor and carnevale wtc is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations a simple modification of the wtc procedure used for partial differential equations enables us to determine the lax pairs hirota equations and auto backlund transformations for ordinary differential equations including a new lax pair for an integrable case of the henon heiles system a detailed study of the kdv hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained the role of the singular branches is examined in detail and important new insights obtained in particular we find that each singular branch is simply a re expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce a parallel analysis is carried out for the akns hierarchy

solitary solutions of some nonlinear evolution equations in this paper the traveling wave reduction combined with the homogeneous balance method hbm is used to find new exact solutions by using less studied nonlinear partial differential equations npdg of higher order the usual starting point is a special transformation converting the npdg in its two variables x and t into an nonlinear ordinary differential equation node in the single variable x using the hyperbolic tangent method new exact solutions for the three npdgs are studied the new feature of this paper is of course the fact that we are dealing with npdgs which cannot be found by studying the relevant literature therefore we believe it is time for publishing some special solutions of this interesting kind of equations clearly we do not use a new formalism but with the aid of the tanh method we are able to calculate at least exact traveling wave solutions solitary waves which can physically interpreted as an action of wave propagation simultaneous we point out the necessity of such sophisticated methods since a general theory of npdgs does not exist the existence of nontrivial noncomplex valued solutions of the ramani equation enables us to generate families of real valued solutions of a nonlinear sixth order system special differential transformations are leading to connections between traveling wave solutions of associated nonlinear evolution equations

a generalized exponential transform method for solving non linear evolution equations of physical relevance the purpose of the present paper is to introduce a new computational algebraic procedure that can easily be applied for solving non linear partial differential equations npde especially the celebrated evolutions equations describing any time depended sequences the crucial step needs an auxiliary variable satisfying special class of ordinary differential equations ode of first order which are introduced new in this field for the first time the validity and reliability of the method is tested by its application to some non linear evolution equations leading to new class of solutions related with some new types of special functions otherwise for practical use in science and engineering the algebraic construction of new class of solutions is of fundamental interest and moreover the proposed approach convinced by its easiness and does not need tedious steps of evaluation and can be used without studying the whole theory the possibility to write a symbolic software using any programming languages is given further the algorithm works efficiently is clear structured and can be used in any applications independently from the order and the non linearity of the underlying npde therefore the given novel algebraic method is suitable for a wider class of npde in order to augment the solution manifold by an alternative approach

on nonlinear diffusion equations with x dependent convection and absorption abstract for the 1 1 dimensional nonlinear diffusion equations with x dependent convection and source terms u t d u u x x q x u u x p x u we obtain conditions under which the equations admit the second order generalized conditional symmetries x u u xx h u u x 2 g x u u x f x u and the first order sign invariants j x u u t a u u x 2 b x u u x c x u on the solutions u x t several different generalized conditional symmetries and first order sign invariants for equations in which the diffusion term offers different possibilities power law exponential mullin fujita are presented exact solutions to the resulting equations corresponding to the generalized conditional symmetries and the first order sign invariants are constructed

the geometry of the critical set of nonlinear periodic sturm liouville operators we study the critical set c of the nonlinear differential operator f u u f u defined on a sobolev space of periodic functions h p s 1 p 1 let r 2 r 3 be the plane z 0 and for n 0 let n be the cone x 2 y 2 tan 2 z z 2 n 0 n for a generic smooth nonlinearity f r r with surjective derivative we show that there is a diffeomorphism between the pairs h p s 1 c and r 3 h where h is a real separable infinite dimensional hilbert space

a rational generalization of fan s method and its application to generalized shallow water wave equations in this paper we employ a rational expansion to generalize fan s method for exact travelling wave solutions for nonlinear partial differential equations pdes to verify the reliability of the proposed method the generalized shallow water wave gsww equation has been investigated as an example kinds of new exact travelling wave solutions of a rational form have been obtained this indicates that the proposed method provides a more general result for exact solution of nonlinear equations

new complex solutions for some special nonlinear partial differential systems abstract in this present work we explore new applications of direct algebraic method for some special nonlinear partial differential systems and equations then new types of complex travelling wave solutions are obtained to the davey stewartson system the coupled higgs system and the perturbed nonlinear schrodinger s equation the balance number of it is not a positive integer

from singular manifold equations to integrable evolution equations the singular manifold equations are used to construct nonlinear integrable partial differential equations a number of well known integrable equations hierarchies of korteveg de vries caudrey dodd gibbon kaup kuperschmidt and harry dym are obtained some new integrable equations are presented a possible approach to classification of integrable evolution equations is discussed

on nonlinear population waves we discuss a model system of partial differential equations for description of the spatio temporal dynamics of interacting populations we are interested in the waves caused by migration of the populations we assume that the migration is a diffusion process influenced by the changing values of the birth rates and coefficients of interaction among the populations for the particular case of one population and one spatial dimension the general model is reduced to analytically tractable pde with polynomial nonlinearity up to 4th order we investigate this particular case and obtain two kinds of solutions i approximate solution for small value of the ratio between the coefficient of diffusion and the wave velocity and ii exact solutions which describe nonlinear kink and solitary waves in an appropriate phase space the kinks correspond to a connection between two states represented by a saddle point and a stable node finally we derive conditions for the asymptotic stability of the obtained solutions

non existence of elliptic travelling wave solutions of the complex ginzburg landau equation abstract we give a simple proof that for generic parameter values the cubic complex one dimensional ginzburg landau equation has no elliptic travelling wave solutions this is contrary to the expectations of musette and conte in physica d 181 2003 70 79 that elliptic solutions of zero codimension should exist the method of proof based on the residue theorem is very general and can be applied to determine necessary conditions for the existence of elliptic travelling waves for any autonomous partial differential equation as another application we prove that kudryashov s codimension one elliptic solution of the generalized kuramoto sivashinsky equation is the only one possible

invariant painleve analysis and coherent structures of two families of reaction diffusion equations exact closed form coherent structures pulses fronts domain walls having the form of complicated traveling waves are constructed for two families of reaction diffusion equations by the use of invariant painleve analysis these analytical solutions which are derived directly from the underlying pde s are investigated in the light of restrictions imposed by the ode that any traveling wave reduction of the corresponding pde must satisfy in particular it is shown that the coherent structures a asymptotically satisfy the ode governing traveling wave reductions and b are accessible to the pde from compact support initial conditions the solutions are compared with each other and with previously known solutions of the equations

analytic solitary waves of nonintegrable equations abstract a major drawback of most methods to find analytic expressions for solitary waves is the a priori restriction to a given class of expressions to overcome this difficulty we present a new method applicable to a wide class of autonomous equations which builds as an intermediate information the first order autonomous ordinary differential equation ode satisfied by the solitary wave we discuss its application to the cubic complex one dimensional ginzburg landau equation and conclude to the elliptic nature of the yet unknown most general solitary wave

temperature correlators of the impenetrable bose gas as an integrable system it is shown that the temperature equal time correlators of impenetrable bosons in one space dimension are described by a classical integrable system partial differential equations for two point as well as for multipoint correlators are obtained the short distance and low density expansions are constructed

regular level sets of averages of nemytskii operators are contractible abstract leth1 s1 be the space of periodic real functions with derivative inl2andf r r be a smooth function with no double roots then there is a diffeomorphism ofh1 s1 taking the setz v h1 s1 s1 f v t dt 0 to a hyperplane in this paper we state and prove a general version of this example we consider a banach spacevof functions from some manifoldmtorn and a functionf m r n r n under suitable hypothesis there is a homeomorphism ofvtakingz v v m f m v m dm 0 to a closed subspace of codimensionn

on non linear evolution equations of higher order the introduction and application of a novel computational approach the purpose of the present paper is to introduce a new computational algebraic procedure that can easily be applied to derive class of solutions of non linear partial differential equations npde especially of higher order the crucial step needs an auxiliary variable satisfying some ordinary differential equations ode of first order containing sine cosine and their hyperbolic varieties introducing to the first time general transformations are given to determine class of solutions explicitly the validity and reliability of the method is tested by its application to some important non linear evolution equations leading to new class of solutions with physical significance nevertheless it should be emphasised that this techniques do not need the solution of complicate nodes as in the case of similarity reduction further the algorithm works efficiently is clear structured and can be used in any applications independent of the order of the npde for computational purposes the method is appropriate to rewrite it in any computer languages therefore the given novel algebraic approach is suitable for a wider class of npde in order to augment the solution manifold by a straightforward alternative approach

connection formulae for painleve v functions we study those painleve v equations which admit a one parameter family of solutions analytic at the origin and solve the global problem of connecting the two parameter asymptotic expansion at t with the two parameter expansion about t 0 this connection is the one needed to study the correlation functions of the transverse ising chain at the critical field strength from these connection formulae we derive a one parameter connection formula for the general painleve ii equation and a two parameter connection formula for a class of painleve iii equations

painleve analysis of new higher dimensional soliton equation in this note we prove that the recently proposed new higher dimensional nonlinear partial differential equation admits the painleve property we briefly discuss the integrability properties of the equation

fronts pulses sources and sinks in generalized complex ginzberg landau equations an important clement in the long time dynamics of pattern forming systems is a class of solutions we will call coherent structures these are states that are either themselves localized or that consist of domains of regular patterns connected by localized defects or interfaces this paper summarizes and extends recent work on such coherent structures in the one dimensional complex ginzburg landau equation and its generalizations for which rather complete information can be obtained on the existence and competition of fronts pulses sources and sinks for the special subclass of uniformly translating structures the solutions are derived from a set of ordinary differential equations that can be interpreted as a flow in a three dimensional phase space fixed points of the flow correspond to the two basic building blocks of coherent structures uniform amplitude states and evanescent waves whose amplitude decreases smoothly to zero a study of the stability of the fixed points under the flow leads to results on the existence and multiplicity of the different coherent structures the dynamical analysis of the original partial differential equation focusses on the competition between pulses and fronts and is expressed in terms of a set of conjectures for front propagation that generalize the marginal stability and pinch point approaches of earlier authors these rules together with an exact front solution whose dynamics plays an important role in the selection of patterns yield an analytic expression for the upper limit of the range of existence of pulse solutions as well as a determination of the regions of parameter space where uniformly translating fron t solutions can exist extensive numerical simulations show consistency with these rules and conjectures for the existence of fronts and pulses in the parameter ranges where no uniformly translating fronts can exist examples are shown of irregularly spreading fronts that generate strongly chaotic regions as well as nonuniformly translating fronts that lead to uniform amplitude states recent perturbative treatments based on expansions about the nonlinear schr6dinger equation are generalized to perturbations of the cubic quintic and derivative schr idinger equations for which both pulses and fronts exist in the unperturbed system comparison of the results with the exact solutions shows that the perturbation theory only yields a subset of the relevant solutions nevertheless those that are obtained are found to be consistent with the general conjectures and in particular they provide an analytic demonstration of front pulse competition while the discussion of the competition between fronts and pulses focusses on the complex ginzburg landau equation with quintic terms and a subcritical bifurcation a number of results are also presented for the cubic equation in particular the existence of a family of moving source solutions derived by bekki and nozaki for this equation contradicts the naive counting arguments we attribute this contradiction to a hidden symmetry of the solution but have not been able to show explicitly how this symmetry affects the phase space orbits

painleve analysis of the two dimensional burgers equation a painleve analysis of the two dimensional burgers equation is carried out and used to obtain a restricted backlund transformation that maps a subclass of the solutions of the 1 2d burgers equation onto a linear heat like equation alternatively the backlund transformation can be expressed as a map onto the derivative of the one dimensional burgers equation in appropriate dependent and independent variables the singularity analysis also yields a further class of solutions obtained by solving a schwarzian differential equation

new explicit solutions of quasilinear heat equations with general first order sign invariants abstract for the one dimensional quasilinear heat equations ut u uxx f u with smooth functions u 0 and f u we derive conditions under which a general quadratic hamilton jacobi operator h u ut h2 u ux 2 h1 u ux h0 u with the functions h2 h1 h0 to be determined preserves both the signs 0 and 0 on the solutions u x t h u is then called a sign invariant of the given parabolic equation several types of different sign invariants are established which are governed by a linear second order ordinary differential equations new explicit solutions via finite dimensional dynamical systems corresponding to zero preserving of the first order operator are constructed