analytical and numerical studying of the perturbed korteweg de vries equation the perturbed korteweg de vries equation is considered this equation is used for the description of one dimensional viscous gas dynamics nonlinear waves in a liquid with gas bubbles and nonlinear acoustic waves the integrability of this equation is investigated using the painleve approach the condition on parameters for the integrability of the perturbed korteweg de vries equation equation is established new classical and nonclassical symmetries admitted by this equation are found all corresponding symmetry reductions are obtained new exact solutions of these reductions are constructed they are expressed via trigonometric and airy functions stability of the exact solutions of the perturbed korteweg de vries equation is investigated numerically

similarity reduction nonlocal and master symmetries of sixth order korteweg devries equation a systematic investigation to derive the lie point symmetries nonlocal and master symmetries of sixth order korteweg de vries equation kdv6 is presented using the obtained point symmetries similarity reductions are derived and constructed their particular solutions wherever possible it is shown that kdv6 admits infinitely many nonlocal and master symmetries the existence of infinitely many master symmetries ensures that kdv6 is completely integrable

new classes of solutions for the schwarzian korteweg de vries equation in 2 1 dimensions we have obtained new classes of solutions for the 2 1 dimensional schwarzian korteweg de vries equation by considering several types of reductions of a system equivalent to this equation the first analysis is done by studying the nonclassical reductions of the system further reductions are attained by means of other types of symmetry reductions or by ansatz based reductions most of the new classes of solutions depend on jacobian elliptic functions and solutions of a riemann wave equation including the cnoidal waves solutions the new classes of solutions can display several types of coherent structures and can exhibit the overturning or intertwining phenomena according to the suitable selection of the functions these solutions depend on

reduction of the n dimensional static vacuum einstein equation and generalized schwarzschild solutions abstract we consider the static vacuum einstein spacetime when the spatial factor is conformal to a n dimensional pseudo euclidean space the most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations is completely described we obtain the entire set of solutions of the reduced system where the classical schwarzschild solution arises as a particular solution in addition we show that the riemannian spatial factors associated to these solutions are foliated by parallel hypersurfaces of constant mean curvature

iterations of the non classical symmetries method and conditional lie backlund symmetries iterations of the non classical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the original equation invariant solutions of these heir equations supply new solutions of the original equation we show that particular cases of such invariant solutions correspond to zhdanov s conditional lie backlund symmetries

nonclassical analysis and painleve property for the kupershmidt equations by means of the nonclassical lie approach and the direct method all the similarity reductions of the kupershmidt equations are found two reduction equations which can also be obtained by the classical lie approach are the painleve type ii and iv respectively the other three reduction equations which cannot be obtained by the classical lie approach are only linear or riccati type

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

partial symmetry of initial value problems abstract it is generally believed that the symmetry of initial value problems ivp must leave both the partial differential equations pdes and initial conditions invariant in this paper we propose partial symmetry of ivp which needs less restrictive conditions on the pdes and initial conditions and only leaves the ivp invariant on some non empty subset of the whole solution set of the governing pdes thus some symmetries which either only leave the pdes invariant or only leave initial conditions invariant are partial symmetries of ivp then considering from two different starting points we define two types of partial symmetry and give the corresponding algorithms where one starts with the pdes and the other is from the initial conditions applications to boussinesq equation and a class of nonlinear heat equation are performed

on new ways of group methods for reduction of evolution type equations new exact solutions of the evolution type equations are constructed by means of a non point contact symmetries also we analyzed the discrete symmetries of maxwell equations in vacuum and decoupled ones to the four independent equations that can be solved independently

direct reduction and differential constraints direct reductions of partial differential equations to systems of ordinary differential equations are in one to one correspondence with compatible differential constraints the differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions but admits a non characteristic second order reduction if and only if it is linearizable

partial lie point symmetries of differential equations when we consider a differential equation 0 whose set of solutions is a lie point exact symmetry of this is a lie point invertible transformation t such that t i e such that any solution to 0 is transformed into a generally different solution to the same equation here we define partial symmetries of 0 as lie point invertible transformations t such that there is a non empty subset such that t i e such that there is a subset of solutions to 0 which are transformed into one another we discuss how to determine both partial symmetries and the invariant set and show that our procedure is effective by means of concrete examples we also discuss relations with conditional symmetries and how our discussion applies to the special case of dynamical systems our discussion will focus on continuous lie point partial symmetries but our approach would also be suitable for more general classes of transformations the discussion is indeed extended to partial generalized or lie backlund symmetries along the same lines and in the appendix we will discuss the case of discrete partial symmetries

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

conditional symmetry and spectrum of the one dimensional schrodinger equation we develop an algebraic approach to studying the spectral properties of the stationary schrodinger equation in one dimension based on its high order conditional symmetries this approach makes it possible to obtain in explicit form representations of the schrodinger operator by n n matrices for any n n and thus to reduce a spectral problem to a purely algebraic one of finding eigenvalues of constant n n matrices the connection to so called quasiexactly solvable models is discussed it is established in particular that the case when conditional symmetries reduce to high order lie symmetries corresponds to exactly solvable schrodinger equations a symmetry classification of schrodinger equation admitting nontrivial high order lie symmetries is carried out which yields a hierarchy of exactly solvable schrodinger equations exact solutions of these are constructed in explicit form possible applications of the technique developed to multidimensional linear and one dimensional nonlinear schrodinger equations are briefly discussed

conditional lie backlund symmetry and reduction of evolution equations we suggest a generalization of the notion of invariance of a given partial differential equation with respect to a lie backlund vector field such a generalization proves to be effective and enables us to construct principally new ansatz reducing evolution type equations to several ordinary differential equations in the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations ut uxx f u ux with poor lie symmetry and obtain their exact solutions it is shown that these solutions cannot be constructed by means of the symmetry reduction procedure

group foliation and non invariant solutions of the heavenly equation the main physical results of this paper are new exact analytical solutions of the heavenly equation of importance in the general theory of relativity these solutions are not invariant under any subgroup of the symmetry group of the equation the main mathematical result is a new method of obtaining non invariant solutions of partial differential equations with infinite dimensional symmetry groups the method involves the compatibility of the given equations with a differential constraint which is automorphic under a specific symmetry subgroup the latter acting transitively on the submanifold of the common solutions by studying the integrability of the resulting conditions one can provide an explicit foliation of the entire solution manifold of the considered equations

on the wavelet transformation of fractal objects the wavelet transformation is briefly presented it is shown how the analysis of the local scaling behavior of fractals can be transformed into the investigation of the scaling behavior of analytic functions over the half plane near the boundary of its domain of analyticity as an example a weierstrass like fractal function is considered for which the wavelet transform is related to a jacobi theta function some of the scalings of this theta function are analyzed and give some information about the scaling behavior of this fractal

differential constraints and exact solutions of nonlinear diffusion equations the differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations certain linear determining equations with parameters are used to find such differential constraints they generalize the determining equations used in the search for classical lie symmetries

similarity reductions of the 1 3 dimensional burgers equation we consider the 1 3 dimensional burgers equation u t u x x u y y u z z uu x which has considerable interest in mathematical physics we complete the list of similarity reductions that are obtained from the lie symmetries admitted by this equation to achieve this goal we employ two and three dimensional subalgebras of the lie symmetry algebra in addition to the one dimensional subalgebras that appears in the literature we solve analytically where it is possible the reduced ordinary differential equations and hence we obtain closed form solutions for the original equation finally we derive certain non lie solutions

conditional symmetries and riemann invariants for hyperbolic systems of pdes this paper contains an analysis of rank k solutions in terms of riemann invariants obtained from interrelations between two concepts that of the symmetry reduction method and of the generalized method of characteristics for first order quasilinear hyperbolic systems of pdes in many dimensions a variant of the conditional symmetry method for obtaining this type of solutions is proposed a lie module of vector fields which are symmetries of an overdetermined system defined by the initial system of equations and certain first order differential constraints is constructed it is shown that this overdetermined system admits rank k solutions expressible in terms of riemann invariants finally examples of applications of the proposed approach to the fluid dynamics equations in k 1 dimensions are discussed in detail several new soliton like solutions among them kinks bumps and multiple wave solutions have been obtained these solutions remain bounded even when the riemann invariants admit a gradient catastrophe some physical interpretation of these results is discussed

new similarity reductions of the steady state boundary layer equations new similarity reductions and exact solutions of the steady state boundary layer equations which are of considerable importance in physics are presented besides similarity reductions obtained by applying the non classical group method due to bluman and cole some similarity solutions which figure neither among classical nor among non classical reductions are determined using an extension of the non classical method

a reduction procedure for generalized riemann problems with application to nonlinear transmission lines generalized simple wave solutions to quasilinear hyperbolic nonhomogeneous systems of pdes are obtained through the differential constraint method these solutions prove to be flexible enough to solve generalized riemann problems where discontinuous initial data are involved within such a theoretical framework the governing model of nonlinear transmission lines is investigated throughout

new conditional symmetries and exact solutions of reaction diffusion convection equations with exponential nonlinearities abstract a complete description of q conditional symmetries for a class of reaction diffusion convection equations with exponential diffusivities is derived it is shown that all the known results for reaction diffusion equations with exponential diffusivities follow as particular cases from those obtained here but not vice versa the symmetries obtained for constructing exact solutions of the relevant equations are successfully applied the exact solutions are compared with those found earlier by means of different approaches an application of the solutions for solving a boundary value problem arising in population dynamics is presented

weak transversality and partially invariant solutions new exact solutions are obtained for several nonlinear physical equations namely the navier stokes and euler systems an isentropic compressible fluid system and a vector nonlinear schrodinger equation the solution method makes use of the symmetry group of the system in situations when the standard lie method of symmetry reduction is not applicable

approximate similarity reduction for singularly perturbed boussinesq equation via symmetry perturbation and direct method we investigate the singularly perturbed boussinesq equation in terms of the approximate symmetry perturbation method and the approximate direct method the similarity reduction solutions and similarity reduction equations of different orders display formal coincidence for both methods series reduction solutions are consequently derived for the approximate symmetry perturbation method similarity reduction equations of the zero order are equivalent to the painleve iv painleve i and weierstrass elliptic equations for the approximate direct method similarity reduction equations of the zero order are equivalent to the painleve iv painleve ii painleve i or weierstrass elliptic equations the approximate direct method yields more general approximate similarity reductions than the approximate symmetry perturbation method

a precise definition of reduction of partial differential equations we give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry classical and non classical reductions of partial differential equations using the introduced definition of reduction of differential equations we establish equivalence of the non classical conditional symmetry and direct ansatz approaches to reduction of partial differential equations as an illustration we give an example of non classical reduction of the nonlinear wave equation in 1 3 dimensions the conditional symmetry approach when applied to the equation in question yields a number of non lie reductions which are far reaching generalizations of the well known symmetry reductions of the nonlinear wave equations

approximate conditional symmetries and approximate solutions of the perturbed fitzhugh nagumo equation we consider a perturbation of the fitzhugh nagumo equation the perturbation is proportional to the electric potential across the cell membrane the purpose of this investigation is to determine the effects of a change in electric potential across the cell membrane exact solutions of the perturbed equation are easily obtained from the well known solutions of the unperturbed fitzhugh nagumo equation the method of approximate conditional symmetries is used to obtain first order approximate solutions of the perturbed fitzhugh nagumo equation the approximate solutions are compared with the exact solutions of the perturbed equation the exact solutions of the perturbed equation do not indicate a change in the wave front connecting one constant state to another there is only a proportional increase or decrease in the constant nonzero state the approximate solutions do show a change in the shape of the wave front connecting two constant states as well as a proportional increase or decrease in the constant nonzero state

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

all solutions of standard symmetric linear partial differential equations have classical lie symmetry abstract it is proven that every solution of any linear partial differential equation with an independent variable deforming classical lie point symmetry is invariant under some classical lie point symmetry this is true for any number of independent variables and for equations of any order higher than one although this result makes use of the infinite dimensional component of the lie symmetry algebra due to linear superposition it is shown that new similarity solutions previously thought not to be classical can be recovered prospectively by allowing symmetries to include superposition of similarity solutions already known from the finite part of the symmetry algebra this result applies to all constant coefficient equations and to many variable coefficient equations such as fokker planck equations

nonclassical symmetry reduction and riemann wave solutions abstract in this paper we employ the nonclassical symmetry method in order to obtain riemann multiple wave solutions of a system of first order quasilinear differential equations we show how to construct a lie module of vector fields which are symmetries of the system supplemented by certain first order differential constraints we demonstrate the usefulness of our approach on several examples

nonclassical symmetry reductions for the kadomtsev petviashvili equation abstract in this paper we discuss dimensional reductions of the physically and mathematically significant kadomtsev petviashvili equation in particular we exhibit some new reductions of the kadomtsev petviashvili equation which are obtained by a generalization of the direct method for determining similarity reductions recently developed by clarkson and kruskal j math phys 30 1989 2201 new solutions of the kadomtsev petviashvili equation are obtained in terms of weierstrass elliptic functions solutions of the lame equations and the first second and fourth painleve transcendents a group theoretical explanation of the new solutions is provided in terms of conditional symmetries leaving only specific subsets of solutions of the kadomtsev petviashvili equation invariant

on symmetries of radially symmetric nonlinear diffusion equations in this paper symmetries are sought for radially symmetric nonlinear diffusion equations of the type ut r1 n rn 1f u ur r the nonlinear forms of this equation which admit lie classical symmetries are completely classified and are shown to exist when f un eu also a trivial symmetry exists for arbitrary f lie backlund symmetries are also considered when f un and shown to exist when n 0 linear case and when n 2 and n 1

the effects of a magnetic field on asymptotics of the trace of the heat kernel abstract this paper looks at the effect of a uniform magnetic field on the trace of the heat kernel for a schrodinger operator with a well type potential using weighted sobolev space techniques and noticing the gauge invariance of the perturbation i show that the magnetic field first appears at a higher term in the small time asymptotic expansion of the trace of the heat kernel than might be naively expected

nonclassical symmetries as special solutions of heir equations in phys d 78 1994 124 we have found that iterations of the nonclassical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the given equation in the present paper we show that special solutions of the right order heir equation correspond to classical and nonclassical symmetries of the original equations an infinite number of nonlinear equations which possess nonclassical symmetries are derived

partially invariant solutions of nonlinear klein gordon and laplace equations the concept of partially invariant solutions is discussed in the framework of the group analysis of systems of partial differential equations using a general and systematic approach based on subgroup classification methods nontrivial partially invariant solutions for specific equations are explicitly obtained belonging to the nlkge and nlle classes in two dimensions it is proven that the obtained partially invariant solutions are distinct from the invariant ones and from solutions obtained by other methods of reducing pdes to odes

constructing conditionally integrable evolution systems in 1 1 dimensions a generalization of invariant modules approach given a generalized lie backlund vector field satisfying certain nondegeneracy assumptions we explicitly describe all 1 1 dimensional evolution systems that admit this vector field as a generalized conditional symmetry the connection with the theory of symmetries of systems of odes and with the theory of invariant modules is discussed

on a new method for constructing exact solutions of the nonlinear differential equations of mathematical physics a simple new method for constructing solutions of multidimensional nonlinear wave equations is proposed

symmetry reductions of a generalized cylindrical nonlinear schrodinger equation in this work symmetry reductions for a generalized cylindrical nonlinear schrodinger equation are presented these are obtained using an extension of the direct method originally developed by clarkson and kruskal 1989 which involves no group theoretic techniques

group invariant solutions of differential equations we introduce the concept of a weak symmetry group of a system of partial differential equations that generalizes the nonclassical method introduced by bluman and cole for finding group invariant