the symmetry and exact solutions of the nonlinear d alembert equation for complex fields the nonlinear wave equations for the complex scalar field invariant under a conformal group are constructed and multiparametrical exact solutions of certain nonlinear complex d alembert equations are found

pseudopotentials and lie symmetries for the generalized nonlinear schrodinger equation the generalized nonlinear schrodinger equation zxx izt f z z is analyzed from the point of view of the existence of pseudopotentials backlund transformations lie symmetries and conservation laws applying the wahlquist estabrook method of closed differential ideals we show that eight classes of nontrivial interaction terms f z z exist for which the equation allows the existence of pseudopotentials five of them simply lead to conservation laws the remaining three to backlund transformations the usual cubic nonlinear schrodinger equation with f z z ez z 2 is obtained as a special case it is also the only case for which the backlund transformation contains a free parameter we show that the real and complex parts of this parameter are generated by the dilation and galilei invariance of the equation

nonlinear equations invariant under the poincare similitude and conformal groups in two dimensional space time all realizations of the lie algebras p 1 1 sim 1 1 and conf 1 1 are classified under the action of the group of local diffeomorphisms of r3 the result is used to obtain all second order scalar differential equations invariant under the corresponding poincare similitude and conformal groups the invariant equations are in general nonlinear and the requirement of linearity turns out to be very restrictive group invariant solutions of some of the conformally invariant equations are obtained either by quadratures or by a linearizing transformation

on separable schrodinger equations we classify 1 3 dimensional schrodinger equations for a particle interacting with the electromagnetic field that are solvable by the method of separation of variables as a result we get 11 classes of the vector potentials of the electromagnetic field a t x a0 t x a t x providing separability of the corresponding schrodinger equations it is established in particular that the necessary condition for the schrodinger equation to be separable is that the magnetic field must be independent of the spatial variables next we prove that any schrodinger equation admitting variable separation into second order ordinary differential equations can be reduced to one of the 11 separable schrodinger equations mentioned above and carry out variable separation in the latter furthermore we apply the results obtained for separating variables in the hamilton jacobi equation

higher symmetries and exact solutions of linear and nonlinear schrodinger equation a new approach for the analysis of partial differential equations is developed which is characterized by a simultaneous use of higher and conditional symmetries higher symmetries of the schrodinger equation with an arbitrary potential are investigated nonlinear determining equations for potentials are solved using reductions to weierstrass painleve and riccati forms algebraic properties of higher order symmetry operators are analyzed combinations of higher and conditional symmetries are used to generate families of exact solutions of linear and nonlinear schrodinger equations

symmetries of the self dual einstein equations i the infinite dimensional symmetry group and its low dimensional subgroups this is the first of two papers in which the authors give a complete classification of symmetry reduced solutions of plebanski s potential equation for self dual einstein spaces in this first part the infinite pseudogroup of symmetries of plebanski s equation is described and the conjugacy classes of all local subgroups of dimensions one two and three over both the real and complex numbers are classified then in the second paper this classification is used to obtain all symmetry reduced solutions

a reverse operation self consistent evaluation approach for singular integrals in the sie analysis of pec targets discussed herein is a novel approach entitled reverse operation self consistent evaluation rose specifically we applied rose for the treatment of weak singular and hypersingular kernels in surface integral equations sies for the computation of electromagnetic em scattering from perfect electrical conductors pecs different from the existing forward methods rose is based on reverse operations the singular entries are not evaluated directly by analytical or numerical evaluation instead they are introduced as unknowns and are solved algebraically for em scattering from pec sies can be set up by using the stratton chu representation formula an auxiliary problem with the same meshed boundary is then formulated of which any plane waves are the self consistent solutions it is shown that by applying a consistent discretization the matrix of the auxiliary problem can be made identical with the original one as a consequence these self consistent solutions can be utilized to solve the singular entries of the original system explicitly a set of plane waves are interpolated using the appropriate basis functions subsequently the corresponding coefficient vectors are utilized to construct small linear systems to solve the troublesome singular entries in the underlying matrix the proposed rose approach is applicable to different singular kernels as well as different discretization schemes in this paper we restrict our discussion to the weak singular and hypersingular kernels using respectively div conformal rao wilton glisson basis functions and nonconformal piecewise constant vector basis functions accelerated by the multilevel fast multipole algorithm the proposed method is applicable for real life applications

riemann functions and the group e 1 1 historically lie algebras of first order symmetry operators have proven to be a useful method for finding equivalence classes of riemann functions here this idea is extended to higher order symmetries the approach is to seek self adjoint linear hyperbolic partial differential equations that separate variables in more than one coordinate system under the action of the group e 1 1 the equations derived admit no nontrivial first order operators and can only be obtained from second order symmetry operators using this symmetry structure a new equivalence class of riemann functions can then be found

reduction operators of linear second order parabolic equations the reduction operators i e the operators of nonclassical conditional symmetry of 1 1 dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described this problem proves to be equivalent in some sense to solving initial equations the no go result is extended to the investigation of point transformations admissible transformations equivalence transformations lie symmetries and lie reductions of the determining equations for the nonclassical symmetries transformations linearizing the determining equations are obtained in the general case and under different additional constraints a nontrivial example illustrating applications of reduction operators to finding exact solutions of equations from the class under consideration is presented an observed connection between reduction operators and darboux transformations is discussed

maximal abelian subalgebras of pseudounitary lie algebras abstract the task of classifying and constructing all maximal abelian subalgebras of su p q p q 1 is reduced to that of classifying orthogonally indecomposable masas these are either maximal abelian nilpotent subalgebras represented by nilpotent matrices in any finite dimensional representation or for p q they can be nonorthogonally decomposable and their study can be reduced to a construction of manss of sl p c two types of manss of su p q are shown to exist one rowed and non one rowed numerous classification theorems are proven and applied to obtain all masas of su p 1 su p 2 and su p q with p q 6 physical applications are discussed

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

iterating the nonclassical symmetries method we show that iterations of the nonclassical symmetries method for any evolution equation give raise to new nonlinear equations which inherit the symmetry algebra of the given equation invariant solutions of these heir equations supply new solutions of the original equation the same method generates heir equations for hyperbolic and elliptic equations some examples are given

on the new approach to variable separation in the time dependent schrodinger equation with two space dimensions we suggest an effective approach to separation of variables in the schrodinger equation with two space variables using it we classify inequivalent potentials v x1 x2 such that the corresponding schrodinger equations admit separation of variables besides that we carry out separation of variables in the schrodinger equation with the anisotropic harmonic oscillator potential v k1x21 k2x22 and obtain a complete list of coordinate systems providing its separability most of these coordinate systems depend essentially on the form of the potential and do not provide separation of variables in the free schrodinger equation v 0

exact solutions of the multidimensional classical φ6 field equations obtained by symmetry reduction the often used 6 model of classical critical phenomena is studied in 3 1 dimensional minkowski and euclidean spaces the euler lagrange equations describing the kinetics of the scalar order parameter are in this case nonlinear klein gordon equations the method of symmetry reduction is systematically applied to derive all the solutions invariant under subgroups with generic orbits of codimension 1 whenever the obtained ordinary differential equations have the painleve property they can be transformed to one of two standard forms these are then solved in terms of elliptic functions or elementary ones this results in a large number of new exact solutions particularly interesting solutions are found in the immediate vicinity of the tricritical point our treatment of the 6 theory is complete only for the four dimensional spaces m 3 1 and e 4 but many of the results are given for the more general cases of m n 1 and e n 1

symmetry classification and exact solutions of the kramers equation the complete classification of lie symmetries for the kramers equation is described group invariance under infinitesimal transformations is used to generate a wide class of exact solutions some of the known solutions are obtained as particular cases the general results can also be used to solve some types of moving boundary problems we researched some q conditional symmetry properties of this equation

coherent structures in strongly interacting many body systems ii classical solutions and quantum fluctuations for pt i see ibid vol 22 p 4877 1989 previously i the author considered a second quantised hamiltonian which can be used to model a large number of important strongly interacting many body systems he has shown that the dynamics of such systems can be described exactly using quantum fields the equations of motion derived in this way are highly nonlinear partial differential equations pde he now uses a standard field theoretical approach where initially the fields are treated as classical functions he studies the solutions of these equations which remarkably may be obtained making full use of very recent mathematical discoveries the symmetry reduction method he finds exact solutions of the equations of motion for the classical field for each of the four main cases separately found in the preceding paper these cases may correspond to different physical situations and the last the second order case corresponds to the most general physical situation subsequently he outlines possible quantisation procedures for these fields he then briefly discusses the type of boundary conditions which may be applied to both the classical field and the quantum fluctuations which arise

non classical reductions of initial value problems for a class of nonlinear evolution equations we classify initial value problems for a class of one dimensional evolution equations which can be reduced to cauchy problems for some systems of first order ordinary differential equations the technique applied relies heavily on higher conditional symmetries of the equations under study which means in particular that the obtained reductions cannot be derived within the framework of the standard lie approach

nearly hypersingular integrals by double arctan transformation in higher order geometry modeling in order to directly utilize the dyadic green s functions in surface integral equations sies a novel double arctan transformation for nearly hypersingular integrals is proposed in this communication this new transformation is flexible and applicable to nearly hypersingular integrals in the forms of hat r hat r r 3 hat r r 3 and 1 r 3 over the curved surfaces by a fully numerical method with the help of the sigmoidal transformation to improve the stability of this new singularly handling method there results an efficient solution for the third order near singularity problems in sies moreover the proposed method is also effective for the lower orders of the nearly singular integral kernels with typical testing cases the performance of this method is fairly evaluated and its validity and stability is well demonstrated

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

analysis of the three dimensional time dependent landau ginzburg equation and its solutions the time dependent landau ginzburg equation of the form eta t d delta eta p eta where delta is the three dimensional laplacian operator and p eta is an odd order polynomial up to fifth order has been used to describe the kinetics of non conservative order parameters eta at and near criticality the symmetry reduction method has been applied to solve this equation when p eta is given by p eta a b eta c eta 3 d eta 5 this can be used to model both first and second order phase transitions which take place with a not 0 or without a 0 an external field when either d not 0 or c not 0 two general cases have been found i a b 0 or a b d 0 where the symmetry group involves translations in 3 1 dimensional spacetime rotations in three dimensional space and a dilation ii otherwise where the symmetry group involves translations in 3 1 dimensional spacetime and rotations in three dimensional space all the reductions to ode and the corresponding symmetry variables have been derived the painleve type reduced ode have been solved exactly while the remaining ones can be analysed numerically or approximately physical properties of the obtained solutions have been discussed including their energies

self similar solutions for a coupled system of nonlinear schrodinger equations this work is devoted to the study of self similar solutions of the 2 1 dimensional coupled nonlinear schrodinger equations which occur in nonlinear optics 1 psi 1 z psi 1 xx psi 1 yy eta mod psi 1 to 2 1 h mod psi 2 mod 2 psi 1 0 and square root 1 psi 2 psi 2 xx psi 2 yy eta mod psi 2 mod 2 1 h mod psi 1 psi 2 0 where psi 1 and psi 2 are complex functions h is a non vanishing real parameter eta or 1 and in or 1 the authors give the point symmetry properties of the model and calculate generic 2 1 dimensional symmetry reductions some exact and approximate solutions are obtained in particular they use a variational approach to determine and classify a set of physically relevant localized nonsingular self similar solutions

lie and conditional symmetries of the three component diffusive lotka volterra system lie and q conditional symmetries of the classical three component diffusive lotka volterra system in the case of one space variable are studied the group classification problems for finding lie symmetries and q conditional symmetries of the first type are completely solved notably non lie symmetries q conditional symmetry operators for a multi component nonlinear reaction diffusion system are constructed for the first time the results are compared with those derived for the two component diffusive lotka volterra system the conditional symmetry obtained for the non lie reduction of the three component system used for modeling competition between three species in population dynamics is applied and the relevant exact solutions are found particularly the exact solution describing different scenarios of competition between three species is constructed

numerical evaluation via singularity cancellation schemes of near singular integrals involving the gradient of helmholtz type potentials in this paper we present a purely numerical procedure to evaluate strongly near singular integrals involving the gradient of helmholtz type potentials for observation points at finite arbitrarily small distances from the source domain in the proposed approach the source domain is subdivided into a disc plus truncated subtriangles and proper variable transformations are applied in each integration domain to exactly cancel the kernel singularity a novel feature of the proposed angular transform is that required discrete values of the inverse transform which is transcendental are determined via a root finding procedure the same idea can also be applied to other transforms that arise in singularity cancellation methods the resulting integral may then evaluated via a low order gauss legendre quadrature scheme

symmetry reduction for the kadomtsev petviashvili equation using a loop algebra the kadomtsev petviashvili kp equation ut 3uux 2 1 4 uxxx x 3 uyy 4 0 allows an infinite dimensional lie group of symmetries i e a group transforming solutions amongst each other the lie algebra of this symmetry group depends on three arbitrary functions of time t and is shown to be related to a subalgebra of the loop algebra a 1 4 low dimensional subalgebras of the symmetry algebra are identified specifically all those of dimension n 3 and also a physically important six dimensional lie algebra containing translations dilations galilei transformations and quasirotations new solutions of the kp equation are obtained by symmetry reduction using the one dimensional subalgebras of the symmetry algebra these solutions contain up to three arbitrary functions of t

considerations on double exponential based cubatures for the computation of weakly singular galerkin inner products highly accurate and efficient cubatures based on the double exponential quadrature rules are presented for the computation of weakly singular integrals arising in galerkin mixed potential integral equation formulations due to their unique ability to handle non smooth kernels the proposed integration schemes can safely replace in a plug n play sense the traditional gauss legendre rules in the existing singularity cancellation and singularity subtraction methods numerical examples using rwg basis functions confirm the excellent performance of the proposed method

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

continuous subgroups of the poincare group p 1 4 an exhaustive description of the nonsplitting subalgebras of the lp 1 4 algebra with respect to p 1 4 conjugation is presented

initial value problems for evolutionary partial differential equations and higher order conditional symmetries we suggest a new approach to the problem of dimensional reduction of initial boundary value problems for evolution equations in one spatial variable the approach is based on higher order generalized conditional symmetries of the equations involved it is shown that reducibility of an initial value problem for an evolution equation to a cauchy problem for a system of ordinary differential equations can be fully characterized in terms of conditional symmetries which leave invariant the equation in question we also give some examples of the solution of initial value problems for second and third order nonlinear differential equations by reduction by their conditional symmetries we give a systematic classification of general second order partial differential equations admitting second order conditional symmetries based on lie s classification of invariant second order ordinary differential equations this yields five classes of principally new initial value problems for nonlinear evolution equations which admit no lie symmetries and are reducible via second order conditional symmetries

symmetry reduction of the wess zumino model and supersymmetric reduced systems an investigation of the supersymmetric reduction of the wess zumino model is carried out for the subgroups of the poincare group the necessary conditions for residual supersymmetry derived from super poincare transformations and a classification of the subalgebras of the poincare algebra which might give rise to supersymmetric residual systems are obtained two supersymmetric reductions of the off shell model with interactions by bidimensional subgroups are presented

extended symmetry analysis of generalized burgers equations using enhanced classification techniques we carry out the extended symmetry analysis of the class of generalized burgers equations of the form ut uux f t x uxx 0 this enhances all the previous results on symmetries of these equations and includes the description of admissible transformations lie symmetries lie and nonclassical reductions hidden symmetries conservation laws potential admissible transformations and potential symmetries the study is based on the fact that the class is normalized and its equivalence group is finite dimensional

on global bifurcation for quasilinear elliptic systems on bounded domains general second order quasilinear elliptic systems with nonlinear boundary conditions on bounded domains are formulated into nonlinear mappings between sobolev spaces it is shown that the linearized mapping is a fredholm operator of index zero this and the abstract global bifurcation theorem of jacobo pejsachowicz patrick j rabier degree theory for c1 fredholm mappings of index 0 j anal math 76 1998 289 319 allow us to carry out bifurcation analysis directly on these elliptic systems at the abstract level we establish a unilateral global bifurcation result that is needed when studying positive solutions finally we supply two examples of cross diffusion population model and chemotaxis model to demonstrate how the theory can be applied

a novel approach for evaluating hypersingular and strongly singular surface integrals in electromagnetics solving electromagnetic em problems by integral equation methods requires an accurate and efficient treatment for the singular integral kernels related to the green s function for surface integral equations sies there are l and k operators which include hypersingular integrals hsis and strongly singular integrals ssis respectively the hsis are generated from the double gradient of the green s function while the ssis come from the single gradient of the green s function although the hsis could be reduced to weakly singular integrals wsis in the method of moments mom implementation with divergence conforming basis function such as the rao wilton glisson rwg basis function they do appear in nystrom method nm or boundary element method bem and one has to tackle them the ssis always exist in the k operator and could also exist in the l operator when the testing function is not the rwg like basis function the treatment for the hsis and ssis is essential because they have a significant influence on the numerical solutions there have been many publications dealing with the singular integrals but they mainly focus on the wiss or ssis and the hsis were seldom addressed in this work we develop a novel approach for evaluating those hsis and ssis based on the stokes theorem the derived formulas are much simpler and more friendly in implementation since no polar coordinates or extra coordinate transformation are involved numerical experiments are presented to demonstrate the effectiveness of the approach

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

the symmetry and some exact solutions of the nonlinear many dimensional liouville d alembert and eikonal equations multiparametrical exact solutions of the many dimensional nonlinear d alembert liouville sine gordon and eikonal equations are obtained the maximally extensive local invariance groups of the equations are determined and invariants of the extended poincare group are found

a projection height independent adaptive radial angular r 2 transformation for singular integrals a new radial angular r 2 adaptive singularity cancellation transformation is proposed this new transformation is flexible and applicable to singular integrals over triangular domains when the height tends to zero and the observation point is in the plane of the source domain the transformation remains stable and approaches the corresponding transformation within the source plane usually the green s function and its gradient appear in the integral kernels leading to first order and second order singularities the newly derived transformation formula provides an efficient solution for second order singular coupling integral kernels and the formula is in particular more efficient than alternative schemes for singular vector integral kernels however it is also effective for the lower order singular integral kernels in electromagnetic boundary integral equation formulations the proposed transformation is efficient for all types of singular integrals

singular reduction modules of differential equations the notion of singular reduction modules i e of singular modules of nonclassical conditional symmetry of differential equations is introduced it is shown that the derivation of nonclassical symmetries for differential equations can be improved by an in depth prior study of the associated singular modules of vector fields the form of differential functions and differential equations possessing parameterized families of singular modules is described up to point transformations singular cases of finding reduction modules are related to lowering the order of the corresponding reduced equations as examples singular reduction modules of evolution equations and second order quasi linear equations are studied reductions of differential equations to algebraic equations and to first order ordinary differential equations are considered in detail within the framework proposed and are related to previous no go results on nonclassical symmetries

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

inclusion of finite temperature effects in a nonlinear description of strongly interacting many body systems abstract we consider a class of quantum many body systems which are coupled by one and two body interactions following our earlier work on deriving nonlinear field equations of motion for such systems we incorporate the effects of finite temperature through a semiclassical free energy functional we first obtain implicit analytical formulas for the entropy density which lend themselves to simple asymptotic expansions we then study the effects of entropy in both homogeneous and inhomogeneous regimes in the former case critical amplitude rescaling transition temperature shifts and crossover effects are found in the latter case saturation effects are shown to promote the emergence of spatial periodicities in 1d

invariant solutions of the supersymmetric sine gordon equation a comprehensive symmetry analysis of the supersymmetric sine gordon equation is performed two different forms of the supersymmetric system are considered we begin by studying a system of partial differential equations corresponding to the coefficients of the various powers of the anticommuting independent variables next we consider the super sine gordon equation expressed in terms of a bosonic superfield involving anticommuting independent variables in each case a lie super algebra of symmetries is determined and a classification of all subgroups having generic orbits of codimension 1 in the space of independent variables is performed the method of symmetry reduction is systematically applied in order to derive invariant solutions of the supersymmetric model several types of algebraic hyperbolic and doubly periodic solutions are obtained in explicit form

fredholm operators semigroups and the asymptotic and boundary behavior of solutions of pdes abstract every semigroup t on a banach space x can be used to define elements u x of exponential type relative to t by requiring that u t s v for some s 0 and v x let then x and y be banach spaces in which the exponential type is characterized by the semigroups t and s respectively and let l l x y be fredholm it is shown that if l satisfies some compatibility conditions with respect to t and s and if f y has exponential type then every solution u x of lu f has exponential type as well when l is a differential operator it is often compatible in this sense and in suitable spaces with semigroups that embody an asymptotic or boundary behavior this yields a way to study such a behavior in solutions of pdes which is technically simple very general and delivers rather sharp results furthermore this approach is easily generalized to the nonlinear setting

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

exponential decay and fredholm properties in second order quasilinear elliptic systems we consider second order quasilinear elliptic systems on un bounded domains in the setting of sobolev spaces we complete our earlier work on the fredholm and properness properties of the associated differential operators by giving verifiable conditions for the linearization to be fredholm of index zero this opens the way to using the degree for c 1 fredholm maps of index zero as a tool in the study of such quasilinear systems our work also enables us to check the fredholm assumption which plays an important role in rabies approach to proving exponential decay to zero at infinity of solutions c 2010 elsevier inc all rights reserved

symmetries of the fisher kolmogorov petrovskii piskunov equation with a nonlocal nonlinearity in a semiclassical approximation abstract the classical group analysis approach used to study the symmetries of integro differential equations in a semiclassical approximation is considered for a class of nearly linear integro differential equations in a semiclassical approximation an original integro differential equation leads to a finite consistent system of differential equations whose symmetries can be calculated by performing standard group analysis the approach is illustrated by the calculation of the lie symmetries in explicit form for a special case of the one dimensional nonlocal fisher kolmogorov petrovskii piskunov population equation

on the reduction of the nonlinear multi dimensional wave equations and compatibility of the d alembert hamilton system abstract the necessary conditions of the compatibility of the d alembert hamilton system in minkowsky space r 1 n are established the problem of reduction of p 1 n invariant wave equations to ordinary differential equations is discussed

application of sigmoidal transformations to weakly singular and near singular boundary element integrals accurate numerical determination of line integrals is fundamental to reliable implementation of the boundary element method for a source point distant from a particular element standard gaussian quadrature is adequate as well as being the technique of choice however when the integrals are weakly singular or nearly singular source point near the element this technique is no longer adequate here a co ordinate transformation technique based on sigmoidal transformations is introduced to evaluate weakly singular and near singular integrals a sigmoidal transformation has the effect of clustering the integration points towards the endpoints of the interval of integration the degree of clustering is governed by the order of the transformation comparison of this new method with existing co ordinate transformation techniques shows that more accurate evaluation of these integrals can be obtained based on observations of several integrals considered guidelines are suggested for the order of the sigmoidal transformations copyright 1999 john wiley sons ltd

new scale invariant nonlinear differential equations for a complex scalar field abstract we describe all complex wave equations of the form u f u u invariant under the extended poincare group as a result we have obtained the five new classes of p 1 3 invariant nonlinear partial differential equations for the complex scalar field

lie symmetries of a generalised nonlinear schrodinger equation i the symmetry group and its subgroups the symmetry group of the generalised non linear schrodinger equation i psi t delta psi a0 psi a1 mod psi mod 2 psi a2 mod psi mod 4 psi in three space dimensions is shown to be the extended galilei group g 3 for a1a2 not 0 and the galilei similitude group gd 3 including a dilation for a1 oor a2 0 all lie subgroups of g 3 and gd 3 are found they will be used in a subsequent paper to obtain group invariant solutions of the equation

new conditional symmetries and exact solutions of the nonlinear wave equation we have constructed the new class of conditional symmetries of the nonlinear wave equation in 1 3 dimensions these symmetries enable us to obtain broad families of new exact solutions of the nonlinear wave equation containing up to the four arbitrary functions

amplitude phase representation for solutions of nonlinear d alembert equations we consider the nonlinear complex d alembert equation square operator psi f mod psi mod psi with psi represented in terms of amplitude and phase in 1 n dimensional minkowski space we exploit a compatible d alembert hamilton system to construct new types of exact solutions for some nonlinearities

a family of nonlinear klein gordon equations and their solutions various forms of the nonlinear klein gordon equation are seen to have exact solitonlike solutions when separation of variables is postulated the family for which these exact solutions are found includes the sine gordon equation as a special case an interesting conclusion obtained is that both the soliton antisoliton and breather solutions hitherto known for the sine gordon equations result from a broader class of klein gordon equations the method can be extended to other equations

on a reduction and solutions of nonlinear wave equations with broken symmetry a generalised definition for invariance of partial differential equations is proposed exact solutions of the equations with broken symmetry are obtained

advanced numerical schemes for the accurate evaluation of 4 d reaction integrals in the method of moments in this paper we present a simple and efficient fully numerical technique for the evaluation of 4 d reaction integrals arising in the method of moments in the proposed approach the accuracy of the source integration which uses a singularity cancellation scheme is retained in the final 4 d reaction integral through the application of an appropriate variable transformation and numerical quadrature scheme to the test integral several numerical results compared to the usual gauss legendre test quadrature demonstrate the accuracy of the method for both planar and curvilinear discretizations moreover the proposed scheme is independent of test basis functions hence it can be used for higher order basis functions as well as curl conforming or buffa christiansen test basis functions

singular reduction operators in two dimensions the notion of singular reduction operators i e of singular operators of nonclassical conditional symmetry of partial differential equations in two independent variables is introduced all possible reductions of these equations to first order odes are exhaustively described as examples properties of singular reduction operators of 1 1 dimensional evolution and wave equations are studied it is shown how to favourably enhance the derivation of nonclassical symmetries for this class by an in depth prior study of the corresponding singular vector fields

a complete basis set of functionally independent invariants of continuous lie groups using the methods of differential equations the author has obtained the complete set of functionally independent bases of invariants for some continuous lie groups of mathematical physics i the conformal group ii the semi direct product of the weyl group and the one parameter group of psi translation iii the direct product of the poincare group and the one parameter group of psi scaling iv the weyl group v the 14 parameter maximal symmetry group of the schrodinger equation for the coulombic system of atoms and molecules and vi su 3 of elementary particle physics the first four groups are the only groups which together with the poincare group are the maximal symmetry groups of all forms of the non linear klein gordon equation

on the general solution of the d alembert equation with a nonlinear eikonal constraint and its applications we construct the general solutions of the system of nonlinear differential equations nu 0 u u 0 in the four and five dimensional complex pseudo euclidean spaces the results obtained are used to reduce the multidimensional nonlinear d alembert equation 4u f u to ordinary differential equations and to construct its new exact solutions

preliminary group classification of a class of fourth order evolution equations we perform preliminary group classification of a class of fourth order evolution equations in one spatial variable following the approach developed by basarab horwath et al acta appl math 69 43 2001 we construct all inequivalent partial differential equations belonging to the class in question which admit semisimple lie groups in addition we describe all fourth order evolution equations from the class under consideration which are invariant under solvable lie groups of dimension n 4 we have constructed all galilei invariant equations belonging to the class of evolution differential equations under study the list of so obtained invariant equations contains both the well known fourth order evolution equations and a variety of new ones possessing rich symmetry and as such may be used to model nonlinear processes in physics chemistry and biology

symmetry properties and explicit solutions of the generalized weierstrass system the method of symmetry reduction is systematically applied to derive several classes of invariant solutions for the generalized weierstrass system inducing constant mean curvature surfaces and to the associated two dimensional nonlinear sigma model a classification of subgroups with generic orbits of codimension one of the lie point symmetry group for these systems provides a tool for introducing symmetry variables and reduces the initial systems to different nonequivalent systems of ordinary differential equations we perform a singularity analysis for them in order to establish whether these ordinary differential equations have the painleve property these ordinary differential equations can then be transformed to standard forms and next solved in terms of elementary and jacobi elliptic functions this results in a large number of new solutions and in some cases new interesting constant mean curvature surfaces are found furthermore this symmetry analysis is extended to include conditional symmetries by subjecting the original system to certain differential constraints in this case several new types of nonsplitting algebraic trigonometric and hyperbolic multisoliton solutions have been obtained in explicit form some physical interpretation of these results in the areas of fluid membranes string theory two dimensionl gravity and cosmology are given

generalized conditional symmetries of evolution equations we analyze the relationship of generalized conditional symmetries of evolution equations to the formal compatibility and passivity of systems of differential equations as well as to systems of vector fields in involution earlier results on the connection between generalized conditional invariance and generalized reduction of evolution equations are revisited this leads to a no go theorem on determining equations for operators of generalized conditional symmetry it is also shown that up to certain equivalences there exists a one to one correspondence between generalized conditional symmetries of an evolution equation and parametric families of its solutions

symmetry reduction for nonlinear relativistically invariant equations symmetry reduction is studied for the relativistically invariant scalar partial differential equation h u u 2 u 0 in n 1 dimensional minkowski space m n 1 the introduction of k symmetry variables 1 k as invariants of a subgroup g of the poincare group p n 1 having generic orbits of codimension k n in m n 1 reduces the equation to a pde in k variables all codimension 1 symmetry variables in m n 1 n arbitrary reducing the equation studied to an ode are found as well as all codimension 2 and 3 variables for the low dimensional cases n 2 3 the type of equation studied includes many cases of physical interest in particular nonlinear klein gordon equations such as the sine gordon equation and hamilton jacobi equations

singular basis functions and curvilinear triangles in the solution of the electric field integral equation basis functions are formulated that account for singularities in the charge density near an edge on a conducting body the formulation is general and the basis functions are valid for planar as well curvilinear geometries in principle singularities of any order can be treated but best results are obtained for so called knife edge singularities results are compared with exact solutions or measurements where available for some simple problems

separation of variables in the kramers equation we consider the problem of separation of variables in the kramers equation admitting a non trivial symmetry group provided the external potential v x is at most quadratic a complete solution of the problem of separation of variables is obtained furthermore we construct solutions of the kramers equation with separated variables in explicit form

conditional symmetry of a porous medium equation abstract we obtain an exhaustive description of conditional symmetries admitted by a nonlinear porous medium equation in one spatial dimension it is proved that solving the determining equations for the conditional symmetries either yields classical lie symmetries or is equivalent to solving the porous medium equation itself the second possibility is shown to be typical for an arbitrary evolution type partial differential equation in one spatial variable we show that this very property is responsible for the well known phenomenon when using non classical symmetries yield exact solutions which prove to be invariant ones

nonlocal ansatze reduction and some exact solutions for the system of the van der waals equations abstract in this paper two types of nonlocal equivalence transformations of the system of convection diffusion equations are found using these transformations we obtained two images of the van der waals system lie symmetries of these images are used to construct nonlocal ansatze and reduce that system owing to this it has become possible to have built classes of exact solutions of the van der waals system lie symmetry of this system is used to construct nonlocal ansatze and reduce its both images

global bifurcation for quasilinear elliptic equations on r n keywords degree for fredholm maps of index zero connected sets of solutions nonexistence of nonzero solutions reference ana article 2001 001doi 10 1007 pl00004863view record in web of science record created on 2008 12 10 modified on 2016 08 08

new non lie ansatze and exact solutions of nonlinear reaction diffusion convection equations a constructive method for obtaining new exact solutions of nonlinear evolution equations is further developed the method is based on the consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high order ordinary differential equation using this method new non lie ansatze and exact solutions are obtained for two classes of diffusion equations with power and exponential nonlinearities which describe real processes in physics chemistry and biology the analysis of the found solutions and the relation of the proposed method to some approaches which have been suggested in several recently published papers are presented

group properties of generalized quasi linear wave equations abstract in this paper complete group classification of a class of 1 1 dimensional generalized quasi linear wave equations is performed by using the lie ovsiannikov method additional equivalent transformation and furcate split method lie reductions of some truly variable coefficient wave equations which are singled out from the classification results are investigated some classes of exact solutions of these variable coefficient wave equations are constructed by means of both the reductions and the additional equivalent transformations the nonclassical symmetries to the generalized quasi linear wave equation are also studied this enabled to obtain some exact solutions of the wave equations which are invariant under certain conditional 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

improved double arctan transformation for the stable evaluation of nearly hypersingular integrals in sie the evaluation of singular or near singular integrals is one of the key techniques in the integral equation method based on the previously proposed double arctan transformation dat for the nearly hypersingular integrals in curved surface modeling this work presents new transformation methods to improve the accuracy and stability of dat for the following two situations i e the field point is 1 extremely close to the source surface and 2 extremely close to the boundary of the source surface the original dat fails in these two situations and hence limits its applications the reasons that cause such an instability are studied two additional transformations are introduced to make the dat stable for these two cases for situation i the exponential transformation et is adopted to optimize the distribution of the gaussian quadrature points for situation ii the shape function transformation sft is further proposed to remove the nearly angular singularity we denote the newly developed dat with et and sft as the improved dat idat in this communication the typical testing cases are provided to validate the proposed method

on subalgebras of the lie algebra of the extended poincare group p 1 n some general results on the subalgebras of the lie algebra ap 1 n of the extended poincare group p 1 n n 2 with respect to p 1 n conjugation have been obtained all subalgebras of ap 1 4 that are nonconjugate to the subalgebras of ap 1 4 are classified with respect to p 1 4 conjugation the list of representatives of each conjugacy class is presented

exact solutions for the nonlinear klein gordon and liouville equations in four dimensional euclidean space a systematic method for constructing particular solutions of the nonlinear klein gordon and liouville equations in four spatial dimensions is developed the method of solution presented here first consists of reducing nonlinear partial differential equations to ordinary differential equations ode s by introducing symmetry variables and then seeking exact solutions for more tractable ode s various exact solutions are presented in which new solutions with nonspherical symmetries are included furthermore the exact method is applied to the above equations in general n spatial dimensions among them a conformally invariant nonlinear klein gordon equation is particularly interesting from the viewpoint of field theories the exact solutions for these equations are generalizations of those for the corresponding equations in four spatial dimensions

separation of variables in the nonlinear wave equation we develop a technique making it possible to handle the problem of separation of variables in nonlinear differential equations using it we obtain a number of new two dimensional nonlinear wave equations admitting separation of variables and construct their exact solutions

special conformal groups of a riemannian manifold and lie point symmetries of the nonlinear poisson equation abstract we obtain a complete group classification of the lie point symmetries of nonlinear poisson equations on generic pseudo riemannian manifolds m using this result we study their noether symmetries and establish the respective conservation laws it is shown that the projection of the lie point symmetries on m are special subgroups of the conformal group of m in particular if the scalar curvature of m vanishes the projection on m of the lie point symmetry group of the poisson equation with critical nonlinearity is the conformal group of the manifold we illustrate our results by applying them to the thurston geometries

a certain class of solutions of the nonlinear wave equation in this paper are investigated some differential geometry methods in the theory of the nonlinear wave equation 2u u u u a special class of solutions is discussed for which u u is constant on each level of the function u it is proved that levels of such solutions form in the space of independent variable s hypersurfaces with all principal curvatures constant the general form of such hypersurfaces is given then it is proved that via the method of characteristics it is possible to construct in principle all the solutions of the discussed class they may be obtained by integration of an ode of second order using a special class of the polynomial functions some new solutions are given for equations v 4av3 3bv2 2cv d v exp v v sin v v cosh v and v sinh v

orthogonal and non orthogonal separation of variables in the wave equation utt uxx v x u 0utt uxx v x u 0 we develop a direct approach to the separation of variables in partial differential equations within the framework of this approach the problem of the separation of variables in the wave equation with time independent potential reduces to solving an over determined system of nonlinear differential equations we have succeeded in constructing its general solution and as a result all potentials v x permitting variable separation have been found for each of them we have constructed all inequivalent coordinate systems providing separability of the equation under study it should be noted that the above approach yields both orthogonal and non orthogonal systems of coordinates

numerical evaluation of singular and near singular potential integrals a simple and efficient numerical procedure using a singularity cancellation scheme is presented for evaluating singular and near singular potential integrals with 1 r singularities the procedure not only has several advantages over singularity subtraction methods but also improves on some aspects of other singularity cancellation methods such as polar and duffy transformations a theoretical analysis is presented for triangles quadrilaterals tetrahedrons bricks and prisms and numerical results are presented for triangles and prisms

the subalgebras of the rank two symplectic lie algebra abstract the semisimple subalgebras of the rank 2 symplectic lie algebra sp 4 c are well known and we recently classified its levi decomposable subalgebras in this article we classify the solvable subalgebras of sp 4 c up to inner automorphism this completes the classification of the subalgebras of sp 4 c more broadly speaking in completing the classification of the subalgebras of sp 4 c we have completed the classification of the subalgebras of the rank 2 semisimple lie algebras

separation of variables in 1 2 dimensional schrodinger equations using our classification of separable schrodinger equations with two space dimensions published in j math phys 36 5506 1995 we give an exhaustive description of the coordinate systems providing their separability furthermore we apply these results to separate variables in the heat hamilton jacobi and fokker planck equations

a family of augmented duffy transformations for near singularity cancellation quadrature a new family of systematically constructed near singularity cancellation transformations is presented yielding quadrature rules for integrating near singular kernels over triangular surfaces this family results from a structured augmentation of the well known duffy transformation the benefits of near singularity cancellation quadrature are that no analytical integral evaluations are required and applicability in higher order basis function and curvilinear settings six specific transformations are constructed for near singularities of orders one two and three two of these transformations are found to be equivalent to existing ones the performance of the new schemes is thoroughly assessed and compared with that of existing schemes results for the gradient of the scalar green function are also presented for simplicity static kernel results are shown the new schemes are competitive with and in some cases superior to the existing schemes considered