symmetries and algebras of the integrable dispersive long wave equations in 2 1 dimensional spaces similarly to the kadomtsev petviashvili kp equation a set of generalized symmetries with arbitrary functions of t is given by a simple constructable formula for the integrable dispersive long wave equations in 2 1 space dimensions these symmetries constitute an infinite dimensional lie algebra which is a generalization to the known winfinity algebra

numerically inverting a class of singular fourier transforms theory and application to mountain waves many partial differential equations of physics are solved using fourier transforms even when solutions to the transformed equations can be found analytically it is rare for their inverse transforms to be known in terms of simple functions instead asymptotic and or numerical approaches are commonly used to approximate the inverse transforms a numerical technique is here developed to invert a one dimensional fourier transform with singularities on the real axis of the complex plane that are at worst simple poles subject to the condition that the inverse transform vanishes as its transform variable becomes infinite in one direction by way of demonstration it is used to compute the respective solutions of long and wurtele for trapped and evanescent lee waves driven by flow over gently sloping hills

low dimensional real lie algebras all real lie algebras of dimension up to 8 that admit a nontrivial levi decomposition are found

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

solvable lie algebras of dimension six six dimensional solvable lie algebras over the field of real numbers that possess nilradicals of dimension four are classified into equivalence classes this completes mubarakzyanov s classification of the real six dimensional solvable lie algebras

the poincare algebra in rank 3 simple lie algebras we classify embeddings of the poincare algebra p 3 1 into the rank 3 simple lie algebras up to inner automorphism we show that there are exactly two embeddings of p 3 1 into sl 4 c which are however related by an outer automorphism of sl 4 c next we show that there is a unique embedding of p 3 1 into so 7 c up to inner automorphism but no embeddings of p 3 1 into sp 6 c all embeddings are explicitly described as an application we show that each irreducible highest weight module of sl 4 c not necessarily finite dimensional remains indecomposable when restricted to p 3 1 with respect to any embedding of p 3 1 into sl 4 c

infinitely many lax pairs and symmetry constraints of the kp equation starting from a known lax pair one can get some infinitely many coupled lax pairs infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways in this paper taking the well known kadomtsev petviashvili kp equation as a special example we show that infinitely many nonhomogeneous linear lax pairs can be obtained by using infinitely many symmetries differentiating the spectral functions with respect to the inner parameters using a known lax pair and the darboux transformations dt infinitely many nonhomogeneous nonlinear lax pairs can also be obtained by means of the infinitely many lax pairs dt and the conformal invariance of the schwartz form of the kp equation infinitely many new nonlocal symmetries can be obtained naturally infinitely many integrable models in 1 1 dimensions 2 1 dimensions 3 1 dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the kp equation related to the infinitely many lax pairs

a fast method for the numerical evaluation of continuous fourier and laplace transforms the fast fourier transform fft is often used to compute numerical approximations to continuous fourier and laplace transforms however a straightforward application of the fft to these problems often requires a large fft to be performed even though most of the input data to this fft may be zero and only a small fraction of the output data may be of interest in this note the fractional fourier transform previously developed by the authors is applied to this problem with a substantial savings in computation

a classification of the embeddings of the diamond lie algebra into sl 3 c and sp 4 c and restrictions of irreducible modules the diamond lie algebra d is a four dimensional solvable lie algebra it is a central extension of the poincare lie algebra in two dimensions we classify the embeddings of d into each of the classical lie algebras of rank 2 namely sl 3 c and sp 4 c up to conjugation by elements of sl 3 c or sp 4 c respectively we then show that the irreducible representations of sl 3 c or sp 4 c remain indecomposable upon restriction to d under any embedding

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

lie symmetry analysis and some new exact solutions of the wu zhang equation the lie symmetry analysis and the basic similarity reductions are performed for the wu zhang equation a 2 1 dimensional nonlinear dispersive wave equation some new exact solutions generated from the similarity transformation are provided they demonstrate some new three dimensional features of a single solitary wave and two interacting solitary waves

lie symmetries of soliton equations with self consistent sources via source generation procedure abstract in this paper the source generation procedure sgp proposed by hu and wang x b hu h y wang construction of dkp and bkp equation with self consistent sources inverse problems 22 2006 1903 1920 is utilized to derive lie symmetries of bilinear soliton equations with self consistent sources sescs such as kpescs bkpescs and differential difference kpescs furthermore it is shown that these lie symmetries constitute generators of the corresponding lie symmetry algebras

pfaffianization of the two dimensional toda lattice pfaffianization procedure due to hirota and ohta is applied to the two dimensional toda lattice as a result a pfaffianized version of the two dimensional toda lattice is found

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

constructing semisimple subalgebras of semisimple lie algebras abstract algorithms are described that help with obtaining a classification of the semisimple subalgebras of a given semisimple lie algebra up to linear equivalence the algorithms have been used to obtain classifications of the semisimple subalgebras of the simple lie algebras of ranks 8 these have been made available as a database inside the sla package of gap 4 the subalgebras in this database are explicitly given as well as the inclusion relations among them

a sextic formalism for three dimensional elastodynamics of cylindrically anisotropic radially inhomogeneous materials elasticity of arbitrary cylindrically anisotropic radially inhomogeneous media is considered the system of six first order ordinary differential equations describing time harmonic cylindrical waves propagating along the cylinder axis is obtained the matrix of the system coefficients has a specific algebraic symmetry underlying the structure of the formalism the matricant solution in the form of the peano expansion is discussed the wentzel kramers brillouin approximation is derived algebraic features of the solution are analysed and their physical content is elucidated the dispersion equation for waves in a hollow cylinder with free or clamped lateral surfaces is formulated in terms of the impedance admittance matrices their analytical properties are established which are useful for analysis of the resulting frequency spectrum the theory is extended to anisotropic radially inhomogeneous piezoelectric cylinders for which the octet generalization of the formalism is developed and studied

spectral method algorithm for modeling dispersion of acoustic modes in elastic cylindrical structures a new spectral method algorithm can be used to study wave propagation in cylindrically layered fluid and elastic structures the cylindrical structure is discretized with chebyshev points in the radial direction whereas differentiation matrices are used to approximate the differential operators we express the problem of determining modal dispersions as a generalized eigenvalue problem that can be solved readily for all eigenvalues corresponding to various axial wavenumbers modal dispersions of guided modes can then be expressed in terms of axial wavenumbers as a function of frequency the associated eigenvectors are related to the displacement potentials that can be used to calcu late radial distributions of modal amplitudes as well as stress components at a given frequency the workflow includes input parameters and the construction of differentiation matrices and boundary conditions that yield the generalized eigenvalue problem results from this algorithm for a fluid filled borehole surrounded by an

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

calculating the spectrum of anisotropic waveguides using a spectral method the computation of the spectrum of a waveguide with arbitrary anisotropy with spatial dependence is a challenging task due to the coupling between axial and azimuthal harmonics this problem is tackled in cylindrical coordinates by extending a spectral method for the general case by considering the matrix representation of the operator on the right hand side of the governing equations the latter are exactly reformulated as an infinite set of integro differential equations essential part of this study is taking into account the coupling of different harmonics which becomes evident from the kernels of these equations provided a waveguide is translationally invariant in the axial direction the coupling of axial harmonics vanishes a practical approximation and truncation procedure yields a generalized eigenvalue problem which can be solved numerically to obtain the entire spectrum of the operator and to construct the dispersion curves for the eigenmodes the spectral method is tested against the results from the measurements of dispersion curves for the monopole dipole and quadrupole normal modes of scaled boreholes in tilted transverse isotropy anisotropic rock sample besides the comparison of dispersion curves calculated by the spectral method and those computed from the synthetic data 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

levi decomposable subalgebras of the symplectic algebra c2 the semisimple subalgebras of the symplectic algebra c2 are well known in this article we classify the levi decomposable subalgebras of the symplectic algebra c2 up to inner automorphism by levi s theorem a full classification of the subalgebras of c2 would be complete with a classification of its solvable subalgebras

some constraints and solutions of the kadomtsev petviashvili equation we illustrate the relations among the symmetry invariant group and constraint for a differential equation applied to the kadomtsev petviashvili equation some constraints and solutions are given in particular the equation associated with symmetry sigma of the kp equation is introduced and discussed

fast computation of multidimensional fourier integrals bailey and swarztrauber siam j sci comput 15 1994 pp 1105 1110 have developed an algorithm that uses fast fourier transforms to numerically evaluate one dimensional fourier integrals whose integration ranges are finite and symmetric returning a symmetric range of values in transform variable space their algorithm is here extended to evaluate multidimensional fourier integrals possessing asymmetric integration and transform variable ranges using a second order open quadrature rule to admit integrands with endpoint singularities

modeling of wave dispersion along cylindrical structures using the spectral method algorithm and code are presented that solve dispersion equations for cylindrically layered media consisting of an arbitrary number of elastic and fluid layers the algorithm is based on the spectral method which discretizes the underlying wave equations with the help of spectral differentiation matrices and solves the corresponding equations as a generalized eigenvalue problem for a given frequency the eigenvalues correspond to the wave numbers of different modes the advantage of this technique is that it is easy to implement especially for cases where traditional root finding methods are strongly limited or hard to realize i e for attenuative anisotropic and poroelastic media the application of the new approach is illustrated using models of an elastic cylinder and a fluid filled tube the dispersion curves so produced are in good agreement with analytical results which confirms the accuracy of the method particle displacement profiles of the fundamental mode in a free solid cylinder are computed for a range of frequencies

the schrodinger langevin equation special solutions and nonexistence of solitary waves two types of solutions of the schrodinger langevin equation are investigated it is proved that a special type of gaussian solutions exist globally in time for the harmonic oscillator hamiltonian furthermore it is shown that the schrodinger langevin equation can have no solitary wave type solutions in the damped free particle case which lie in l2

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

negative flows of the potential kp hierarchy we construct a grassmannian like formulation for the potential kp hierarchy including additional negative flows our approach will generalize the notion of a r function to include negative flows we compare the resulting hierarchy with results by hirota satsuma and bogoyavlenskii

maximal abelian subalgebras of real and complex symplectic lie algebras we provide guidelines for classifying maximal abelian subalgebras masa s of the symplectic lie algebras sp 2n r and sp 2n c into conjugacy classes under the lie groups sp 2n r and sp 2n c respectively the task of classifying all masa s is reduced to the classification of orthogonally indecomposable oid masa s two types of orthogonally indecomposable masa s of sp 2n c exist 1 indecomposable maximal abelian nilpotent subalgebras mans s 2 decomposable masa s their classification reduces to a classification of mans s of sl n c four types of orthogonally indecomposable masa s of sp 2n r exist 1 absolutely indecomposable masa s mans s 2 relatively indecomposable masa s their classification reduces to a classification of mans s of su p q for p q n 3 decomposable absolutely oid masa s involving mans s of sl n r 4 decomposable relatively oid masa s involving mans s of sl n 2 c for n even low dimensional cases of sp 2n f n 1 2 3 f r or c are treated exhau

on the infinite dimensional symmetry group of the davey stewartson equations the lie algebra of the group of point transformations leaving the davey stewartson equations dse s invariant is obtained the general element of this algebra depends on four arbitrary functions of time the algebra is shown to have a loop structure a property shared by the symmetry algebras of all known 2 1 dimensional integrable nonlinear equations subalgebras of the symmetry algebra are classified and used to reduce the dse s to various equations involving only two independent variables

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

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

ermakov and non ermakov systems in quantum dissipative models via the hydrodynamical formulation of quantum mechanics a unified protocol to treat the quantum time dependent harmonic oscillator with friction is presented described by two different models an explicitly time dependent linear schrodinger equation caldirola kanai model and a logarithmic nonlinear schrodinger equation kostin model for the former model an ermakov system that makes it possible to obtain an invariant of ermakov lewis type is derived for the latter model a non ermakov system is derived instead and it is shown that neither an exact nor an approximate invariant of ermakov lewis type exists

algebra of pseudodifferential operators and symmetries of equations in the kadomtsev petviashvili hierarchy point symmetries are obtained for all equations in the kp hierarchy the lie algebra for each equation is infinite dimensional and involves several arbitrary functions of the corresponding time tn the symmetry algebra is a semidirect sum of a virasoro algebra and a kac moody one the positive part of this algebra is embedded into the known w algebra of kp symmetries and into the free fermion algebra l the corresponding action on the tau function is presented the negative part of the point symmetries does not fit into the free fermion algebra but is embedded into a p algebra based on the algebra of pseudodifferential operators

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

the euclidean algebra in rank 2 classical lie algebras we classify the embeddings of the euclidean algebra e 2 into the rank 2 semisimple classical lie algebras the classifications are up to inner automorphism we also examine the finite dimensional irreducible representations of the rank 2 semisimple classical lie algebras restricted to e 2 with respect to various embeddings all lie algebras and representations are over the complex numbers

closed subsets of root systems and regular subalgebras abstract we describe an algorithm for classifying the closed subsets of a root system up to conjugation by the associated weyl group our algorithm is implemented in the language of the computer algebra system gap 4 we discuss the implementation and give runtimes on some sample inputs the classification of the closed subsets of an irreducible root system is closely related to the classification of the regular subalgebras up to inner automorphism of the corresponding simple lie algebra we show how to obtain the regular subalgebras corresponding to a given closed set for the root systems of rank 3 the version of this paper on the arxiv has tables describing the subalgebras that were obtained

maurer cartan equations for lie symmetry pseudogroups of differential equations a new method of constructing structure equations of lie symmetry pseudogroups of differential equations dispensing with explicit solutions of the infinitesimal determining systems of the pseudogroups is presented and illustrated by the examples of the kadomtsev petviashvili and korteweg de vries equations

variable coefficient davey stewartson system with a kac moody virasoro symmetry algebra we study the symmetry group properties of the variable coefficient davey stewartson vcds system the lie point symmetry algebra with a kac moody virasoro kmv structure is shown to be isomorphic to that of the usual constant coefficient ds system if and only if the coefficients satisfy some conditions these conditions turn out to coincide with those for the vcds system to be transformable to the ds system by a point transformation the equivalence group of the vcds system is applied to pick out the integrable subsystems from a class of non integrable ones additionally the full symmetry group of the ds system is derived explicitly without exponentiating its symmetry algebra lump solutions rationally localized in all directions in 2 introduced by ozawa for the ds system are shown to hold even for the vcds system precisely when the system belongs to the integrable class i e equivalent to the ds system these solutions can be used for establishing exact blow up solutions in finite time in the s

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

analysis of the symmetry group and exact solutions of the dispersionless kp equation in n 1 dimensions the lie algebra of the symmetry group of the n 1 dimensional generalization of the dispersionless kadomtsev petviashvili equation is obtained and identified as a semi direct sum of a finite dimensional simple lie algebra and an infinite dimensional nilpotent subalgebra group transformation properties of solutions under the subalgebra sl 2 r are presented known explicit analytic solutions in the literature are shown to be actually group invariant solutions corresponding to certain specific infinitesimal generators of the symmetry group

the fourth painleve equation and associated special polynomials in this article rational solutions and associated polynomials for the fourth painleve equation are studied these rational solutions of the fourth painleve equation are expressible as the logarithmic derivative of special polynomials the okamoto polynomials the structure of the roots of these okamoto polynomials is studied and it is shown that these have a highly regular structure the properties of the okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials further representations are given of the associated rational solutions in the form of determinants through schur functions

equations of arbitrary order invariant under the kadomtsev petviashvili symmetry group by means of a simple new approach a general kadomtsev petviashvili kp family with an arbitrary function of group invariants of arbitrary order is proposed it is proved that the general kp family possesses a common infinite dimensional kac moody virasoro lie point symmetry algebra the known fourth order one can be re obtained as a special example the finite transformation group is presented in a clearer form the kac moody virasoro group invariant solutions and the kac moody group invariant solutions of the kp family are determined by the boussinesq and kdv families respectively

levi decomposable algebras in the classical lie algebras abstract we examine embeddings of an important class of levi decomposable algebras into the classical lie algebras in particular we classify the embeddings of a n c n 1 b n c 2 n 1 c n c 2 n and d n c 2 n into the complex simple lie algebras a n 1 b n 1 c n 1 and d n 1 respectively up to inner automorphism

painleve analysis lie symmetries and invariant solutions of potential kadomstev petviashvili equation with time dependent coefficients in this paper the painleve analysis of 2 1 dimensional variable coefficients potential kadomstev petviashvili vcpkp equation s x email protected t s email protected x email protected t s x x x email protected t s y y 0 has been performed to study the painleve properties the vcpkp equation is also reduced to a 1 1 dimensional partial differential equation via classical lie group method and reduced equations are further studied to obtain certain general solutions

symmetry full symmetry groups and some exact solutions to a generalized davey stewartson system the lie symmetry algebra of a generalized davey stewartson gds system is obtained the general element of this algebra depends on eight arbitrary functions of time which has a kac moody virasoro loop algebra structure and is isomorphic to that of the standard integrable davey stewartson equations under certain conditions imposed on parameters and arbitrary functions then based on the symmetry group direct method proposed by lou and ma j phys a 38 l129 2005 the full symmetry groups of the gds system are obtained from the full symmetry groups both the lie symmetry group and a group of discrete transformations can be obtained finally some exact solutions involving sech sech2 sech2 and tanh tanh2 tanh2 type solitary wave solutions are presented by a generalized subequation expansion method

full symmetry groups painleve integrability and exact solutions of the nonisospectral bkp equation abstract based on the generalized symmetry group method presented by lou and ma lou and ma non lie symmetry groups of 2 1 dimensional nonlinear systems obtained from a simple direct method j phys a math gen 38 2005 l129 firstly both the lie point groups and the full symmetry group of the nonisospectral bkp equation are obtained at the same time a relationship is constructed between the new solutions and the old ones of equation secondly the nonisospectral bkp can be proved to be painleve integrability by combining the standard wtc approach with the kruskal s simplification some solutions are obtained by using the standard truncated painleve expansion finally based on the relationship by the generalized symmetry group method and some solutions by using the standard truncated painleve expansion some interesting solution are constructed

kdv extensions with painleve property by means of the conformal invariance mobious transformation invariance the well known kdv equation is extended to i a 1 1 dimensional space time symmetric form ii two 2 1 dimensional space isotropic forms and iii general 3 1 dimensional and n dimensional forms the extensions are proven to be integrable under the meaning that they possess the painleve property