resonant algebras and gravity the s expansion framework is analyzed in the context of a freedom in closing the multiplication tables for the abelian semigroups including the possibility of the zero element in the resonant decomposition and associating the lorentz generator with the semigroup identity element leads to a wide class of the expanded lie algebras introducing interesting modifications to the gauge gravity theories among the results we find all the maxwell algebras of type and the recently introduced the additional new examples complete the resulting generalization of the bosonic enlargements for an arbitrary number of the lorentz like and translational like generators some further prospects concerning enlarging the algebras are discussed along with providing all the necessary constituents for constructing the gravity actions based on the obtained results

split step orthogonal spline collocation methods for nonlinear schrodinger equations in one two and three dimensions abstract split step orthogonal spline collocation osc methods are proposed for one two and three dimensional nonlinear schrodinger nls equations with time dependent potentials firstly the nls equation is split into two nonlinear equations and one or more one dimensional linear equations commonly the nonlinear subproblems could be integrated directly and accurately but it fails when the time dependent potential cannot be integrated exactly in this case we propose three approximations by using quadrature formulae but the split order is not reduced discrete time osc schemes are applied for the linear subproblems in numerical experiments many tests are carried out to prove the reliability and efficiency of the split step osc ssosc methods solitons in one two and three dimensions are well simulated and conservative properties and convergence rates are demonstrated we also apply the ways of solving the nonlinear subproblems to the split step finite difference ssfd methods and the time splitting spectral tssp methods and the approximate ways still work well finally we apply the ssosc methods to solve some problems of bose einstein condensates

optimal error bounds for two grid schemes applied to the navier stokes equations abstract we consider two grid mixed finite element schemes for the spatial discretization of the incompressible navier stokes equations a standard mixed finite element method is applied over the coarse grid to approximate the nonlinear navier stokes equations while a linear evolutionary problem is solved over the fine grid the previously computed galerkin approximation to the velocity is used to linearize the convective term for the analysis we take into account the lack of regularity of the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data optimal error bounds are obtained

s expansion of higher order lie algebras by means of a generalization of the s expansion method we construct a procedure to obtain expanded higher order lie algebras it is shown that the direct product between an abelian semigroup s and a higher order lie algebra g is also a higher order lie algebra from this s expanded lie algebra are obtained resonant submultialgebras and reduced multialgebras of a resonant submultialgebra

a numerical energy conserving method for the dnls equation an implicit numerical energy conserving method is developed for the derivative nonlinear schrodinger dnls equation for periodic boundary conditions we find no numerical high frequency modulational instabilities in addition to the modulational instability from a linear analysis around a nonlinear state for the dnls equation if the modulation is small and k0 a22 2 t k0 is the wavenumber and a the amplitude the numerical scheme is used to follow the nonlinear behavior of the dnls modulational instability the numerical code is also tested by the evolution for one soliton initial data these tests show that if the modulation is not small compared to the background wave amplitude new nonlinear numerical instabilities are introduced

expanding lie super algebras through abelian semigroups we propose an outgrowth of the expansion method introduced by de azcarraga et al nucl phys b 662 185 2003 the basic idea consists in considering the direct product between an abelian semigroup s and a lie algebra g general conditions under which relevant subalgebras can systematically be extracted from s g are given we show how for a particular choice of semigroup s the known cases of expanded algebras can be reobtained while new ones arise from different choices concrete examples including the m algebra and a d auria fre like superalgebra are considered finally we find explicit nontrace invariant tensors for these s expanded algebras which are essential ingredients in e g the formulation of supergravity theories in arbitrary space time dimensions

a numerical study of the large period limit of a zakharov shabat eigenvalue problem with periodic potentials deconinck and kutz 2006 j comput phys 219 296 321 developed an efficient algorithm for solving the zakharov shabat eigenvalue problem with periodic potentials numerically it is natural to use the same algorithm for solving the problem for non periodic potential decaying potentials defined over the whole real line using large periods in this paper we propose the use of a specific value of the floquet exponent our numerical results indicate that it can produce accurate results long before the period becomes large enough for the analytical convergence results of gardner 1997 j reine angew math 491 149 81 to be valid we also illustrate the rather complicated path to convergence of some nonlinear schr dinger potentials

bianchi spaces and their three dimensional isometries as s expansions of two dimensional isometries in this paper we show that certain three dimensional isometry algebras specifically those of type i ii iii and v according to bianchi s classification can be obtained as expansions of the isometries in two dimensions in particular we use the so called s expansion method which makes use of the finite abelian semigroups because it is the most general procedure known until now also it is explicitly shown why it is impossible to obtain the algebras of type iv vi ix as expansions from the isometry algebras in two dimensions all the results are checked with computer programs this procedure shows that the problem of how to relate by an expansion two lie algebras of different dimensions can be entirely solved in particular the procedure can be generalized to higher dimensions which could be useful for diverse physical applications as we discuss in our conclusions

pseudo spectral solution of nonlinear schro dinger equations abstract we compare four pseudo spectral split step methods for solving a class of nonlinear schrodinger nls equations the importance of observing the l 2 invariance of the continuous problem is demonstrated through numerical experiments the best performance is obtained by transforming the given equation to an nls equation where two of the coefficients satisfy a simple algebraic relationship the problem can be solved efficiently in terms of the new variables and the l 2 norm of the computed solution is time invariant

split step methods for the solution of the nonlinear schro dinger equation a split step method is used to discretize the time variable for the numerical solution of the nonlinear schrodinger equation the space variable is discretized by means of a finite difference and a fourier method these methods are analyzed with respect to various physical properties represented in the nls in particular it is shown how a conservation law dispersion and instability are reflected in the numerical schemes analytical and numerical instabilities of wave train solutions are identified and conditions which avoid the latter are derived these results are demonstrated by numerical examples

dual formulation of the lie algebra s expansion procedure the expansion of a lie algebra entails finding a new bigger algebra g through a series of well defined steps from an original lie algebra g one incarnation of the method the so called s expansion involves the use of a finite abelian semigroup s to accomplish this task in this paper we put forward a dual formulation of the s expansion method which is based on the dual picture of a lie algebra given by the maurer cartan forms the dual version of the method is useful in finding a generalization to the case of a gauge free differential algebra which in turn is relevant for physical applications in e g supergravity it also sheds new light on the puzzling relation between two chern simons lagrangians for gravity in 2 1 dimensions namely the einstein hilbert lagrangian and the one for the so called exotic gravity

multisoliton solution of a class of coupled soliton equations abstract the n soliton solutions of the korteweg devries kdv and boussinesq equations with a source are obtained explicitly by using the bilinear transformation method also the interactions of two soliton solutions are studied and the explicit forms for the one and two soliton solutions are obtained for the two systems

generating higher order lie algebras by expanding maurer cartan forms by means of a generalization of the maurer cartan expansion method we construct a procedure to obtain expanded higher order lie algebras the expanded higher order maurer cartan equations for the case g v0 v1 are found a dual formulation for the s expansion multialgebra procedure is also considered the expanded higher order maurer cartan equations are recovered from s expansion formalism by choosing a special semigroup this dual method could be useful in finding a generalization to the case of a generalized free differential algebra which may be relevant for physical applications in e g higher spin gauge theories

geometrical aspects of the lie algebra s expansion procedure in this article it is shown that s expansion procedure affects the geometry of a lie group changing it and leading us to the geometry of another lie group with higher dimensionality a method for determining the semigroup which would provide a lie algebra from another is outlined via an example finally it is proved that a lie algebra obtained from another lie algebra via s expansion is a non simple lie algebra

order estimates in time of splitting methods for the nonlinear schrodinger equation in this paper we consider the nonlinear schrodinger equation u t i delta u f u 0 in two dimensions we show by an operator theoretic proof that the well known lie and strang formulae which are splitting methods are approximations of the exact solution of order 1 and 2 in time

analytical and numerical aspects of certain nonlinear evolution equations i analytical abstract nonlinear partial difference equations are obtained which have as limiting forms the nonlinear schrodinger korteweg devries and modified korteweg devries equations these difference equations have a number of special properties they are constructed by methods related to the inverse scattering transform they can be used as a basis for numerical schemes to the associated nonlinear evolution equations experiments have shown that they compare very favorably with other known numerical methods papers ii iii in paper ii the ablowitz ladik scheme for the nonlinear schrodinger equation is compared to other known numerical schemes and generally proved to be faster than all utilized finite difference schemes but somewhat slower than the finite fourier pseudospectral methods in paper iii a proposed scheme for the korteweg de vries equation proved to be faster than both the finite difference and finite fourier methods already considered

numerical investigation of the generalized lubrication equation abstract explicit implicit explicit and crank nicolson implicit explicit numerical schemes for solving the generalized lubrication equation are derived we prove that the implicit explicit and crank nicolson implicit explicit numerical schemes are unconditionally stable numerical solutions obtained from both schemes are compared initial curves with both zero and finite contact angles are considered

integrable discretization of time and its application on the fourier pseudospectral method to the korteweg de vries equation this paper presents a new integrable discretization of the korteweg de vries kdv equation different from other discrete analogues we discretize the variable time and obtain an integrable differential difference system this system has the original kdv equation as a standard limit when the step size tends to zero the main idea is based on hirota s bilinear method and backlund transformation and can be applied to other integrable systems by applying the fourier pseudospectral method to the space variable we derive a new numerical scheme for the kdv equation numerical results are found to agree with the exact solution and the first five conservation quantities are preserved quite well

computational synergetics and mathematical innovation it is demonstrated how the computer used in a heuristic mode has greatly augmented our understanding of the mathematics of nonlinear dynamical process examples are given of recent work in soliton mathematics mainly but also nonlinear iterative mappings and low order nonlinear ordinary differential equation systems the role of good graphics in enhancing the discovery and retention of new mathematical properties of equations is illustrated several examples are given of the present frontier of research in two space dimensional nonlinear evolutionary problems

spurious resonance in semidiscrete methods for the korteweg de vries equation a multiple scales analysis of semidiscrete methods for the korteweg de vries equation is conducted methods that approximate the spatial derivatives by finite differences with arbitrary order accuracy and the limiting method the fourier pseudospectral method are considered the analysis reveals that a resonance effect can occur in the semidiscrete solution but not in the solution of the continuous equation it is shown for the fourier pseudospectral discretization that resonance can only be caused by aliased modes the spurious semidiscrete solutions are investigated in numerical experiments and we suggest methods for avoiding spurious resonance

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

regular article finite difference schemes for u t x α δg δu that inherit energy conservation or dissipation property we propose a new procedure for designing by rote finite difference schemes that inherit energy conservation or dissipation property from nonlinear partial differential equations such as the korteweg de vries kdv equation and the cahn hilliard equation the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly since the inherited properties are kept even if the time mesh size changes in the time evolution process we can use some appropriate time mesh adaptive methods to obtain numerical solutions through the derived schemes because of these properties the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions and sufficiently flexible to treat the inheritance of the energy conservation and dissipation properties are verified numerically for the kdv equation and the cahn hilliard equation

infinite s expansion with ideal subtraction and some applications according to the literature the s expansion procedure involving a finite semigroup is valid no matter what the structure of the original lie super algebra is however when something about the structure of the starting super algebra is known and when certain particular conditions are met the s expansion method with its features of resonance and reduction is able not only to lead to several kinds of expanded super algebras but also to reproduce the effects of the standard as well as the generalized inonu wigner contraction in the present paper we propose a new prescription for s expansion involving an infinite abelian semigroup s and the subtraction of an infinite ideal subalgebra we show that the subtraction of the infinite ideal subalgebra corresponds to a reduction our approach is a generalization of the finite s expansion procedure presented in the literature and it offers an alternative view of the generalized inonu wigner contraction we then show how to write the invariant tensors of

dissipative or conservative finite difference schemes for complex valued nonlinear partial differential equations abstract we propose a new procedure for designing finite difference schemes that inherit energy conservation or dissipation property from complex valued nonlinear partial differential equations pdes such as the nonlinear schrodinger equation the ginzburg landau equation and the newell whitehead equation the procedure is a complex version of the procedure that furihata has recently presented for real valued nonlinear pdes furthermore we show that the proposed procedure can be modified for designing linearly implicit finite difference schemes that inherit energy conservation or dissipation property

contractions of lie algebras generalized inonu wigner contractions versus graded contractions recently a new notion of contraction has been introduced the purely algebraically defined graded contractions that fall into two disjoint classes continuous and discrete our main result is that all continuous graded contractions can be realized by generalized inonu wigner contractions furthermore we give a complete characterization of the discrete graded contractions via certain higher order identities inherent to the grading group

the multiscale expansions of difference equations in the small lattice spacing regime and a vicinity and integrability test i we propose an algorithmic procedure i to study the distance between an integrable pde and any discretization of it in the small lattice spacing regime and at the same time ii to test the asymptotic integrability properties of such discretization this method should provide in particular useful and concrete information on how good is any numerical scheme used to integrate a given integrable pde the procedure illustrated on a fairly general ten parameter family of discretizations of the nonlinear schr dinger equation consists of the following three steps i the construction of the continuous multiscale expansion of a generic solution of the discrete system at all orders in following degasperis et al 1997 physica d 100 187 211 ii the application to such an expansion of the degasperis procesi dp integrability test degasperis a and procesi m 1999 asymptotic integrability symmetry and perturbation theory spt98 ed a degasperis and g gaeta singapore world scientific pp 23 37 degasperis a 2001 multiscale expansion and integrability of dispersive wave equations lectures given at the euro summer school what is integrability isaac newton institute cambridge uk 13 24 august integrability lecture notes in physics vol 767 ed a mikhailov berlin springer to test the asymptotic integrability properties of the discrete system and its distance from its continuous limit iii the use of the main output of the dp test to construct infinitely many approximate symmetries and constants of motion of the discrete system through novel and simple formulas

numerical approximation of quadratic observables of schrodinger type equations in the semi classical limit we apply wigner transform techniques to the analysis of difference methods for schrodinger type equations in the case of a small planck constant in this way we are able to obtain sharp conditions on the spatial temporal grid which guarantee convergence for average values of observables as the planck constant tends to zero the theory developed in this paper is not based on local and global error estimates and does not depend on whether caustics develop or not numerical test examples are presented to help interpret the theory

on fully discrete galerkin methods of second order temporal accuracy for the nonlinear schrodinger equation we approximate the solutions of an initial and boundary value problem for nonlinear schrodinger equations with emphasis on the cubic nonlinearity by two fully discrete finite element schemes based on the standard galerkin method in space and two implicit crank nicolson type second order accurate temporal discretizations for both schemes we study the existence and uniqueness of their solutions and provel 2 error bounds of optimal order of accuracy for one of the schemes we also analyze one step of newton s method for solving the nonlinear systems that arise at every time step we then implement this scheme using an iterative modification of newton s method that at each time stept n requires solving a number of sparse complex linear systems with a matrix that does not change withn the effect of this inner iteration is studied theoretically and numerically

a systematic method for constructing time discretizations of integrable lattice systems local equations of motion we propose a new method for discretizing the time variable in integrable lattice systems while maintaining the locality of the equations of motion the method is based on the zero curvature lax pair representation and the lowest order conservation laws in contrast to the pioneering work of ablowitz and ladik our method allows the auxiliary dependent variables appearing in the stage of time discretization to be expressed locally in terms of the original dependent variables the time discretized lattice systems have the same set of conserved quantities and the same structures of the solutions as the continuous time lattice systems only the time evolution of the parameters in the solutions that correspond to the angle variables is discretized the effectiveness of our method is illustrated using examples such as the toda lattice the volterra lattice the modified volterra lattice the ablowitz ladik lattice an integrable semi discrete nonlinear schrodinger system and the lattice heisenberg ferromagnet model for the modified volterra lattice we also present its ultradiscrete analogue

a generalized action for 2 1 dimensional chern simons gravity we show that the so called semi simple extended poincare ssep algebra in d dimensions can be obtained from the anti de sitter algebra by means of the s expansion procedure with an appropriate semigroup s a general prescription is given for computing casimir operators for s expanded algebras and the method is exemplified for the ssep algebra the s expansion method also allows us to extract the corresponding invariant tensor for the ssep algebra which is a key ingredient in the construction of a generalized action for chern simons gravity in 2 1 dimensions

pseudospectral method for the good boussinesq equation we prove the nonlinear stability and convergence of a fully discrete pseudospectral scheme for the good boussinesq equation utt uxxxx uxx u2 xx numerical comparisons with finite difference schemes are also reported

general properties of the expansion methods of lie algebras the study of the relation between lie algebras and groups and especially the derivation of new algebras from them is a problem of great interest in mathematics and physics because finding a new lie group from an already known one also means that a new physical theory can be obtained from a known one one of the procedures that allow us to do so is called expansion of lie algebras and has been recently used in different physical applications particularly in gauge theories of gravity here we report on further developments of this method required to understand in a deeper way their consequences in physical theories we have found theorems related to the preservation of some properties of the algebras under expansions that can be used as criteria and more specifically as necessary conditions to know if two arbitrary lie algebras can be related by some expansion mechanism formal aspects such as the cartan decomposition of the expanded algebras are also discussed finally an instructive example that allows us to check explicitly all our theoretical results is also provided