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

symplectic methods for the nonlinear schrodinger equation abstract in this paper we show that the spatial discretization of the nonlinear schrodinger equation leads to a hamiltonian system which can be simulated with symplectic numerical schemes in particular we apply two symplectic integrators to the nonlinear schrodinger equation and we demonstrate that they are able to produce accurate results and to preserve very well the invariants of the original system such as the energy and charge

efficient and accurate numerical methods for the klein gordon schrodinger equations in this paper we present efficient unconditionally stable and accurate numerical methods for approximations of the klein gordon schrodinger kgs equations with without damping terms the key features of our methods are based on i the application of a time splitting spectral discretization for a schrodinger type equation in kgs ii the utilization of fourier pseudospectral discretization for spatial derivatives in the klein gordon equation in kgs iii the adoption of solving the ordinary differential equations odes in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying crank nicolson leap frog for linear nonlinear terms for time derivatives the numerical methods are either explicit or implicit but can be solved explicitly unconditionally stable and of spectral accuracy in space and second order accuracy in time moreover they are time reversible and time transverse invariant when there is no damping terms in kgs conserve or keep the same decay rate of the wave energy as that in kgs without or with a linear damping term keep the same dynamics of the mean value of the meson field and give exact results for the plane wave solution extensive numerical tests are presented to confirm the above properties of our numerical methods for kgs finally the methods are applied to study solitary wave collisions in one dimension 1d as well as dynamics of a 2d problem in kgs

nonlinear stability and convergence of finite difference methods for the good boussinesq equation the good boussinesq equationu tt u xxxx u xx u 2 xx has recently been found to possess an interesting soliton interaction mechanism in this paper we study the nonlinear stability and the convergence of some simple finite difference schemes for the numerical solution of problems involving the good boussinesq equation numerical experimentas are also reported

smooth solutions to a class of free boundary parabolic problems we establish existence uniqueness and regularity results for solutions to a class of free boundary parabolic problems including the free boundary heat equation which arises in the so called focusing problem in the mathematical theory of combustion such solutions are proved to be smooth with respect to time for positive t if the data are smooth

a general approach to stability in free boundary problems abstract every solution of a linear elliptic equation on a bounded domain may be considered as an equilibrium of a free boundary problem the free boundary problem consists of the corresponding parabolic equation on a variable unknown domain with free boundary conditions prescribing both dirichlet and neumann data we establish a rigorous stability analysis of such equilibria including the construction of stable and unstable manifolds for this purpose we transform the free boundary problem to a fully nonlinear and nonlocal parabolic problem on a fixed domain with fully nonlinear lateral boundary conditions and we develop the general theory for such problems as an illustration we give two examples the second being the focussing flame problem in combustion theory

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

extension of the ablowitz ladik method to the derivative nonlinear schro dinger equation abstract the derivative nonlinear schrodinger equation is solved by application of the ablowitz ladik scheme to an equivalent equation the variations of the results due to modifications in the spatial grid size and time step are analyzed the scheme maintains the main properties of the original equation and allows the use of rather large time steps

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

global phase synchronization in an array of time delay systems we report the identification of global phase synchronization gps in a linear array of unidirectionally coupled mackey glass time delay systems exhibiting highly non phase coherent chaotic attractors with complex topological structure in particular we show that the dynamical organization of all the coupled time delay systems in the array to form gps is achieved by sequential synchronization as a function of the coupling strength further the asynchronous ones in the array with respect to the main sequentially synchronized cluster organize themselves to form clusters before they achieve synchronization with the main cluster we have confirmed these results by estimating instantaneous phases including phase difference average phase average frequency frequency ratio and their differences from suitably transformed phase coherent attractors after using a nonlinear transformation of the original non phase coherent attractors the results are further corroborated using two other independent approaches based on recurrence analysis and the concept of localized sets from the original non phase coherent attractors directly without explicitly introducing the measure of phase

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

recovery of the absorption coefficient from diffused reflected light using a discrete diffusive model an algorithm based on a discrete model for photon migration is proposed for recovering the absorption coefficient in a semi infinite slab a pulsed light source is modeled and only reflected signals are used in the reconstruction the diffusion coefficient and speed of light are assumed constant and known the algorithm is based on the result that the earliest reflected signals obtained from a particular voxel depend either in a linear or in a quadratical fashion on the square of the absorption coefficient in that voxel by solving the discrete model in the forward direction in a region where the absorption coefficient has been calculated already the absorption coefficients in the voxels on the boundary of the region are calculated using interpolation results show that simple absorptive regions can be accurately recovered

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

a free boundary problem stemmed from combustion theory part ii stability instability and bifurcation results we deal with a free boundary problem depending on a real parameter in a infinite strip in r2 providing stability instability and bifurcation

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

stability in a two dimensional free boundary combustion model we prove rigorously the stability of the planar travelling wave solution to a free boundary system in r2 arising as a model in combustion theory for appropriate values of the reduced lewis number after changements of coordinates and unknowns the problem is reduced to the stability of the null solution of a fixed boundary parabolic system the difficulties consist in the fact that the final system is fully nonlinear and the spectrum of the linearized system contains the halffine 0 they are overcome setting the problem in a suitably weighted holder space

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

integrable discretizations for toda type lattice soliton equations from a proper 2 2 discrete isospectral problem a new integrable lattice soliton system is proposed integrable discretizations of a general toda type lattice soliton equation associated with the discrete isospectral problem are established the lagrangian and newtonian forms of integrable discretizations of toda type lattice equations which occur in the literature are given uniformly and some new integrable discretizations of the toda type lattice are obtained

operator splitting methods for generalized korteweg de vries equations we apply the method of operator splitting on the generalized korteweg de vries kdv equation ut f u x uxxx 0 by solving the nonlinear conservation law ut f u x 0 and the linear dispersive equation ut uxxx 0 sequentially we prove that if the approximation obtained by operator splitting converges then the limit function is a weak solution of the generalized kdv equation convergence properties are analyzed numerically by studying the effect of combining different numerical methods for each of the simplified problems

linearly implicit methods for the nonlinear schrodinger equation in nonhomogeneous media the nonlinear schrodinger equation in one dimensional cartesian coordinates is studied numerically by means of three linearly implicit finite difference methods that result in block tridiagonal matrices at each time level as a function of the damping pumping and changes in the refraction index the numerical experiments performed to determine the effects of the time step spatial step size and implicitness parameters of the dispersion and nonlinear terms show that the most accurate results are obtained when both the dispersion and nonlinear terms are treated by means of a second order accurate trapezoidal discretization and full linearization of the nonlinear terms it is also shown that if either the dispersion or the nonlinear terms are treated with a first order accurate approximation substantial errors are found and these errors are largest when the nonlinear terms are discretized by means of a first order difference and do not decrease substantially as the number of grid points is increased it has been observed that when solitons move from a medium to another one characterized by a higher refraction index the amplitude and speed of the soliton increase whereas its width decreases upon approaching the interface the amplitude of the soliton was found to increase downstream while some radiation was observed upstream both the increase in amplitude and radiation were found to increase with the change in the refraction index the effect of the distance where the index of refraction changes was found to be small if this distance is smaller than the width of the soliton for solitons propagating from a medium to another one of lower refraction index the amplitude and speed of the soliton decrease whereas its width increases the amplitude of the soliton increases downstream and spreads towards the boundary where it is reflected when the index of refraction decreases below a threshold it was found that the soliton may become trapped at the interface the results presented in this paper indicate that theories based on treating solitons as particles perturbation methods inverse scattering and variational formulations can provide accurate results only when the change in the refraction index is sufficiently small

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

on the abstract cauchy problem of parabolic type in spaces of continuous functions on etudie l existence l unicite et les proprietes de regularite des solutions du probleme de cauchy u t au r f t o t t u o u 0 ou a d a ce e est un operateur lineaire clos dans l espace de banach e

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

order of convergence of splitting schemes for both deterministic and stochastic nonlinear schrodinger equations we first prove the second order convergence of the strang type splitting scheme for the nonlinear schrodinger equation the proof does not require commutator estimates but crucially relies on an integral representation of the scheme it reveals the connection between strang type splitting and the midpoint rule we then show that the integral representation idea can also be used to study the stochastic nonlinear schrodinger equation with multiplicative noise of stratonovich type even though the nonlinear term there is not globally lipschitz we prove the first order convergence of a splitting scheme of it both schemes preserve the mass they are very efficient because they use explicit formulas to solve the subproblems containing the nonlinear or the nonlinear plus stochastic terms

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

hopscotch method the numerical solution of the frank kamenetskii partial differential equation numerical solutions to the frank kamenetskii partial differential equation modelling a thermal explosion in a cylindrical vessel are obtained using the hopscotch scheme we observe that a nonlinear source term in the equation leads to numerical difficulty and hence adjust the scheme to accommodate such a term numerical solutions obtained via matlab mathematica and the crank nicolson implicit scheme are employed as a means of comparison to gain insight into the accuracy of the hopscotch scheme the solution is compared to a power series solution obtained via the lie group method the numerical solution is also observed to converge to a well known steady state solution a linear stability analysis is performed to validate the stability of the results obtained

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

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

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

a free boundary problem stemmed from combustion theory part i existence uniqueness and regularity results abstract we deal with a free boundary problem depending on a real parameter in a infinite strip in r 2 which admits a planar travelling wave solution for every r we prove existence uniqueness and regularity results for the solutions near the travelling wave

a numerical scheme for the simulation of blow up in the nonlinear schrodinger equation we present a self adaptive multigrid version of a conservative finite difference scheme useful for the study of collapse processes in nonlinear schrodinger equations nlses as an example we study the character of the focusing singularity of the two dimensional critical nlse

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

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

optimal error estimates of finite difference methods for the gross pitaevskii equation with angular momentum rotation we analyze finite difference methods for the gross pitaevskii equation with an angular momentum rotation term in two and three dimensions and obtain the optimal convergence rate for the conservative crank nicolson finite difference cnfd method and semi implicit finite difference sifd method at the order of o h2 2 in the l2 norm and discrete h1 norm with time step and mesh size h besides the standard techniques of the energy method the key technique in the analysis for the sifd method is to use the mathematical induction and resp for the cnfd method is to obtain a priori bound of the numerical solution in the l norm by using the inverse inequality and the l2 norm error estimate in addition for the sifd method we also derive error bounds on the errors between the mass and energy in the discretized level and their corresponding continuous counterparts respectively which are at the same order of the convergence rate as that of the numerical solution itself finally numerical results are reported to confirm our error estimates of the numerical methods

instabilities in a two dimensional combustion model with free boundary we prove instability of the planar travelling wave solution in a two dimensional free boundary problem modelling the propagation of near equidiffusional premixed flames in the whole plane we reduce the problem to a fixed boundary fully nonlinear parabolic system the spectrum of the linearized operator contains an interval 0 c c 0 so we cannot construct backward solutions we use an argument about stability of dynamical systems in banach spaces in order to prove pointwise instability of the moving front

uniform error estimates of finite difference methods for the nonlinear schrodinger equation with wave operator we establish uniform error estimates of finite difference methods for the nonlinear schrodinger equation nls perturbed by the wave operator nlsw with a perturbation strength described by a dimensionless parameter varepsilon varepsilon in 0 1 when varepsilon to0 nlsw collapses to the standard nls in the small perturbation parameter regime i e 0 varepsilon ll1 the solution of nlsw is perturbed from that of nls with a function oscillating in time with o varepsilon 2 wavelength at o varepsilon 4 and o varepsilon 2 amplitudes for well prepared and ill prepared initial data respectively this high oscillation of the solution in time brings significant difficulties in establishing error estimates uniformly in varepsilon of the standard finite difference methods for nlsw such as the conservative crank nicolson finite difference cnfd method and the semi implicit finite difference sifd method we obtain error bounds uniformly in varepsilon at the order of o h 2 tau and o h 2 tau 2 3 with time step tau and mesh size h for well prepared and ill prepared initial data respectively for both cnfd and sifd in the l 2 norm and discrete semi h 1 norm our error bounds are valid for general nonlinearity in nlsw and for one two and three dimensions to derive these uniform error bounds we combine varepsilon dependent error estimates of nlsw varepsilon dependent error bounds between the numerical approximate solutions of nlsw and the solution of nls together with error bounds between the solutions of nlsw and nls other key techniques in the analysis include the energy method cut off of the nonlinearity and a posterior bound of the numerical solutions by using the inverse inequality and discrete semi h 1 norm estimate finally numerical results are reported to confirm our error estimates of the numerical methods and show that the convergence rates are sharp in the respective parameter regimes