einstein maxwell solitons the application of the inverse scattering method to the einstein maxwell equations for stationary axisymmetric exterior fields leads the authors to an explicit formula for the ernst and electromagnetic potentials of new exact solutions generated from an arbitrary seed solution

an adaptive moving mesh method for forced curve shortening flow we propose a novel adaptive moving mesh method for the numerical solution of a forced curve shortening geometric evolution equation control of the mesh quality is obtained using a tangential mesh velocity derived from a mesh equidistribution principle where a positive adaptivity measure or monitor function is approximately equidistributed along the evolving curve central finite differences are used to discretize in space the governing evolution equation for the position vector and a second order implicit scheme is used for the temporal integration simulations are presented indicating the generation of meshes which resolve areas of high curvature and are of second order accuracy furthermore the new method delivers improved solution accuracy compared to the use of uniform arc length meshes

explicit description of ansatz en for the ernst equation in general relativity for the ernst equation a hierachy of ansatz that generates determinantal solutions of the ernst equation is proposed the ansatz is described explicitly in the inverse scattering formalism and it is shown that the corresponding exact solutions are determinantal solutions that have been constructed by kyriakopoulos phys rev d 30 1158 1984 and vein class quantum gravit 2 899 1985

transformation group acting on a self dual yang mills hierarchy a gl n c self dual yang mills hierarchy is introduced it is an infinite system of self dual yang mills equations having an infinite number of independent variables cauchy problems for the hierarchy are formally solved by using lie transforms of a wave matrix a relationship between the kadomtsev petviashvili hierarchy and the self dual yang mills hierarchy is discussed furthermore it is shown that an infinite dimensional transformation group acts on a solution space to the n 2 self dual yang mills hierarchy a parametric solution to the hierarchy is also given as a representation of the transformation group

on the link between umbilic geodesics and soliton solutions of nonlinear pdes in this paper we describe a new class of soliton solutions called umbilic solitons for certain nonlinear integrable pdes these umbilic solitons have the property that as the space variable x tends to infinity the solution tends to a periodic wave and as x tends to minus infinity it tends to a phase shifted wave of the same shape the equations admitting solutions in this new class include the dym equation and equations in its hierarchy the methods used to find and analyse these solutions are those of algebraic and complex geometry we look for classes of solutions by constructing associated finite dimensional integrable hamiltonian systems on riemann surfaces in particular in this setting we use geodesics on n dimensional quadrics to find the spatial or x flow which together with the commuting t flow given by the equation itself defines new classes of solutions amongst these geodesics particularly interesting ones are the umbilic geodesics which then generate the class of umbilic soliton solutions this same setting also enables us to introduce another class of solutions of dym like equations which are related to elliptic and umbilic billiards

a method for the integration in time of certain partial differential equations abstract a method for the numerical solution of ordinary differential equations is analyzed that is explicit and yet can conserve the quadratic quantities conserved by the equations the method can be a useful alternative to the usual leapfrog technique in that it does not suffer from the occurrence of blowup phenomena numerical examples concerning the korteweg de vries equation and the nonlinear schrodinger equation are given

existence and regularity of the global attractor for a weakly damped forced shallow water equation in h1 r abstract in this paper we prove the existence of the global attractor for a weakly damped forced shallow water equation in h 1 r moreover we show that the global attractor is actually compact in h 4 r the main method is the combination of the energy equation a suitable splitting of the solutions and bilinear estimates in bourgain spaces

on simple moving grid methods for one dimensional evolutionary partial differential equations abstract two moving grid algorithms for the numerical integration of evolutionary one dimensional partial differential equations are considered both algorithms discretize the equation on trapezoidal space time elements and combine a prediction step with a remeshing technique in order to determine the orientation of the sides of the trapezoids joining nodes at consecutive time levels the behavior of the discretizations employed is forecast by means of the modified equation technique and then studied in a series of numerical experiments

blow up global existence and infinite propagation speed for the weakly dissipative camassa holm equation in this paper we consider the weakly dissipative camassa holm equation first we try to improve other authors results and get some new criterion on blow up then discuss the global existence of the solution finally we intend to establish sufficient conditions on the propagation speed for the weakly dissipative camassa holm equation

finite difference solutions of a non linear schrodinger equation abstract this paper describes a finite difference method to approximate a schrodinger equation with a power non linearity a special case of this equation is currently used to model the propagation of a laser beam in a plasma the main feature of the method we present is that it satisfies a discrete analogue of some important conservation laws of the equations we present numerical results which show in particular the propagation and the formation of solitions in the one dimensional case

an hr adaptive method for the cubic nonlinear schrodinger equation abstract the nonlinear schrodinger equation nlse is one of the most important equations in quantum mechanics and appears in a wide range of applications including optical fibre communications plasma physics and biomolecule dynamics it is a notoriously difficult problem to solve numerically as solutions have very steep temporal and spatial gradients adaptive moving mesh methods r adaptive attempt to optimise the accuracy obtained using a fixed number of nodes by moving them to regions of steep solution features this approach on its own is however limited if the solution becomes more or less difficult to resolve over the period of interest adaptive mesh refinement h adaptive where the mesh is locally coarsened or refined is an alternative adaptive strategy which is popular for time independent problems in this paper we consider the effectiveness of a combined method h r adaptive to solve the nlse in one space dimension simulations are presented indicating excellent solution accuracy compared to other moving mesh approaches the method is also shown to control the spatial error based on the user s input error tolerance evidence is also presented indicating second order spatial convergence using a novel monitor function to generate the adaptive moving mesh

local well posedness of the ostrovsky stepanyams and tsimring equation in sobolev spaces of negative indices abstract in this paper we prove by using the method of bilinear estimate in the bourgain type spaces that the initial value problem of the ost equation u t u x x x h u x h u x x x u u x 0 x r t 0 where 0 and h denotes the usual hilbert transformation is locally well posed in the sobolev space h s r when s 1

the anti self dual yang mills equation and the painleve iii equation the classical transcendental solutions to the painleve iii equation are derived from a family of solutions to the anti self dual yang mills equation it is also shown that the affine weyl group symmetry of piii is recovered from the symmetry of yang s equation

five dimensional axisymmetric stationary solutions as harmonic maps the complete scheme of the application of one and two dimensional subspaces and the subgroups method to five dimensional gravity with a g3 group of motion are presented here in space time and in potential space formalisms from this method one obtains the kramer belinsky ruffini dobiasch maison clement gross perry sorkin solutions etc as special cases

global existence for 2d nonlinear schrodinger equations via high low frequency decomposition method abstract we study global existence of solutions for the cauchy problem of the nonlinear schrodinger equation i u t u u 2 m u in the 2 dimension case where m is a positive integer m 2 using the high low frequency decomposition method we prove that if 10 m 6 10 m 5 s 1 then for any initial value h s r 2 the cauchy problem has a global solution in c r h s r 2 and it can be split into u t e i t y t with y c r h 1 r 2 satisfying y t h 1 1 t 2 1 s 10 m 5 s 10 m 6 where is an arbitrary sufficiently small positive number

almost optimal local well posedness for the 3 1 dimensional maxwell klein gordon equations this paper contains a detailed study of the local in time regularity properties of the maxwell klein gordon mkg equations the mkg equations represent a physical model for the interaction of a spin 0 particle sometimes referred to as a higgs or scalar field with the classical equations of electrodynamics i e maxwell s equations in terms of the calculus of variations the mkg system consists of the euler lagrange equations associated to the density l 4 f f 1 2 d d where f a a is the electro magnetic field associated to the set a of real valued potentials on the minkowski space d 1a is the corresponding covariant derivative with respect to those potentials and r c is a complex scalar field in terms of the field f the euler lagrange equations read f d 1a d d 0 1b

an envelope soliton problem the direct scattering problem of zakharov and shabat for the nonlinear schrodinger equation iu t u xx kappa u 2 u 0 is solved for the initial condition u text sech x the asymptotic solution t uparrow infty is a bound state of n centered solitons all moving at the same speed where n is the integral part of kappa 2 1 2

wave breaking for a periodic shallow water equation the focus of this paper is on the blow up of a recently derived one dimensional shallow water equation which is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equations in the shallow water regime some new criteria guaranteeing the development of singularities in finite time for strong solutions with regular initial data are obtained for the periodic case

analyticity and smoothing effect for the korteweg de vries equation with a single point singularity we show that a solution of the cauchy problem for the kdv equation cases partial tv partial 3 x v partial x v 2 0 qquad t in t t x in bbb r v 0 x phi x endcases has a drastic smoothing effect up to real analyticity if the initial data only have a single point singularity at x 0 it is shown that for h s mathbb r s 3 4 data satisfying the condition sum k 0 infty frac a 0 k k x partial x k phi h s infty the solution is analytic in both space and time variable the above condition allows us to take as initial data the dirac delta measure or the cauchy principal value of 1 x the argument is based on the recent progress on the well posedness result by bourgain 2 and kenig ponce vega 20 and a systematic use of the dilation generator 3t partial t x partial x

exact boundary controllability of the korteweg de vries equation we consider boundary control of the distributed parameter system described by the korteweg de vries kdv equation posed on a finite interval alpha leq x leq beta left begin array l u t u x uu x u xxx 0 u alpha t h 1 t qquad u beta t h 2 t qquad u x beta t h 3 t u end array right qquad for t geq 0 it is shown that by choosing appropriate control inputs h j t j 1 2 3 one can always guide the system from a given initial state phi in h s alpha beta to a given terminal state psi in h s alpha beta in the time period 0 t so long as phi and psi satisfy phi cdot w cdot 0 h s alpha beta leq delta quad mbox and quad psi cdot w cdot t h s alpha beta leq delta for some delta 0 independent of phi and psi where s geq 0 and w equiv w x t is a given smooth solution of the kdv equation this exact boundary controllability is established by considering a related initial value control problem of the kdv equation posed on the whole line r various recently discovered smoothing properties of the kdv equation have played important roles in our approach

a supraconvergent scheme for the korteweg de vries equation a finite difference method for the integration of the korteweg de vries equation on irregular grids is analyzed under periodic boundary conditions the method is shown to be supraconvergent in the sense that though being inconsistent it is second order convergent however such a convergence only takes place on grids with an odd number of points per period when a grid with an even number of points is used the inconsistency of the method leads to divergence numerical results backing the analysis are presented

initial boundary value problem and asymptotic stabilization of the camassa holm equation on an interval abstract we investigate the nonhomogeneous initial boundary value problem for the camassa holm equation on an interval we provide a local in time existence theorem and a weak strong uniqueness result next we establish a result on the global asymptotic stabilization problem by means of a boundary feedback law

on the characterization of finite differences optimal meshes abstract regridding methods has become an important tool in the integration of pde systems whose solutions exhibit sharp transitions in spatial derivatives this paper improves the results presented in an earlier contribution of the author and f oliveira 1988 theoretical justifications of finite differences regridding criteria for the transport and heat equations are presented the nonuniform meshes in the physical space are generated by the use of coordinate transforms which map them into uniform meshes in the computational space after the two mesh systems have been generated two approaches are used for solving the pde to construct the approximations on the uniform mesh in the computational space or to construct the finite difference approximations on the nonuniform mesh in the physical space in this paper we are concerned with the question of the relationship between the two approaches namely the characterization of the mesh density coordinate transform which improves the spatial accuracy of the approximation in the physical computational space

global well posedness of the cauchy problem of the fifth order shallow water equation abstract by using the i method we prove that the cauchy problem of the fifth order shallow water equation t u x 2 t u x 3 u 3 u x u 2 x u x 2 u u x 3 u x 5 u 0 is globally well posed in the sobolev space h s r provided s 5 7 10 4

contour dynamics for the euler equations curvature controlled initial node placement and accuracy abstract we have performed a systematic study of several contour dynamical algorithms for the euler equations for short times we have used the kirchhoff elliptical vortex alone and subject to weak perturbations we have found that if the initial placement of nodes is such that the internodal distance is proportional to curvature p where p 1 3 then errors in short time calculations are minimized this follows because the node density is invariant in time

an adaptive moving grid method for one dimensional systems of partial differential equations we describe a fully adaptive moving grid method for solving initial boundary value problems for systems of one space dimensional partial differential equations whose solutions exhibit rapid variations in space and time the method based on finite differences is of the lagrangian type and has been derived through a co ordinate transformation which leads to equidistribution in space of the second derivative our technique is intermediate between static regridding methods where nodes remain fixed for intervals of time and continuously moving grid methods where the node movement and the pde integration are fully coupled in our approach the computation of the moving grids and the solution on these grids are carried out separately while the nodes are moved at each time step two error monitors have been implemented one to govern the time step selection and the other to eventually adapt the number of moving nodes the method allows the use of different moving grids for different components in the pde system numerical experiments are presented for a set of five sample problems from the literature including two problems from combustion

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

methods for the numerical solution of the nonlinear schroedinger equation optimal l2 rates of convergence are established for several fully discrete schemes for the numerical solution of the nonlinear schroedinger equation both finite differences and finite elements are considered for the discretization in space while the integration in time is treated either by the leap frog technique or by a modified crank nicolson procedure which generalizes the one suggested by delfour fortin and payne and possesses two useful conserved quantities

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

remarks on hirota s d operators the double complex function method and the ernst equation in this paper a double matrix equation with hirota s d operators is given besides the original solutions available this equation provides half the exact solution lost in the common hirota direct method for solving the ernst equation using this double matrix equation we obtain a realization of the double ehlers transformation group in addition we discuss the inverse problem about the equation

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

well posedness for the stochastic higher order modified camassa holm equation abstract in this article the stochastic higher order modified camassa holm equation with additive noise is considered using the fourier restriction norm method and bilinear estimates the local existence and uniqueness result is proved by the contraction mapping principle the global existence of the h 1 r solution of this problem is also obtained by energy inequality

on the numerical solution of initial boundary value problems in one space dimension the numerical solution of initial boundary value problems of the form a u x t u t b u x t u x c u x t is considered particular emphasis is placed on the solution of problems with large gradients e g shocks and boundary layers a mesh selection technique is described that accurately places points in regions where the solution is rapidly changing this is accomplished by a transformation of the original equations to a new coordinate system finite difference solutions for two sample problems are calculated

symmetries of stationary axially symmetric vacuum einstein equations and the new family of exact solutions a new family of exact solutions of the stationary axially symmetric vacuum einstein equations is presented the internal symmetries sl 2 r rotation and duality of parametrization are combined to construct a backlund transformation for the special ansatz of the field equations the backlund transformation can be integrated and the hierarchy of ansatz is generated recursively the riemann hilbert problem is also discussed for the inverse scattering formulas generating nonlocal symmetries

controllability and asymptotic stabilization of the camassa holm equation we investigate the problems of exact controllability and asymptotic stabilization of the camassa holm equation on the circle by means of a distributed control the results are global and in particular the control prevents the solution from blowing up

toda molecule and tomimatsu sato solution towards the complete proof of nakamura s conjecture we discuss the nakamura s conjecture stating that the tomimatsu sato black hole solution with an integer deformation parameter n is composed of the special solutions of the toda molecule equation at the nth lattice site from the previous work in which the conjecture was partly analytically proved we go further towards the final full proof by rearranging the rotation parameter the proof is explicitly performed for the highest and lowest orders though the proof for all orders still remains unsolved the prospect to the full proof becomes transparent and workable by our method

solution of evolutionary partial differential equations using adaptive finite differences with pseudospectral post processing a coordinate transformation approach is described that enables pseudospectral methods to be applied efficiently to unsteady differential problems with steep solutions the work is an extension of a method presented by mulholland huang and sloan for the adaptive pseudospectral solution of steady problems a coarse grid is generated by a moving mesh finite difference method that is based on equidistribution and this grid is used to construct a time dependent coordinate transformation a sequence of spatial transformations may be generated at discrete points in time or a single transformation may be generated as a continuous function of space and time the differential problem is transformed by the coordinate transformation and then solved using a method that combines pseudospectral discretisation in space with a suitable integrator in time numerical results are presented for unsteady problems in one space dimension

an adaptive method of lines solution of the korteweg de vries equation abstract following a method of lines formulation the korteweg de vries equation is solved using a static spatial remeshing algorithm based on the equidistribution principle which allows the number of nodes to be significantly reduced as compared to a fixed grid solution several finite difference schemes including direct and stagewise procedures are compared and the results of a large number of computational experiments are presented which demonstrate that the selection of a spatial approximation scheme for the third order derivative term is the primary determinant of solution accuracy

method of lines study of nonlinear dispersive waves in this study we consider partial differential equation problems describing nonlinear wave phenomena e g a fully nonlinear third order korteweg de vries kdv equation the fourth order boussinesq equation the fifth order kaup kupershmidt equation and an extended kdv5 equation first we develop a method of lines solution strategy using an adaptive mesh refinement algorithm based on the equidistribution principle and spatial regularization techniques on the resulting highly nonuniform spatial grids the computation of high order derivative terms appears particularly delicate and we focus attention on the selection of appropriate approximation techniques finally we solve several illustrative problems and compare our computational approach to conventional solution techniques

a variable node finite element method abstract a new finite element method with variable nodes for integrating partial differential equations in one space dimension is presented the method utilizes the approximation power of piecewise polynomial functions better than a fixed node finite element method because it distributes grid points in a nonuniform way to suit the peculiarities of the solution at each instant of time the method uses the gradient and curvature of the approximating piecewise polynomial itself to determine the position of the nodes rather than physical properties which vary from problem to problem the method has been implemented in a program vfe which has the exceptional feature of a variable space step as well as a variable time step since the method concentrates nodes in regions where small inter nodal spacing is needed it is very efficient in integrating equations with shock like solutions finally the method can be extended to systems of partial differential equations in one dimension and to equations in two dimensions

local well posedness for the dispersion generalized periodic kdv equation abstract in this paper we set up the local well posedness of the initial value problem for the dispersion generalized periodic kdv equation t u x d x u x u 2 u 0 for 2 s 4 and h s t and we show that the 4 is a lower endpoint to obtain the bilinear estimates 1 2 and 1 3 which are the crucial steps to obtain the local well posedness by picard iteration the case 2 was studied in kenig et al 1996 10

review of some adaptive node movement techniques in finite element and finite difference solutions of partial differential equations abstract this article summarizes the results of a literature search for adaptive numerical methods of solving partial differential equations the methods discussed involve the adaptive movement of nodes so as to obtain a low level of solution truncation error while minimizing the number of nodes used in the calculation such methods are applicable to the solution of nonstationary flow problems that contain moving regions of rapid change in the flow variables surrounded by regions of relatively smooth variation flows with shock waves contact surfaces slip streams phase change interfaces and boundary layers can be modelled with great precision by these methods it will be shown that significant economies of execution can be attained if nodes are moved so that they remain concentrated in regions of rapid variation of the flow variables

the global attractor for a weakly damped forced shallow water equation in sobolev spaces of low regularity abstract we consider the global attractor for the weakly damped forced shallow water equation t u x 3 u 1 2 x u 2 1 x 2 1 x u 2 1 2 x u 2 u f in sobolev spaces h s t for s 1 under the assumption that f h 1 t we prove the existence of the global attractor for the equation in h s t for 2 3 s 1 which is identical to that in h 1 t our method is a combination of the i method and decomposing the solution into two parts one of which is uniformly bounded in h 1 t while the other decays exponentially in h s t in the argument we establish some bilinear estimates involving the operator i to make the index s as low as possible

a simplified adaptive mesh technique derived from the moving finite element method an adaptive mesh technique is derived as a constrained minimization of the functional used in the moving finite element mfe method the result is a matrix equation for mesh velocities alone i e a pure adaptive mesh prescription the method retains many attractive features of mfe including the ability to independently control mesh motion through penalty terms

numerical experience with the nonlinear schrodinger equation abstract increasing the magnitude of the parameter multiplying the nonlinear term of the nonlinear schrodinger equation without changing the initial condition u x 0 sech x leads to bound states of an increasing number of solitons this results in very steep gradients in space and time and so provides a more severe test of numerical methods than before in particular we find that methods which satisfy various conservation laws theoretically may now fail to do so in practice various analytical and numerical results relevant to this situation are discussed and illustrated by numerical examples

well posedness of the cauchy problem for a shallow water equation on the circle in this paper we consider the periodic cauchy problem for a fifth order modification of the camassa holm equation we prove local well posedness in appropriate bourgain spaces for initial data in a sobolev space hs t s 1 2 we also prove global well posedness for data in h1 t and of arbitrary size the proofs are based on a priori estimates using fourier analysis techniques microlocalization in phase space an interpolation argument and a fixed point theorem

the ernst equation as a motion on a universal grassmann manifold we find a linearization of the ernst equation by means of universal grassmann manifold ugm techniques all local analytic solutions defined at the origin are obtained by solving an initial value problem for a linear differential equation on a ugm we give an explicit formula which represents solutions of the ernst equation by using this formula we generate several special solutions

well posedness and ill posedness of kdv equation with higher dispersion abstract first the cauchy problem for kdv equation with 2 n 1 order dispersion is studied and the local well posedness result for the initial data in sobolev spaces h s r with s n 1 4 is established via the fourier restriction norm method second we prove that the kdv equation with 2 n 1 order dispersion is ill posed for the initial data in h s r with s n 1 4 n 2 n n if the flow map is c 2 differentiable at zero form h s r to c 0 t h s r finally we obtain the sharp regularity requirement for the kdv equation with 2 n 1 order dispersion s n 1 4

well posedness for stochastic camassa holm equation abstract the camassa holm equation describes the unidirectional propagation of waves at the free surface of shallow water under the influence of gravity due to uncertainty in the modelling and external environment this modelling could be subject to random fluctuations in this article the stochastic camassa holm equation with additive noise is considered using regularization a local existence and uniqueness result in the sobolev space h s r with s 3 2 of stochastic camassa holm equation is obtained with the help of priori estimates the local solution will blow up in h q r in finite time for any q 3 2 when initial value satisfies some conditions

a grid redistribution method for singular problems many physical phenomena develop singular or nearly singular behavior in localized regions e g boundary layers or blowup solutions using uniform grids for such problems becomes computationally prohibitive as the solution approaches singularity ren and wang developed a semi static adaptive grid method w ren x p wang an iterative grid redistribution method for singular problems in multiple dimensions j comput phys 159 2000 246 273 for the solution of these problems known as the iterative grid redistribution igr method in this study we develop a theoretical basis for semi static adaptive grid method for singular problems based on this theory we obtain the key result of this study a methodology for designing robust weight functionals which ensures grid resolution in the singular region as well as control of the maximal grid spacing in the outer region using this methodology we introduce a semi static adaptive grid method which does not involve an iterative procedure for grid redistribution as in the igr method we demonstrate the efficacy of this method with numerical examples of solutions which localize by more than nine orders of magnitude

an adaptive mesh refinement method for nonlinear dispersive wave equations abstract adaptive mesh refinement techniques are often essential for solving nonlinear partial differential equations numerically a new method for spatial grid refinement is developed and implemented several numerical experiments are performed to compare the method with results obtained using a uniform grid the new method has the following properties it is simple to implement and requires little modification of existing code to use the solutions achieved as a result of using these methods prove to be accurate and the stability of the numerical methods is affected minimally the effect of the grid refinement on essential properties of some of the equations such as conservation is minimized through the use of piecewise uniformity

a finite difference formula for the discretization of d 3 dx 3 on nonuniform grids we analyze the use of a five point difference formula for the dis cretization of the third derivative operator on nonuniform grids the formula was derived so as to coincide with the standard five point formula on regular grids and to lead to skew symmetric schemes it is shown that under periodic boundary conditions the formula is supraconvergent in the sense that in spite of being inconsistent it gives rise to schemes with second order convergence however such a supraconvergence only takes place when the number of points in the grid is odd for grids with an even number of points the inconsistency of the formula results in lack of convergence both stationary and evolutionary problems are considered and the analysis is backed by numerical experiments

the cauchy problem for a higher order shallow water type equation on the circle abstract in this paper we investigate the cauchy problem for a higher order shallow water type equation u t u t x x x 2 j 1 u x 2 j 3 u 3 u u x 2 u x u x x u u x x x 0 where x t r 2 and j n firstly we prove that the cauchy problem for the shallow water type equation is locally well posed in h s t with s j 2 2 for arbitrary initial data by using the i method we prove that the cauchy problem for the shallow water type equation is globally well posed in h s t with 2 j 1 j 2 2 j 1 s 1 our results improve the results of a a himonas g misiolek himonas and misiolek 1998 11 2000 12

global well posedness for a fifth order shallow water equation in sobolev spaces abstract the cauchy problem of a fifth order shallow water equation t u x 2 t u x 3 u 3 u x u 2 x u x 2 u u x 3 u x 5 u 0 is shown to be globally well posed in sobolev spaces h s r for s 6 10 17 4 the proof relies on the i method developed by colliander keel staffilani takaoka and tao for this equation lacks scaling invariance we reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data we prove the almost conservation law and combine it with the local result to obtain the global well posedness

asymptotic smoothing and the global attractor of a weakly damped kdv equation on the real line abstract the existence of the global attractor of a weakly damped forced korteweg de vries equation in the phase space l 2 r is proved an optimal asymptotic smoothing effect of the equation is also shown namely that for forces in l 2 r the global attractor in the phase space l 2 r is actually a compact set in h 3 r the energy equation method is used in conjunction with a suitable splitting of the solutions the dispersive regularization properties of the equation in the context of bourgain spaces are extensively exploited

well posedness for a higher order modified camassa holm equation abstract we show that the cauchy problem for a higher order modification of the camassa holm equation is locally well posed for initial data in the sobolev space h s r for s s where 1 4 s 1 2 and the value of s depends on the order of equation employing harmonic analysis methods we derive the corresponding bilinear estimate and then use a contraction mapping argument to prove existence and uniqueness of solutions

the cauchy problem of a periodic higher order kdv equation in analytic gevrey spaces abstract this paper studies the periodic cauchy problem for a kdv equation whose dispersion is of order m 2 j 1 where j is a positive integer kdvm using bourgain gevrey type analytic spaces and appropriate bilinear estimates it is shown that local in time well posedness holds when the initial data belong to an analytic gevrey spaces of order this implies that in the space variable the regularity of the solution remains the same with that of the initial data it also implies that the size of the uniform radius of analyticity is preserved moreover the solution is not necessarily g in time however it belongs to g m r near zero for every x on the circle

the moving finite element method applications to general partial differential equations with multiple large gradients abstract the moving finite element mfe method has been reduced to practice in the automatic solution program dyla for general systems of transient partial differential equations pdes in 1 d several test examples are presented which illustrate the unique node movement and systematic control features which are intrinsic in the mfe method computational dilemmas of numerical diffusion gibbs overshooting and undershooting zone tangling and grid remap or re connection aliasing which occur frequently in conventional pde methods are essentially eliminated in the mfe method arbitrarily large gradients or shocks can be solved with extremely high resolution and accuracy for non coincident or even counterdirected propagating wavefronts boundary layers of arbitrarily small dimensions are solved with high accuracy simultaneously with the large scale structures in reactive and non reactive fluid calculations the mfe method requires a small fraction of the grid nodes which are used in conventional pde solution methods because the nodes migrate continuously and systematically to those positions where they are most needed in order to yield accurate pde solutions on entire problem domains courant friedrichs lewy time step limits are exceeded by wide margins by factors of two to several thousand finally the extension of the mfe method to 2 d is briefly discussed

strichartz estimates for dispersive equations and solvability of the kawahara equation abstract we first establish a series of strichartz estimates for a general class of linear dispersive equations by applying the theory of oscillatory integrals established by kenig ponce and vega next we use such estimates to study solvability of the cauchy problem of the kawahara equation t u a u x u x 3 u x 5 u 0 in the class c r h s r local existence is proved for s 1 4 and global existence is proved for s 2

some observations on a static spatial remeshing method based on equidistribution principles this paper proposes a method of lines solution procedure with time and space adaptation for one dimensional systems of partial differential equations whose solutions display steep moving fronts the spatial remeshing algorithm which is a variation of the method published by sanz serna and christie and an extension suggested by revilla is a static remeshing method based on equidistribution principles the selection of the several algorithm components i e grid placement criterion spatial discretization scheme time integrator adaptation frequency and parameter tuning are investigated and illustrated with several test examples i e the cubic schrodinger equation a model of a single step reaction with diffusion and a model of flame propagation

an engineer s guide to soliton phenomena application of the finite element method abstract the paper attempts an elementary survey of the physical and mathematical background appertaining to solitons and discusses in particular the numerical solution of three types of dispersive nonlinear partial differential equations exhibiting soliton type solutions namely the korteweg de vries equation the nonlinear schrodinger equation and the sine gordon equation throughout this study a semidiscrete galerkin method is applied using a finite element discretization in space and a step by step time integration of the resulting system of nonlinear ordinary differential equations depending upon the special type of the evolutionary equation the application of a petrov galerkin procedure may increase significantly the numerical stability accuracy and effectivity of the different approaches are demonstrated on a series of computer plots

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

a simple adaptive technique for nonlinear wave problems abstract a method for the numerical integration of the nonlinear schrodinger equation is derived which uses variable time steps and a moving spatial grid the benefits of adaptation are clearly demonstrated in the numerical experiments reported the simple technique employed to move the nodes can be applied with little coding effort to general one dimensional systems of pdes

neugebauer kramer solutions of the ernst equation in hirota s direct method we prove analytically the sasa satsuma conjecture which states that their solution of bilinear form of the ernst equation gives the neugebauere kramer solution in particular cases this proof relates hirota s direct method with the backlund transformation method and opens the way towards the comprehensive interpretation of the ernst equation

global solutions of the klein gordon schrodinger system with rough data in r 2 1 abstract in this paper we consider the klein gordon schrodinger system with quadratic yakuwa coupling and cubic autointeractions in r 2 1 and prove the existence and uniqueness of global solution for rough data the techniques to be used are adapted from a general scheme originally introduced by j bourgain to split the data into the low and high frequency parts

orthogonal spline collocation methods for schr o dinger type equations in one space variable we examine the use of orthogonal spline collocation for the semi discreti za tion of the cubic schr o dinger equation and the two dimensional parabolic equation of tappert in each case an optimal order l 2 estimate of the error in the semidiscrete approximation is derived for the cubic schr o dinger equation we present the results of numerical experiments in which the integration in time is performed using a routine from a software library

the cauchy problem for higher order modified camassa holm equations on the circle abstract in this paper we investigate the cauchy problem for the shallow water type equation u t x 2 n 1 u 1 2 x u 2 x 1 x 2 1 u 2 1 2 u x 2 0 with low regularity data in the periodic settings firstly we proved that the bilinear estimate related to the nonlinear term of the equation in space w s defined in page 5 is invalid with s n 2 1 then the locally well posed of the cauchy problem for the periodic shallow water type equation is obtained in h s t with s n 3 2 n 2 for arbitrary initial data thus our result improves the result of himonas and misiolek commun partial differ equ 23 1998 123 139 where they have proved that the problem is locally well posed for small initial data in h s t with s n 2 1 n n with the aid of the standard fourier restriction norm method