temporal derivatives in the finite element method on continuously deforming grids when solving certain time dependent partial differential equations using a finite element technique on a deforming grid it is shown that there is a need to differentiate the trial solution with respect to the position of each of the moveable node points a result is presented which enables these nonstandard derivatives to be expressed in terms of the standard spatial derivatives of the trial functions provided that certain conditions are satisfied these conditions are investigated and shown to be satisfied for a large class of trial spaces even when the necessary conditions are not satisfied by all of the trial functions being used the result is shown to be still of great use in evaluating these awkward nodal derivatives 1 u x t y u x t a u

a solution of the generalised non linear schrodinger equation the author first presents a non linear schrodinger equation that describes wave propagation in fluids and plasmas with sharp boundaries and dissipation then the author shows that an exact solution can be found

study of the solutions to a model porous medium equation with variable exponent of nonlinearity abstract the authors of this paper study the dirichlet problem of the following equation u t div u x t u f u p x t 1 u the existence and uniqueness of weak solutions are proven also the properties of the solutions are studied which include the property of finite speed of propagation of disturbances localization property and the property of vanishing at a finite time etc

convergence of the finite element method for the porous media equation with variable exponent in this work we study the convergence of the finite element method when applied to the following parabolic equation u t div u gamma mathbf x nabla u f mathbf x t mathbf x in omega subset r m t in 0 t since the problem may be of degenerate type we utilize an approximate problem regularized by introducing a parameter varepsilon we prove under certain conditions on gamma and f that the weak solution of the approximate problem converges to the weak solution of the initial problem when the parameter varepsilon tends to zero discrete solutions are built using the finite element method and the convergence of these for the weak solution of the approximate problem is proved finally we present some numerical results of a matlab implementation of the method

moving mesh strategy based on a gradient flow equation for two dimensional problems in this paper we introduce a moving mesh method for solving pdes in two dimensions it can be viewed as a higher dimensional generalization of the moving mesh pde mmpde strategy developed in our previous work for one dimensional problems w huang y ren and r d russell siam j numer anal 31 1994 pp 709 730 the mmpde is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps geometrical interpretations are given for the gradient equation and functional and basic properties of this mmpde are discussed numerical examples are presented where the method is used both for mesh generation and for solving time dependent pdes the results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality

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

on a class of homogeneous nonlinear schrodinger equations a class of homogeneous norm conserving nonlinear wave equations of the schrodinger type is studied it is shown that those equations which derive from a lagrangian can be linearized but have no regular confined solutions whereas the equations which cannot be obtained from a local lagrangian do admit such confined solutions the latter however are unstable against small perturbations of the initial data

numerical study of the porous medium equation with absorption variable exponents of nonlinearity and free boundary abstract in this paper we study an application of the moving mesh method to the porous medium equation with absorption and variable exponents of nonlinearity in 2d domains with moving boundaries the boundary s movement is governed by an equation prompted by the darcy law and the spatial discretization is defined by a triangulation of the domain at each finite element the solution is approximated by piecewise polynomial functions of degree r 1 using lagrange interpolating polynomials in area coordinates the vertices of the triangles move according to a system of differential equations which is added to the equations of the problem the resulting system is converted into a system of ordinary differential equations in time variable which is solved using a suitable integrator the integrals that arise in the system of ordinary differential equations are calculated using the gaussian quadrature finally we present some numerical results of application of this technique

modulational instabilities and soliton solutions of a generalized nonlinear schrodinger equation previous forms of the generalized nonlinear schrodinger equation are ambiguously related to the corresponding nonlinear dispersion relation the authors therefore propose a hamiltonian equation which is determined in a unique way soliton solutions of the nonlinear schrodinger equations are found and modulational instabilities are investigated

design and application of a gradient weighted moving finite element code ii in two dimensions in part i the authors reported on the design of a robust and versatile gradient weighted moving finite element gwmfe code in one dimension and on its application to a variety of pdes and pde systems this companion paper does the same for the two dimensional 2d case these moving node methods are especially suited to problems which develop sharp moving fronts especially problems where one needs to resolve the fine scale structure of the fronts the many potential pitfalls in the design of gwmfe codes and the special features of the implicit one dimensional 1d and 2d codes which contribute to their robustness and efficiency are discussed at length in part i this paper concentrates on issues unique to the 2d case brief explanations are given of the variational interpretation of gwmfe the geometrical mechanical interpretation simplified regularization terms and the treatment of systems a catalog of inner products which occur in gwmfe is given with particular attention paid to those involving second order operators after presenting an example of the 2d phenomenon of grid collapse and discussing the need for long time regularization the paper reports on the application of the 2d code to several nontrivial problems nonlinear arsenic diffusion in the manufacture of semiconductors the drift diffusion equations for semiconductor device simulation the buckley leverett black oil equations for reservoir simulation and the motion of surfaces by mean curvature

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

variational mesh adaptation isotropy and equidistribution we present a new approach for developing more robust and error oriented mesh adaptation methods specifically assuming that a regular in cell shape and uniform in cell size computational mesh is used as is commonly done in computation we develop a criterion for mesh adaptation based on an error function whose definition is motivated by the analysis of function variation and local error behavior for linear interpolation the criterion is then decomposed into two aspects the isotropy or conformity and uniformity or equidistribution requirements each of which can be easier to deal with the functionals that satisfy these conditions approximately are constructed using discrete and continuous inequalities a new functional is finally formulated by combining the functionals corresponding to the isotropy and uniformity requirements the features of the functional are analyzed and demonstrated by numerical results in particular unlike the existing mesh adaptation functionals the new functional has clear geometric meanings of minimization a mesh that has the desired properties of isotropy and equidistribution can be obtained by properly choosing the values of two parameters the analysis presented in this article also provides a better understanding of the increasingly popular method of harmonic mapping in two dimensions

a model porous medium equation with variable exponent of nonlinearity existence uniqueness and localization properties of solutions abstract we study the localization properties of weak solutions to the dirichlet problem for the degenerate parabolic equation u t div u x t u f with variable exponent of nonlinearity we prove the existence and uniqueness of weak solutions and establish conditions on the problem data and the exponent x t sufficient for the existence of such properties as finite speed of propagation of disturbances the waiting time effect finite time vanishing of the solution it is shown that the solution may instinct in a finite time even if x 0 in the problem domain but max 0

algebraic bethe ansatze and eigenvalue based determinants for dicke jaynes cummings gaudin quantum integrable models in this work we construct an alternative formulation to the traditional algebraic bethe ans tze for quantum integrable models derived from a generalized rational gaudin algebra realized in terms of a collection of spins 1 2 coupled to a single bosonic mode the ensemble of resulting models which we call dicke jaynes cummings gaudin models are particularly relevant for the description of light matter interaction in the context of quantum optics having two distinct ways to write any eigenstate of these models we then combine them in order to write overlaps and form factors of local operators in terms of partition functions with domain wall boundary conditions we also demonstrate that they can all be written in terms of determinants of matrices whose entries only depend on the eigenvalues of the conserved charges since these eigenvalues obey a much simpler set of quadratic bethe equations the resulting expressions could then offer important simplifications for the numerical treatment of these models

properties of nonlinear schrodinger equations associated with diffeomorphism group representations the authors recently derived a family of nonlinear schrodinger equations on r3 from fundamental considerations of generalized symmetry ih cross delta t psi h cross 2 2m del 2 psi f psi psi psi ih cross d del 2 psi mod del psi mod 2 mod psi mod 2 psi where f is an arbitrary real functional and d a real continuous quantum number these equations descriptive of a quantum mechanical current that includes a diffusive term correspond to unitary representations of the group diff m parametrized by d where m r3 is the physical space in the present paper we explore the most natural ansatz for f which is labelled by five real coefficients we discuss the invariance properties describe the stationary states and some non stationary solutions and determine the extra dissipative terms that occur in the ehrenfest theorem we identify an interesting galilean invariant subfamily whose properties we investigate including the case where the dissipative terms vanish

interfaces in multidimensional diffusion equations with absorption terms we study the properties of interfaces in solutions of the cauchy problem for the nonlinear degenerate parabolic equation ut um aup in rn 0 t n 1 2 3 with the parameters m 1 p 0 a 0 satisfying the condition m p 2 we show that the velocity of the interface t supp u x t is given by the formula v m m 1 um 1 t where is the solution of the degenerate elliptic equation div u aup 0 0 on t the first term expresses the classical darcy law while the second one models the presence of an external force that makes the interface to move in the inward direction we give explicit formulas which represent the interface t as a bijection from 0 it is proved that the solution u and its interface t are analytic functions of time t and that they preserve the initial regularity in the spatial variables we also show that the regularity of the interface velocity v with respect to the spatial variables is better than it was at the initial instant the analysis is based on a special coordinate transformation a local version of lagrangian coordinates that renders the free boundary stationary

invariants in a resonant derivative nonlinear schrodinger model a broad class of derivative nonlinear schrodinger equations which incorporate a de broglie bohm type quantum potential is considered it is shown that a wave packet ansatz inserted into these equations leads to an integrable hamiltonian dynamical sub system

ermakov ray reid systems in nonlinear optics a hydrodynamics type system incorporating a madelung bohm type quantum potential as derived by wagner et al via maxwell s equations and the paraxial approximation in nonlinear optics is reduced to a nonlinear schrodinger canonical form a two parameter nonlinear ermakov ray reid system that arises from this model and which governs the evolution of beam radii in an elliptically polarised medium is shown to be reducible to a classical poschl teller equation a class of exact solutions to the ermakov type system is constructed in terms of elliptic dn functions it is established that integrable two component ermakov ray reid subsystems likewise arise in a coupled 2 1 dimensional nonlinear optics model descriptive of the two pulse interaction in a kerr medium the hamiltonian structure of these subsystems allows their complete integration

galilei invariant nonlinear systems of evolution equations all systems of n 1 dimensional quasilinear second order evolution equations invariant under chain of algebras ag 1 n contained in implied by ag1 1 n contained in implied by ag2 1 n are described the results obtained are illustrated by the examples of the nonlinear schrodinger equations hamilton jacobi type systems and reaction diffusion equations

a study on moving mesh finite element solution of the porous medium equation an adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption the method is based on the moving mesh partial differential equation approach and employs its newly developed implementation the implementation has several improvements over the traditional one including its explicit compact form of the mesh velocities ease to program and less likelihood of producing singular meshes three types of metric tensor that correspond to uniform and arclength based and hessian based adaptive meshes are considered the method shows first order convergence for uniform and arclength based adaptive meshes and second order convergence for hessian based adaptive meshes it is also shown that the method can be used for situations with complex free boundaries emerging and splitting of free boundaries and the porous medium equation with variable exponents and absorption two dimensional numerical results are presented