unilateral global bifurcation for a class of quasilinear elliptic systems and applications abstract in this paper we establish a unilateral bifurcation result for a class of quasilinear elliptic system strongly coupled extending the bifurcation theorem of 1 to this aim we use several results such that fredholm operator of index 0 theory eigenvalues of elliptic operators and the krein rutman theorem lastly we apply this result to some particular systems arising from population dynamics and determine a region of existence of coexistence states

multisymplectic geometry and multisymplectic preissman scheme for the kp equation the multisymplectic structure of the kp equation is obtained directly from the variational principal using the covariant de donder weyl hamilton function theories we reformulate the kp equation to the multisymplectic form which was proposed by bridges from the multisymplectic equation we can derive a multisymplectic numerical scheme of the kp equation which can be simplified to the multisymplectic 45 points scheme

backward error analysis for multi symplectic integration methods a useful method for understanding discretization error in the numerical solution of odes is to compare the system of odes with the modified equations obtained through backward error analysis and using symplectic integration for hamiltonian odes provides more incite into the modified equations in this paper the ideas of symplectic integration are extended to hamiltonian pdes and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations in particular local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution these results are also applied to the nonlinear wave equation

the midpoint scheme and variants for hamiltonian systems advantages and pitfalls the implicit midpoint scheme like higher order gauss collocation schemes is algebraically stable and symplectic and it preserves quadratic integral invariants it may appear particularly suitable for the numerical solution of highly oscillatory hamiltonian systems such as those arising in molecular dynamics or structural mechanics because there is no stability restriction when it is applied to a simple harmonic oscillator although it is well known that the midpoint scheme may also exhibit instabilities in various stiff situations one might still hope for good results when resonance type instabilities are avoided in this paper we investigate the suitability of the midpoint scheme for highly oscillatory frictionless mechanical systems where the step size k is much larger than the system s small parameter varepsilon in the case that the solution remains bounded as varepsilon rightarrow 0 we show that in general one must require that k 2 varepsilon be small enough or else even the errors in slowly varying quantities like the energy may grow undesirably especially when fast and slow modes are tightly coupled or worse the computation may yield misleading information in some cases this may already happen when k o varepsilon the same holds for higher order collocation at gaussian points the encountered restrictions on k are typically still better than the corresponding ones for explicit schemes

multiplicity results for some nonlinear schrodinger equations with potentials multiple semiclassical standing waves for a class of nonlinear schrodinger equations with potentials are found by means of a perturbative variational method

construction of multisymplectic schemes of any finite order for modified wave equations in this paper we analyze several symplectic discretizations of linear wave equations with periodic boundary conditions and show the discretizations are multisymplectic of arbitrary finite order for wave equations when the periodic boundary condition vanishes

symplectic energy momentum preserving variational integrators the purpose of this paper is to develop variational integrators for conservative mechanical systems that are symplectic and energy and momentum conserving to do this a space time view of variational integrators is employed and time step adaptation is used to impose the constraint of conservation of energy criteria for the solvability of the time steps and some numerical examples are given

explicit lie poisson integration and the euler equations we give a wide class of lie poisson systems for which explicit lie poisson integrators preserving all casimirs can be constructed the integrators are extremely simple and have widespread applicability examples are the free rigid body a moment truncation and a new fast algorithm for the sine bracket truncation of the 2d euler equations

difference schemes for solving the generalized nonlinear schrodinger equation this paper studies finite difference schemes for solving the generalized nonlinear schrodinger gnls equationiut uxx q u 2 u f x t u a new linearlized crank nicolson type scheme is presented by applying an extrapolation technique to the real coefficient of the nonlinear term in the gnls equation several schemes including crank nicolson type schemes hopscotch type schemes split step fourier scheme and pseudospectral scheme are adopted for solving three model problems of gnls equation which arise from many physical problems withq s s2 q s ln 1 s andq s 4s 1 s respectively the numerical results demonstrate that the linearized crank nicolson scheme is efficient and robust

geometric integrators for the nonlinear schrodinger equation abstract recently an interesting new class of pde integrators multisymplectic schemes has been introduced for solving systems possessing a certain multisymplectic structure some of the characteristic features of the method are its local nature independent of boundary conditions and an equal treatment of spatial and temporal variables the nonlinear schrodinger equation nls has a multisymplectic formulation and in this paper we discuss the performance of both symplectic and multisymplectic integrators for the nls in the numerical experiments the multisymplectic concatenated midpoint scheme a centered cell discretization is shown to preserve the local conservation laws extremely well over long times and to preserve global invariants such as the norm and momentum within roundoff on the other hand an integrable hamiltonian semi discretization of nls from ablowitz and ladik al possesses a full set of global conservation laws and a noncanonical symplectic structure we generalize the generating function technique to develop symplectic integrators of arbitrary order for a general class of noncanonical systems carrying a symplectic structure of the al type another approach examined in the paper is the introduction of transformations to reduce the al system to either 1 separable form or 2 canonical form and then apply standard schemes in the new coordinates all of the discretizations are tested numerically using initial data for spatially periodic multiphase solutions the performance of the schemes as well as interrelations among various geometric features are discussed

the semiclassical limit of the defocusing nls hierarchy we establish the semiclassical limit of the one dimensional defocusing cubic nonlinear schrodinger nls equation complete integrability is exploited to obtain a global characterization of the weak limits of the entire nls hierarchy of conserved densities as the field evolves from reflectionless initial data under all the associated commuting flows consequently this also establishes the zero dispersion limit of the modified korteweg de vries equation that resides in that hierarchy we have adapted and clarified the strategy introduced by lax and levermore to study the zero dispersion limit of the korteweg de vries equation expanding it to treat entire integrable hierarchies and strengthening the limits obtained a crucial role is played by the convexity of the underlying log determinant with respect to the times associated with the commuting flows c 1999 john wiley sons inc

effects of a degeneracy in the competition model part i classical and generalized steady state solutions abstract we study the competition model where the coefficient functions are strictly positive over the underlying spatial region except b x which vanishes in a nontrivial subdomain of and is positive in the rest of we show that there exists a critical number such that if then the model behaves similarly to the well studied classical competition model where all the coefficient functions are positive constants but when new phenomena occur our results demonstrate the fact that heterogeneous environmental effects on population models are not only quantitative but can be qualitative as well in part i here we mainly study two kinds of steady state solutions which determine the dynamics of the model one consists of finite functions while the other consists of generalized functions which satisfy u v 0 on the part of the domain that b x vanishes but are positive and finite on the rest of the domain and are determined by certain boundary blow up systems the research is continued in part ii where these two kinds of steady state solutions will be used to determine the dynamics of the model

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

symplectic and multi symplectic methods for the nonlinear schrodinger equation the hamiltonian and the multi symplectic formulations of the nonlinear schrodinger equation are considered for the multi symplectic formulation a new six point difference scheme which is equivalent to the multi symplectic preissman integrator is derived numerical experiments are also reported c 2002 elsevier science ltd all rights reserved

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

stability and instability of solitary waves of the fifth order kdv equation a numerical framework abstract the spectral problem associated with the linearization about solitary waves of the generalized fifth order kdv equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to eigenvalues a numerical framework based on a fast robust shooting algorithm on exterior algebra spaces is introduced the complete algorithm has several new features including a rigorous numerical algorithm for choosing starting values a new method for numerical analytic continuation of starting vectors the role of the grassmannian g 2 c 5 in choosing the numerical integrator and the role of the hodge star operator for relating 2 c 5 and 3 c 5 and deducing a range of numerically computable forms for the evans function the algorithm is illustrated by computing the stability and instability of solitary waves of the fifth order kdv equation with polynomial nonlinearity

verlet i r respa impulse is limited by nonlinear instabilities this paper shows that in molecular dynamics md when constant energy nve simulations of newton s equations of motion are attempted using the multiple time stepping mts integrator verlet i r respa impulse there are nonlinear instabilities when the longest step size is a third or possibly a fourth of the period s of the fastest motion s in the system this is demonstrated both through a thorough set of computer experiments and through the analysis of a nonlinear model problem the numerical experiments include not only the unconstrained dynamics simulation of a droplet of flexible water and a flexible protein but also the constrained dynamics simulation of a solvated protein representing a range of simulation protocols commonly in use by biomolecular modelers the observed and predicted instabilities match exactly previous work has identified and explained a linear instability for verlet i r respa impulse at around half the period of the fastest motion mandziuk and schlick discovered nonlinear resonances in single time stepping md integrators but unstable nonlinear resonances for mts integrators are reported here for the first time this paper also offers an explanation on the instability of mts constrained molecular dynamics simulations of explicitly solvated proteins more aggressive multiple step sizes are possible with mild langevin coupling or targeted langevin coupling and its combination with the mollified impulse method permits step sizes 3 to 4 times larger thanverlet i r respa impulse while still retaining some accuracy

elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation abstract we derive a result on the limit of certain sequences of principal eigenvalues associated with some elliptic eigenvalue problems this result is then used to give a complete description of the global structure of the curves of positive steady states of a parameter dependent diffusive version of the classical logistic equation in particular we characterize the bifurcation values from infinity to positive steady states the stability of the positive steady states as well as the asymptotic behaviour of positive solutions is also discussed

on the symbiotic lotka volterra model with diffusion and transport effects abstract in this work we analyze the existence stability and multiplicity of coexistence states for a symbiotic lotka volterra model with general diffusivities and transport effects global bifurcation theory blowing up arguments for a priori bounds singular perturbation results singularity theory and fixed point index in cones are among the techniques used to get our results and to explain the drastic change of behavior exhibited by the dynamics of the model between the cases of weak and strong mutualism between the species our methodology works out to treat much more general classes of symbiotic models

optimal uniqueness theorems and exact blow up rates of large solutions in this paper we prove some optimal uniqueness results for large solutions of a canonical class of semilinear equations under minimal regularity conditions on the weight function in front of the non linearity and combine these results with the localization method introduced in lopez gomez the boundary blow up rate of large solutions j differential equations 195 2003 25 45 to prove that any large solution l of u a x up p 1 a 0 must satisfy limx x0l x fx0 dist x i0 pp 1p 1p 1p 1p 1foreachx0 where fx0 t t 0sfx01p 1 p 1p 1ds i0 limt 0fx0 t fx0 t fx0 t 2 and fx0 is any smooth extension of the boundary normal section of a at x0 i e fx0 t a x0 tnx0 t 0 t 0 subsequently nx0 stands for the outward unit normal at x0 therefore the theory can be extended to cover the general case when fx0 l1p 1

the life span of backward error analysis for numerical integrators backward error analysis is a useful tool for the study of numerical approximations to ordinary differential equations the numerical solution is formally interpreted as the exact solution of a perturbed differential equation given as a formal and usually divergent series in powers of the step size for a rigorous analysis this series has to be truncated in this article we study the influence of this truncation to the difference between the numerical solution and the exact solution of the perturbed differential equation results on the long time behaviour of numerical solutions are obtained in this way we present applications to the numerical phase portrait near hyperbolic equilibrium points to asymptotically stable periodic orbits and hopf bifurcation and to energy conservation and approximation of invariant tori in hamiltonian systems

numerical study of time splitting spectral discretizations of nonlinear schrodinger equations in the semiclassical regimes in this paper we study the performance of time splitting spectral approximations for general nonlinear schrodinger equations nls in the semiclassical regimes where the planck constant varepsilon is small the time splitting spectral approximation under study is explicit unconditionally stable and conserves the position density in l 1 moreover it is time transverse invariant and time reversible when the corresponding nls is extensive numerical tests are presented for weak strong focusing defocusing nonlinearities for the gross pitaevskii equation and for current relaxed quantum hydrodynamics the tests are geared towards the understanding of admissible meshing strategies for obtaining correct physical observables in the semiclassical regimes furthermore comparisons between the solutions of the nls and its hydrodynamic semiclassical limit are presented

approximate momentum conservation for spatial semidiscretizations of semilinear wave equations we prove that a standard second order finite difference uniform space discretization of the semilinear wave equation with periodic boundary conditions analytic nonlinearity and analytic initial data conserves momentum up to an error which is exponentially small in the stepsize our estimates are valid for as long as the trajectories of the full semilinear wave equation remain real analytic the method of proof is that of backward error analysis whereby we construct a modified equation which is itself lagrangian and translation invariant and therefore also conserves momentum this modified equation interpolates the semidiscrete system for all time and we prove that it remains exponentially close to the trigonometric interpolation of the semidiscrete system these properties directly imply approximate momentum conservation for the semidiscrete system we also consider discretizations that are not variational as well as discretizations on non uniform grids through numerical example as well as arguments from geometric mechanics and perturbation theory we show that such methods generically do not approximately preserve momentum

multisymplectic geometry and multisymplectic preissmann scheme for the kdv equation the multisymplectic structure of the kdv equation is presented directly from the variational principle from the numerical view point we give a multisymplectic twelve points scheme which is equivalent to the multisymplectic preissmann scheme finally we test the twelve points scheme on solitary waves over long time intervals

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

multisymplectic conservation laws for differential and differential difference equations many well known partial differential equations can be written as multisymplectic systems such systems have a structural conservation law from which scalar conservation laws can be derived these c

multi symplectic runge kutta methods for nonlinear dirac equations in this paper we consider the multi symplectic runge kutta msrk methods applied to the nonlinear dirac equation in relativistic quantum physics based on a discovery of the multi symplecticity of the equation in particular the conservation of energy momentum and charge under msrk discretizations is investigated by means of numerical experiments and numerical comparisons with non msrk methods numerical experiments presented reveal that msrk methods applied to the nonlinear dirac equation preserve exactly conservation laws of charge and momentum and conserve the energy conservation in the corresponding numerical accuracy to the method utilized it is verified numerically that msrk methods are stable and convergent with respect to the conservation laws of energy momentum and charge and msrk methods preserve not only the inner geometric structure of the equation but also some crucial conservative properties in quantum physics a remarkable advantage of msrk methods applied to the nonlinear dirac equation is the precise preservation of charge conservation law

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

derivation of the discrete conservation laws for a family of finite difference schemes abstract we present conservative schemes in finite differences applied to nonlinear odes and nonlinear klein gordon schrodinger and dirac field equations a systematic approach is used and an explicit derivation of the corresponding conservation laws is given

nondegeneracy and uniqueness for boundary blow up elliptic problems abstract in this paper we use for the first time linearization techniques to deal with boundary blow up elliptic problems after introducing a convenient functional setting we show that the problem u a x u p g x u in with u on has a unique positive solution for large enough and determine its asymptotic behavior as here p 1 a x is a continuous function which can be singular near and g x u is a perturbation term with potential growth near zero and infinity we also consider more general problems obtained by replacing u p by e u or a logistic type function f u

total variation in hamiltonian formalism and symplectic energy integrators we present a discrete total variation calculus in hamiltonian formalism in this paper using this discrete variation calculus and generating functions for the flows of hamiltonian systems we derive symplectic energy integrators of any finite order for hamiltonian systems from a variational perspective the relationship between the symplectic integrators derived directly from the hamiltonian systems and the variationally derived symplectic energy integrators is explored

the effect of a singular perturbation on nonconvex variational problems we study the effect of a singular perturbation on certain nonconvex variational problems the goal is to characterize the limit of minimizers as some perturbation parameter 0 the technique utilizes the notion of convergence of variational problems developed by de giorgi the essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem in so doing one characterizes the limit of minimizers as the solution of a new variational problem for the cases considered here these new problems have a simple geometric nature involving minimal surfaces and geodesics

on time splitting spectral approximations for the schrodinger equation in the semiclassical regime in this paper we study time splitting spectral approximations for the linear schrodinger equation in the semiclassical regime where the planck constant e is small in this regime the equation propagates oscillations with a wavelength of o e and finite difference approximations require the spatial mesh size h o e and the time step k o e in order to obtain physically correct observables much sharper mesh size constraints are necessary for a uniform l2 approximation of the wave function the spectral time splitting approximation under study will be proved to be unconditionally stable time reversible and gauge invariant it conserves the position density and gives uniform l2 approximation of the wave function for k o e and h o e extensive numerical examples in both one and two space dimensions and analytical considerations based on the wigner transform even show that weaker constraints e g k independent of e and h o e are admissible for obtaining correct observables finally we address the application to nonlinear schrodinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

haar wavelet approximation for the solution of cubic nonlinear schrodinger equations abstract in this study haar wavelet collocation method is used for the numerical solution of 1d and 2d cubic nonlinear schrodinger equations with initial and dirichlet boundary conditions the space derivatives are estimated through haar wavelet collocation method whereas for time derivative we have used crank nicolson scheme the proposed method is implemented upon several test problems and the numerical results of these test problems establish that the proposed method is accurate

natural discretizations for the divergence gradient and curl on logically rectangular grids abstract this is the first in series of papers creating a discrete analog of vector analysis on logically rectangular nonorthogonal nonsmooth grids we introduce notations for 2 d logically rectangular grids describe both cell valued and nodal discretizations for scalar functions and construct the natural discretizations of vector fields using the vector components normal and tangential to the cell boundaries we then define natural discrete analogs of the divergence gradient and curl operators based on coordinate invariant definitions and interpret these formulas in terms of curvilinear coordinates such as length of elements of coordinate lines areas of elements of coordinate surfaces and elementary volumes we introduce the discrete volume integral of scalar functions the discrete surface integral and a discrete analog of the line integral and prove discrete versions of the main theorems relating these objects these theorems include the following the discrete analog of relationship div a 0 if and only if a curl b curl a 0 if and only if a grad if a grad then the line integral does not depend on path and if the line integral of a vector function is equal to zero for any closed path then this vector is the gradient of a scalar function last we define the discrete operators div grad and curl in terms of primitive differencing operators based on forward and backward differences and primitive metric operators related to multiplications of discrete functions by length of edges areas of surfaces and volumes of 3 d cells these formulations elucidate the structure of the discrete operators and are useful when investigating the relationships between operators and their adjoints

high order multi symplectic schemes for the nonlinear klein gordon equation in this paper we construct multi symplectic schemes with any order of accuracy for the nonlinear klein gordon equation by concatenating the symplectic schemes for odes some existing schemes such as the preissman scheme and the leap frog scheme and new multi symplectic schemes are constructed we also show that the composition method which plays a crucial role in finding the high order symplectic integrators for the odes can also be applied to construct high order multi symplectic schemes for pdes extension of the concatenating method to more than one space dimension is also discussed numerical experiments are presented to show the order and the efficiency of the constructed multi symplectic schemes

finite mode analogs of 2d ideal hydrodynamics coadjoint orbits and local canonical structure abstract the algebraic finite mode hydrodynamic type systems that have o n integrals of motion for o n n modes and are intrinsically connected with two dimensional ideal fluid flows are studied special emphasis is given on the structure of their phase space and the problem of finding proper dynamical variables for its description in connection with appropriate problems for the original hydrodynamic equations various possibilities for embedding in the space of canonical variables are demonstrated on the simplest example of the su 3 model giving the finite dimensional analogs of clebsch variables

the symplectic evans matrix and the instability of solitary waves and fronts hamiltonian evolution equations which are equivariant with respect to the action of a lie group are models for physical phenomena such as oceanographic flows optical fibres and atmospheric flows and such systems often have a wide variety of solitary wave or front solutions in this paper we present a new symplectic framework for analyzing the spectral problem associated with the linearization about such solitary waves and fronts at the heart of the analysis is a multi symplectic formulation of hamiltonian partial differential equations where a distinct symplectic structure is assigned for the time and space directions with a third symplectic structure with two form denoted by omega associated with a coordinate frame moving at the speed of the wave this leads to a geometric decomposition and symplectification of the evans function formulation for the linearization about solitary waves and fronts we introduce the concept of the symplectic evans matrix a matrix consisting of restricted omega symplectic forms by applying hodge duality to the exterior algebra formulation of the evans function we find that the zeros of the evans function correspond to zeros of the determinant of the symplectic evans matrix based on this formulation we prove several new properties of the evans function restricting the spectral parameter lambda to the real axis we obtain rigorous results on the derivatives of the evans function near the origin based solely on the abstract geometry of the equations and results for the large lambda behaviour which use primarily the symplectic structure but also extend to the non symplectic case the lie group symmetry affects the evans function by generating zero eigenvalues of large multiplicity in the so called systems at infinity we present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation by combining all these results a new rigorous sufficient condition for instability of solitary waves and fronts is obtained the theory applies to a large class of solitary waves and fronts including waves which are biasymptotic to a nonconstant manifold of states as x tends to infinity to illustrate the theory it is applied to three examples a boussinesq model from oceanography a class of nonlinear schrodinger equations from optics and a nonlinear klein gordon equation from atmospheric dynamics

stability of finite difference models containing two boundaries or interfaces it is known that the stability of finite difference models of hyperbolic initial boundary value problems is connected with the propagation and reflection of parasitic waves here the waves point of view is applied to models containing two boundaries or interfaces where repeated reflection of trapped packets is a potential new source of instability our analysis accounts for various known instability phenomena in a unified way and leads to several new results three of which are as follows 1 dissipativity does not ensure stability when three or more formulas are concatenated at a boundary or internal interface 2 algebraic gks instabilities can be converted by a second boundary to exponential instabilities only when an infinite numerical reflection coefficient is present 3 gks stability and p stability can be established in certain problems by showing that the numerical reflection coefficient matrices have norm less than one

on the energy functionals derived from a non homogeneous p laplacian equation γ convergence local minimizers and stable transition layers abstract in this paper we consider a family of singularly perturbed non homogeneous p laplacian problems p div k x u p 2 u k x g u 0 in r n subject to neumann boundary conditions we establish the convergence of the energy functionals associate to this family of problems as an application we obtain the existence and profile asymptotic of a family of local minimizers in the one dimensional case i e 0 1 in particular these minimizers are stable solutions which develop inner transition layer in 0 1

geometric integrators for classical spin systems practical structure preserving methods for integrating classical heisenberg spin systems are discussed two new integrators are derived and compared including 1 a symmetric energy and spin length preserving integrator based on a red black splitting of the spin sites combined with a staggered timestepping scheme and 2 a lie poisson symplectic integrator based on hamiltonian splitting the methods are applied to both 1d and 2d lattice models and are compared with the commonly used explicit runge kutta projected runge kutta and implicit midpoint schemes on the bases of accuracy conservation of invariants and computational expense it is shown that while any of the symmetry preserving schemes improves the integration of the dynamics of solitons or vortex pairs compared to runge kutta or projected runge kutta methods the staggered red black scheme is far more efficient than the other alternatives

multi symplectic spectral discretizations for the zakharov kuznetsov and shallow water equations abstract the time evolution of multi symplectic pdes with periodic boundary conditions in one and two space dimensions is considered we introduce the idea of a multi symplectic fourier transform for systems on periodic domains leading to a semi discretization on fourier space and to the concept of multi symplecticity on fourier space the spatial discretization leads to a discrete lattice model with certain discrete conservation laws imposed on it a discrete wave model a by product of the semi discretization is that it leads automatically to a system of hamiltonian odes in time when truncated we show that the one dimensional shallow water equations and the two dimensional zakharov kuznetsov equation are multi symplectic and derive spectral discretizations for these systems and present numerical experiments

numerical implementation of the multisymplectic preissman scheme and its equivalent schemes we analyze the multisymplectic preissman scheme for the kdv equation with the periodic boundary condition and show that the unconvergence of the widely used iterative methods to solve the resulting nonlinear algebra system of the preissman scheme is due to the introduced potential function a artificial numerical condition is added to the periodic boundary condition the added boundary condition makes the numerical implementation of the multisymplectic preissman scheme practical and is proved not to change the numerical solutions of the kdv equation based on our analysis we derive some new schemes which are not restricted by the artificial boundary condition and more efficient than the preissman scheme because of less computing cost and less computer storages by eliminating the auxiliary variables we also derive two schemes for the kdv equation one is a 12 point scheme and the other is an 8 point scheme as the byproducts we present two new explicit schemes which are not multisymplectic but still have remarkable numerical stable property numerical experiments on soliton collisions are also provided to confirm our conclusion and to show the benefits of the multisymplectic schemes with comparison of the spectral method and zabusky kruskal scheme

nonlinear stability analysis of area preserving integrators linear stability analysis is inadequate for integrators designed for nondissipative systems such as hamiltonian systems in which nonlinear effects are often decisive mathematical theory exists kam theory for rigorous analysis of small perturbations from equilibria but it needs to be expressed in a form that is more easily applicable to the study of area preserving maps we have pursued this obtaining a completely rigorous nonlinear stability analysis for elliptic equilibria based on the moser twist theorem and a result of cabral and meyer nonlinearity 12 1999 pp 1351 1362 together with the theory of normal forms for hamiltonian systems the result is a determination of necessary and sufficient conditions for stability these conditions are sharpened for the case of reversible maps and applied to the symplectic members of the newmark family of integrators which includes the leapfrog the implicit midpoint and the stormer cowell methods nonlinear stability limits are more severe than those of linear theory as an example the leapfrog scheme actually has a step size limitation of 71 of that predicted by linear analysis

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

the maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems abstract in this work we deal with the problem of the existence and uniqueness of principal eigenvalues for some linear weighted boundary value problems associated to a general second order uniformly elliptic operator for a large class of sign definited weights we characterize whether the boundary value problem admits a principal eigenvalue or not

the multi symplecticity of partitioned runge kutta methods for hamiltonian pdes in this article we consider partitioned runge kutta prk methods for hamiltonian partial differential equations pdes and present some sufficient conditions for multi symplecticity of prk methods of hamiltonian pdes

an angiogenesis model with nonlinear chemotactic response and flux at the tumor boundary abstract in this paper we consider a parabolic problem as well as its stationary counterpart of a model arising in angiogenesis the problem includes a chemotaxis type term and a nonlinear boundary condition at the tumor boundary we show that the parabolic problem admits a unique positive global in time solution moreover by bifurcation methods we show the existence of coexistence states and also we study the local stability of the semi trivial states

symplectic methods for the ablowitz ladik discrete nonlinear schrodinger equation using several kinds of coordinate transformations we standardize the noncanonical symplectic structure of the ablowitz ladik model a l model of nonlinear schrodinger equation nlse then we employ some symplectic scheme to simulate the solitons motion and test the evolution of the discrete invariants of the a l model and also the conserved quantities of the original nlse in comparison with a higher order non symplectic scheme applied directly to the a l model we show the overwhelming superiorities of the symplectic method we also compare the implementation of the same symplectic scheme to different standardized hamiltonian systems resulting from different coordinate transformations and show that the symmetric coordinate transformation improves the numerical results obtained via the asymmetric one in preserving the invariants of the a l model and the original nlse

on multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in rn abstract in this work we study the existence multiplicity and concentration of positive solutions for the following class of quasilinear problems n n u v x u n 2 u f u in r n u x 0 x r n where n u is the n laplacian operator is a positive parameter n 2 v r n r is a continuous function and f r r is a c 1 function having critical exponential growth

finite difference calculus invariant structure of a class of algorithms for the nonlinear klein gordon equation in a previous work the authors have presented a formalism for deriving systematically invariant symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities this formalism is herein cast in the context of exact finite difference calculus the algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus in the same sense as that of the corresponding differential equation being derivable from its associated energy function a conserved quantity a clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities which are the consistent counterpart of the invariant quantities in the continuous case results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism and that preserve energy or linear momentum are discussed in the context of finite difference calculus some

multi layered stationary solutions for a spatially inhomogeneous allen cahn equation abstract we consider stationary solutions of a spatially inhomogeneous allen cahn type nonlinear diffusion equation in one space dimension the equation involves a small parameter e and its nonlinearity has the form h x 2 f u where h x represents the spatial inhomogeneity and f u is derived from a double well potential with equal well depth when e is very small stationary solutions develop transition layers we first show that those transition layers can appear only near the local minimum and local maximum points of the coefficient h x and that at most a single layer can appear near each local minimum point of h x we then discuss the stability of layered stationary solutions and prove that the morse index of a solution coincides with the total number of its layers that appear near the local maximum points of h x we also show the existence of stationary solutions having clustering layers at the local maximum points of h x

symplectic integration of hamiltonian wave equations the numerical integration of a wide class of hamiltonian partial differential equations by standard symplectic schemes is discussed with a consistent hamiltonian approach we discretize the hamiltonian and the poisson structure separately then form the the resulting ode s the stability accuracy and dispersion of different explicit splitting methods are analyzed and we give the circumstances under which the best results can be obtained in particular when the hamiltonian can be split into linear and nonlinear terms many different treatments and examples are compared

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

study of an elliptic system arising from angiogenesis with chemotaxis and flux at the boundary the main goal of this paper is to study a stationary problem arising from angiogenesis including terms of chemotaxis and flux at the boundary of the tumor we give sufficient conditions on terms of the data of the problems assuring the existence of positive solutions

dispersion relation equation preserving fdtd method for nonlinear cubic schrodinger equation in this study we aim to solve the cubic nonlinear schrodinger cnls equation by the method of fractional steps over a time step from t n to t n 1 the linear part of the schrodinger equation is solved firstly through four time integration steps in this part of the simulation the explicit symplectic scheme of fourth order accuracy is adopted to approximate the time derivative term the second order spatial derivative term in the linear schrodinger equation is approximated by centered scheme the resulting symplectic and space centered difference scheme renders an optimized numerical dispersion relation equation in the second part of the simulation the solution of the nonlinear equation is computed exactly thanks to the embedded invariant nature within each time increment the proposed semi discretized difference scheme underlying the modified equation analysis of second kind and the method of dispersion error minimization has been assessed in terms of the spatial modified wavenumber or the temporal angular frequency resolution several problems have been solved to show that application of this new finite difference scheme for the calculation of one and two dimensional schrodinger equations can deemed conserve hamiltonian quantities and preserve dispersion relation equation dre

properties of the principal eigenvalues of a general class of non classical mixed boundary value problems in this paper we characterize the existence of principal eigenvalues for a general class of linear weighted second order elliptic boundary value problems subject to a very general class of mixed boundary conditions our theory is a substantial extension of the classical theory by p hess and t kato 1980 comm partial differential equations5 999 1030 in obtaining our main results we must give a number of new results on the continuous dependence of the principal eigenvalue of a second order linear elliptic boundary value problem with respect to the underlying domain and the boundary condition itself these auxiliary results complement and in some sense complete the theory of d daners and e n dancer 1997 j differential equations138 86 132 the main technical tool used throughout this paper is a very recent characterization of the strong maximum principle in terms of the existence of a positive strict supersolution due to h amann and j lopez gomez 1998 j differential equations146 336 374

on the stability of symplectic and energy momentum algorithms for non linear hamiltonian systems with symmetry abstract this paper presents a detailed comparison of two implicit time integration schemes for a simple non linear hamiltonian system with symmetry the motion of a particle in a central force field the goal is to establish analytical and numerical results pertaining to the stability properties of the implicit mid point rule the proto typical implicit symplectic method and a particular energy momentum conserving scheme and to compare the two schemes with respect to accuracy while all results presented herein are within the context of a simple model problem the problem was constructed so as to exhibit key features typical of more complex systems with symmetry such as those arising in non linear solid mechanics namely the presence of large and relatively slow overall motions together with high frequency internal motions