the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

one symmetry does not imply integrability we show that bakirov s counter example which had been checked by computer algebra methods up to order 53 to the conjecture that one nontrivial symmetry of an evolution equation implies infinitely many is indeed a counter example to prove this we use thesymbolic methodof gel fand dikii andp adic analysis we also formulate a conjecture to the effect that almost all equations in the family considered by bakirov have at most finitely many symmetries this conjecture depends on the solution of a diophantine problem which we explicitly state

painleve classification of coupled korteweg de vries systems in this work we give a classification of coupled korteweg de vries equations we found new systems of equations that are completely integrable in the sense of painleve

optimal error estimate for energy preserving splitting schemes for maxwell s equations abstract two efficient splitting schemes are developed for 3d maxwell s equations the schemes are energy preserving and unconditionally stable while being implemented explicitly rigorous optimal error estimates are established for the proposed schemes and especially the constant in the error estimates is only o t numerical results confirm the theoretical analysis and numerical comparison with some existing methods shows the good performance of the present schemes

soliton fission and fusion of a new two component korteweg de vries kdv equation abstract in this paper the painleve test is performed for a new two component korteweg de vries kdv equation proposed by foursov it is shown that this equation passes the integrability test and is p integrable by means of the truncated singular expansion some explicit solutions from the trivial zero solution are derived the phenomena of soliton fission and fusion are studied in detail

classification of certain integrable coupled potential kdv and modified kdv type equations in this article we describe the classification of integrable symmetrically coupled potential kdv and modified kdv type equations that possess higher symmetries restricting our attention to the systems that cannot be decoupled by a change of dependent variables we obtain 11 previously unknown classes of integrable equations in some cases we present hamiltonian or bi hamiltonian formulations

symplectic and multisymplectic numerical methods for maxwell s equations in this paper we compare the behaviour of one symplectic and three multisymplectic methods for maxwell s equations in a simple medium this is a system of pdes with symplectic and multisymplectic structures we give a theoretical discussion of how some numerical methods preserve the discrete versions of the local and global conservation laws and verify this behaviour in numerical experiments we also show that these numerical methods preserve the divergence furthermore we extend the discussion on dispersion for multi symplectic methods applied to pdes with one spatial dimension to include anisotropy when applying multi symplectic methods to maxwell s equations in two spatial dimensions lastly we demonstrate how varying the courant friedrichs lewy cfl number can cause the multi symplectic methods in our comparison to behave differently which can be explained by the study of backward error analysis for pdes

on the integrability of homogeneous scalar evolution equations abstract we determine the existence of infinitely many symmetries for equations of the formut uk f u uk 1 when they are homogeneous with respect to the scalinguk k with 0 algorithms are given to determine whether a system has a symmetry also independent oftandx if it has one generalized symmetry we prove it has infinitely many and these can be found using recursion operators or master symmetries the method of proof uses the symbolic method and results from diophantine approximation theory we list the 10 integrable hierarchies the methods can in principle be applied to the 0 case as we illustrate for one example with 0 which seems to be new in principle they can also be used for systems of evolution equations but so far this has only been demonstrated for one class of examples

energy conserved splitting finite difference time domain methods for maxwell s equations in three dimensions in this paper we develop and analyze efficient energy conserved splitting finite difference time domain fdtd schemes for solving three dimensional maxwell s equations in electromagnetic computations all proposed energy conserved splitting finite difference time domain ec s fdtd algorithms are strictly proved to be energy conserved and unconditionally stable and they can be computed efficiently rigorous convergence results are obtained for the schemes the ec s fdtdii schemes are proved to have second order in both time step and spatial steps while the ec s fdtdi schemes have second order in spatial steps and first order in time step the error estimates are optimal and especially the constant in the error estimates is proved to be only o t numerical experiments confirm the theoretical analysis results

new multisymplectic self adjoint scheme and its composition scheme for the time domain maxwell s equations in this paper we investigate euler box scheme for bridges multisymplectic form of maxwell s equations a new multisymplectic scheme is derived for maxwell s equations we prove that it is also a self adjoint scheme in time direction the multisymplecticity of composition schemes based on the new scheme is also discussed two numerical examples are proposed to indicate that the derived multisymplectic schemes are effective when used to integrate the 2 1 dimensional maxwell s equations

the spatial fourth order energy conserved s fdtd scheme for maxwell s equations abstract in this paper we develop a new spatial fourth order energy conserved splitting finite difference time domain method for maxwell s equations based on the staggered grids the splitting technique is applied to lead to a three stage energy conserved splitting scheme at each stage using the spatial fourth order difference operators on the strict interior nodes by a linear combination of two central differences one with a spatial step and the other with three spatial steps we first propose the spatial high order near boundary differences on the near boundary nodes which ensure the scheme to preserve energy conservations and to have fourth order accuracy in space step the proposed scheme has the important properties energy conserved unconditionally stable non dissipative high order accurate and computationally efficient we first prove that the scheme satisfies energy conversations and is in unconditional stability we then prove the optimal error estimates of fourth order in spatial step and second order in time step for the electric and magnetic fields and obtain the convergence and error estimate of divergence free as well numerical dispersion analysis and numerical experiments are presented to confirm our theoretical results

integrable coupled kdv systems we give the conditions for a system of n coupled korteweg de vries kdv type of equations to be integrable we find the recursion operators of each subclass and give all examples for n 2

the hamiltonian structure of the maxwell vlasov equations morrison 25 has observed that the maxwell vlasov and poisson vlasov equations for a collisionless plasma can be written in hamiltonian form relative to a certain poisson bracket we derive another poisson structure for these equations by using general methods of symplectic geometry the main ingredients in our construction are the symplectic structure on the co adjoint orbits for the group of canonical transformations and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory our poisson bracket satisfies the jacobi identity whereas morrison s does not 37 our construction also shows where canonical variables can be found and can be applied to the yang mills vlasov equations and to electromagnetic fluid dynamics

a strange recursion operator demystified we show that a new integrable two component system of kdv type studied by karasu kalkanli et al 2004 acta appl math 83 85 94 is bi hamiltonian and its recursion operator which has a highly unusual structure of nonlocal terms can be written as a ratio of two compatible hamiltonian operators found by us using this we prove that the system in question possesses an infinite hierarchy of local commuting generalized symmetries and conserved quantities in involution and the evolution systems corresponding to these symmetries are bi hamiltonian as well we also show that upon introduction of suitable nonlocal variables the nonlocal terms of the recursion operator under study can be written in the usual form with the integration operator d 1x appearing in each term at most once

towards the complete classification of homogeneous two component integrable equations in this article we suggest an improved method for classifying general two component integrable evolution equations homogeneous in a given weighting scheme this method relies on linear changes of variables and on an appropriate splitting of the solution space to illustrate the method we implement the classification of coupled kdv type equations we show that there are five nontriangular systems possessing higher order generalized symmetries one of these systems is previously unknown this seems to be the first classification of coupled integrable equations homogeneous in a given weighting scheme without any restrictions on the form of the main matrix

integrable evolution equations on associative algebras this paper surveys the classification of integrable evolution equations whose field variables take values in an associative algebra which includes matrix clifford and group algebra valued systems a variety of new examples of integrable systems possessing higher order symmetries are presented symmetry reductions lead to an associative algebra valued version of the painleve transcendent equations the basic theory of hamiltonian structures for associative algebra valued systems is developed and the bihamiltonian structures for several examples are found

splitting multisymplectic integrators for maxwell s equations in the paper we describe a novel kind of multisymplectic method for three dimensional 3 d maxwell s equations splitting the 3 d maxwell s equations into three local one dimensional lod equations then applying a pair of symplectic runge kutta methods to discretize each resulting lod equation it leads to splitting multisymplectic integrators we say this kind of schemes to be lod multisymplectic scheme lod ms the discrete conservation laws convergence dispersive relation dissipation and stability are investigated for the schemes theoretical analysis shows that the schemes are unconditionally stable non dissipative and of first order accuracy in time and second order accuracy in space as a reduction we also consider the application of lod ms to 2 d maxwell s equations numerical experiments match the theoretical results well they illustrate that lod ms is not only efficient and simple in coding but also has almost all the nature of multisymplectic integrators

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

a class of coupled kdv systems and their bi hamiltonian formulation a hamiltonian pair is proposed and thus a type of hereditary operators results all the corresponding systems of evolution equations possess local bi hamiltonian formulation and a special choice of the systems leads to the kdv hierarchy illustrative examples are given

energy dissipation splitting finite difference time domain method for maxwell equations with perfectly matched layers abstract in this paper we develop a novel kind of energy dissipation splitting finite difference time domain scheme for solving two dimensional maxwell equations with perfectly matched layers the discrete energy dissipation law convergence dispersion relation and stability are investigated for the scheme theoretical analysis shows that the scheme is unconditionally stable and of second order accuracy both in time and space numerical experiments confirm the theoretical results

solutions of elliptic equations with critical sobolev exponent in dimension three theorem 1 for every a 72 4 there exists a solution of i the previously known results concerning this problem are the following 1 if 1 5 0 i has no solution this follows from pohozaev s identity see lo and from the principle of unique continuation when a 0 we will suppose in the following that i 0 2 every positive solution of i is radial see gidas et al 9 3 let a be the first eigenvalue of a in b with dirichlet boundary condition and p1 the associated eigenfunction then for every a 2 i i has no positive solution this result follows from multiplying i by cp 4 for every a e n2 4 rr there exists a positive solution of i see brezis and nirenberg 5 5 for i 5 n2 4 i has no radial solution see 5 6 for every k e n if 1 e zk with zk k t n k2n2 i has a radial solution with k 1 zeros see atkinson and peletier 2 31 7 the analogue of theorem 1 in dimension n 2 4 holds for general domains see capozzi et al 6 and ambrosetti and struwe l our method is very different and we do not know whether theorem 1 is valid for general domains in ir3

energy conserved splitting spectral methods for two dimensional maxwell s equations in this paper the energy conserved splitting legendre galerkin method and energy conserved splitting legendre collocation method for maxwell s equations in two dimensions are proposed the schemes are energy conserved unconditionally stable and can be implemented efficiently the both methods are of second order convergence in time the high order error estimates in space for the two methods are also obtained for the ec s lc method an identity of the discrete divergence of electric fields is also obtained numerical examples are provided to confirm the theoretical analysis

two component coupled kdv equations and its connection with the generalized harry dym equations it is shown that three different lax operators in the dym hierarchy produce three generalized coupled harry dym equations these equations transform via the reciprocal link to the coupled two component korteweg de vries kdv system the first equation gives us known integrable two component kdv system while the second reduces to the known symmetrical two component kdv equation the last one reduces to the drienfeld sokolov equation this approach gives us new lax representation for these equations

the splitting finite difference time domain methods for maxwell s equations in two dimensions in this paper we consider splitting methods for maxwell s equations in two dimensions a new kind of splitting finite difference time domain methods on a staggered grid is developed the corresponding schemes consist of only two stages for each time step which are very simple in computation the rigorous analysis of the schemes is given by the energy method it is proved that the scheme is unconditionally stable and convergent for the problems with perfectly conducting boundary conditions numerical dispersion analysis and numerical experiments are presented to show the efficient performance of the proposed methods furthermore the methods are also applied to solve a scattering problem successfully