the singular set of solutions to non differentiable elliptic systems we estimate the hausdorff dimension of the singular set of solutions to elliptic systems of the type if the vector fields a and b are holder continuous with respect to the variable x with exponent then the hausdorff dimension of the singular set of any weak solution is at most n 2

new method for deriving nonlinear integrable systems a new approach is proposed to derive nonlinear integrable systems it is used to obtain several new nonlinear integrable systems the above results are relevant to some problems of hydrodynamics plasma physics solid state physics etc

integration of the soliton hierarchy with self consistent sources in contrast with the soliton equations the evolution of the eigenfunctions in the lax representation of soliton equation with self consistent sources sescs possesses singularity we present a general method to treat the singularity to determine the evolution of scattering data the akns hierarchy with self consistent sources the mkdv hierarchy with self consistent sources the nonlinear schrodinger equation hierarchy with self consistent sources the kaup newell hierarchy with self consistent sources and the derivative nonlinear schrodinger equation hierarchy with self consistent sources are integrated directly by using the inverse scattering method the n soliton solutions for some sescs are presented it is shown that the insertion of a source may cause the variation of the velocity of soliton this approach can be applied to all other 1 1 dimensional soliton hierarchies

mixed l p l 2 norms of the lattice point discrepancy we estimate some mixed lp l2 norms of the discrepancy between the volume and the number of integer points in r x a dilation by a factor r and a translation by a vector x of a convex body in r d with smooth boundary with nonvanishing gaussian curvature int t d 1 h int r r h sum kin z d r x k r d dr p 2 dx 1 p we obtain estimates for fixed values of h and r and also asymptotic estimates when h

capture and confinement of solitons in nonlinear integrable systems the system describing the interaction of a long wave with a short wave packet on thex y plane is considered the solutions are found to describe a soliton that comes from infinity and then is captured into a conditionally periodical oscillatory regime the solutions are also found that describe a soliton coming from infinity and then decaying into two solitons one goes to infinity and the other is captured into a conditionally periodical oscillatory regime the obtained results are relevant to some problems of hydrodynamics plasma physics solid state physics etc

the n soliton solutions of the sine gordon equation with self consistent sources abstract the hierarchy of the sine gordon equation with self consistent sources is derived by using the eigenfunctions of recursion operator the bilinear form of the sine gordon equation with self consistent sources is given and the n soliton solutions are obtained through hirota method and wronskian technique respectively some novel determinantal identities are presented to treat the nonlinear term in the time evolution and finish the wronskian verifications

sign changes of the error term in weyl s law for heisenberg manifolds let r t be the error term in weyl s law for the 2l 1 dimensional heisenberg manifold hl gl in this paper several results on the sign changes and odd moments of r t are proved in particular it is proved that for some sufficiently large constant c r t changes sign in the interval t t c t for all large t moreover for a small constant c1 there exist infinitely many subintervals in t 2t of length c1 t log 5 t such that r t c1tl 1 4 holds on each of these subintervals

bi hamiltonian structure of jm hierarchy with self consistent sources by taking the space variable x as the evolution parameter and introducing the jacobi ostrogradiski coordinates we present t type hamiltonian description of soliton equations with self consistent sources which have not x type hamiltonian formulation when considering t as evolution parameter the t type miura map generates the second hamiltonian structure for these t type hamiltonian systems the two compatible hamiltonian operators are used to construct a hereditary operator and to obtain new hierarchies of infinite dimensional integrable hamiltonian systems the reduction of these equations with sources gives rise to the constrained flows of soliton equations and their bi hamiltonian structure we use the jaulent miodek hierarchy with self consistent sources to illustrate the method c 1999 elsevier science b v all rights reserved

two binary darboux transformations for the kdv hierarchy with self consistent sources two binary integral type darboux transformations for the kdv hierarchy with self consistent sources are proposed in contrast with the darboux transformation for the kdv hierarchy one of the two binary darboux transformations provides non auto backlund transformation between two nth kdv equations with self consistent sources with different degrees the formula for the m times repeated binary darboux transformations are presented this enables us to construct the n soliton solution for the kdv hierarchy with self consistent sources

the singular set of minima of integral functionals in this paper we provide upper bounds for the hausdorff dimension of the singular set of minima of general variational integrals open image in new window where f is suitably convex with respect to dv and holder continuous with respect to x v in particular we prove that the hausdorff dimension of the singular set is always strictly less than n where open image in new window

the use of backlund transformations in obtaining n soliton solutions in wronskian form the wronskian formulation of the n soliton solutions of various nonlinear evolution equations modified korteweg de vries with both zero and non zero asymptotic conditions sine gordon korteweg de vries kadomtsev petviashvili and boussinesq equations are obtained by inductive use of backlund transformations in hirota s 1980 bilinear notation

on the pfaffianized kp equation with self consistent sources abstract a new procedure called source generation is applied to the pfaffianized kp equation as a result the pfaffianized kp equation with self consistent sources escs is obtained this coupled system cannot only be reduced to the pfaffianized kp equation but also reduced to the kp equation with self consistent sources kpescs so the pfaffianized kp escs can be viewed as a pfaffian version of the kpescs which indicates the commutativity of the source generation procedure and pfaffianization

constructing hierarchy of sinh gordon type equations from soliton hierarchy with self consistent source abstract a method for constructing hierarchy of sinh gordon shg type equations from soliton hierarchy with self consistent source is proposed the lax representation for the shg type equation is naturally deduced from that for soliton equation with self consistent sources several new hierarchies of shg type equations are presented

on the non isospectral kadomtsev petviashvili equation with self consistent sources abstract a new procedure called source generation is applied to construct non isospectral soliton equations with self consistent sources as results the non isospectral kadomtsev petviashvili equation with self consistent sources kpescs and its gramm type determinant solutions are obtained furthermore the non isospectral pfaffianized kp equation with self consistent sources is constructed this coupled system can not only be reduced to the non isospectral pfaffianized kp equation but also reduced to the non isospectral kpescs

higher integrability of very weak solutions of systems of p x laplacean type abstract we prove higher integrability of very weak solutions of a class of nonlinear elliptic systems whose model is the p x laplacean system div d u p x 2 d u div f p x 2 f i e we show that for small enough a function which solves the system and has bounded p x 1 energy satisfies d u p x l loc 1

a modified kdv equation with self consistent sources in non uniform media and soliton dynamics two non isospectral modified kdv equations with self consistent sources are derived which correspond to the time dependent spectral parameter satisfying t and t 3 respectively gauge transformation between the first non isospectral equation corresponding to t and its isospectral counterpart is given from which exact solutions and conservation laws for the non isospectral one are easily listed besides solutions to the two nonisospectral modified kdv equations with self consistent sources are derived by means of the hirota method and the wronskian technique respectively nonisospectral dynamics and source effects including one soliton characteristics in non uniform media two solitons scattering and special behaviours related to sources for example the ghost solitons in the degenerate two soliton case are investigated analytically

higher power moments of δ x e t and p x let x denote the error term in the asymptotic formula for the divisor sum using a voronoi type formula for x we prove that 2 x x 3 dx cx 4 2 and 2 x x 4 dx c x 2 where c and c are some positive constants the estimate for the third power moment shows that 3 x is strongly biased towards the positive side we also obtain similar results for e t the error term in the mean square formula for the riemann zeta function on the critical line and for p x the error term in the circle problem

a maximum principle for non linear elliptic systems boundary fundamental estimates on demontre un principe de maximum pour des operateurs elliptiques non lineaires eu i d i a i du ou a i sont des vecteurs de r n

boundary regularity and sufficient conditions for strong local minimizers abstract in this paper we present a new proof of the sufficiency theorem for strong local minimizers concerning c 1 extremals at which the second variation is strictly positive the results are presented in the quasiconvex setting in accordance with the original statement by grabovsky and mengesha 31 the strategy that we follow relies on a decomposition theorem that allows to split a sequence of variations into its oscillating and its concentrating parts as well as on a sufficiency result according to which smooth extremals are spatially local minimizers furthermore we prove partial regularity up to the boundary for strong local minimizers in the non homogeneous case and a full regularity result for lipschitz extremals with gradient of vanishing mean oscillation as a consequence we also establish a sufficiency result for this class of extremals the regularity results are established via a blow up argument

partial holder continuity for solutions of subquadratic elliptic systems in low dimensions abstract we consider weak solutions of second order nonlinear elliptic systems in divergence form under standard subquadratic growth conditions with boundary data of class c 1 in dimensions n 2 3 we prove that u is locally holder continuous for every exponent 0 1 n 2 p outside a singular set of hausdorff dimension less than n p this result holds up to the boundary both for non degenerate and degenerate systems in the proof we apply the direct method and classical morrey type estimates introduced by campanato

integration of the modified korteweg de vries hierarchy with an integral type of source the modified korteweg de vries hierarchy with an integral type of source mkdvhws which consists of the reduced akns eigenvalue problem with r q and the mkdv hierarchy with an extra term of the integration of a square eigenfunction is investigated we propose a method to find the explicit evolution equation for the eigenfunction of the auxiliary linear problems of the mkdvhws then we determine the evolution equations of scattering data corresponding to the mkdvhws and solve the equation in the mkdvhws by inverse scattering transformation

pfaffian solution of a semi discrete bkp type equation and its source generation version the pfaffian solution of a semi discrete bkp type equation is obtained at first then utilizing the source generation procedure this equation with self consistent sources bkpescs is presented and its pfaffian solutions are derived finally a bilinear backlund transformation for the semi discrete bkpescs is given

the average behaviour of the error term in a new kind of the divisor problem abstract we define a new kind of divisor function d 1 n by the nth coefficient of the dirichlet series s 2 and denote by 1 x the error term in the asymptotic formula for n x d 1 n then we compute the first moment of 1 x that is 1 x 1 x d x and consider discrete mean values n x 1 k n k 1 2 and deduce asymptotic formulas which are analogous to the results obtained by voronoi hardy and furuya

solution of an initial boundary value problem for coupled nonlinear waves the authors derive and study a hierarchy of nonlinear coupled evolution equations among which is the coupled korteveg de vries schrodinger equation for which they prove that a mixed initial boundary value problem is solvable they give the method of solution together with the backlund transformation and establish the infinite set of conserved densities they finally discuss the applicability of such equations in plasma physics and hydrodynamics