decay estimates for the wave equation in two dimensions abstract we establish strichartz estimates both reversed and some direct ones pointwise decay estimates and weighted decay estimates for the linear wave equation in dimension two with an almost scaling critical potential in the case when there is no resonance or eigenvalue at the edge of the spectrum we also prove some simple applications

rational recursion operators for integrable differential difference equations in this paper we introduce the concept of prehamiltonian pairs of difference operators demonstrate their connections with nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential difference equations we begin with a rigorous setup of the problem in terms of the skew field of rational pseudo difference operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational pseudo difference operators in particular we give a criteria for a rational operator to be weakly nonlocal a difference operator is called prehamiltonian if its image is a lie subalgebra with respect to the lie bracket on the difference field two prehamiltonian operators form a prehamiltonian pair if any linear combination of them is prehamiltonian then we show that a prehamiltonian pair naturally leads to a nijenhuis operator and a nijenhuis operator can be represented in terms of a prehamiltonian pair this provides a systematic method to check whether a rational operator is nijenhuis as an application we construct a prehamiltonian pair and thus a nijenhuis recursion operator for the differential difference equation recently discovered by adler and postnikov the nijenhuis operator obtained is not weakly nonlocal we prove that it generates an infinite hierarchy of local commuting symmetries we also illustrate our theory on the well known examples including the toda the ablowitz ladik and the kaup newell differential difference equations

nonexistence of global solutions to critical semilinear wave equations in exterior domain in high dimensions abstract the aim of this paper is to study the long time behavior of solutions to the initial boundary value problem of critical semilinear wave equations with small data in exterior domain in high dimensions n 5 we prove that solutions cannot exist globally in time the novelty is that we first use a cutoff function to transform the exterior problem to a cauchy problem then we divide the time into two intervals in one time interval we establish an energy estimate to control the error terms while in the other interval the asymptotic behavior of two test functions is used furthermore we obtain the upper bound of the lifespan by following the idea of zhou and han 2014

why nonlocal recursion operators produce local symmetries new results and applications it is well known that integrable hierarchies in 1 1 dimensions are local while the recursion operators that generate these hierarchies usually contain nonlocal terms we resolve this apparent discrepancy by providing simple and universal sufficient conditions for a nonlocal recursion operator in 1 1 dimensions to generate a hierarchy of local symmetries these conditions are satisfied by virtually all recursion operators known today and are much easier to verify than those found in earlier work we also give explicit formulae for the nonlocal parts of higher recursion hamiltonian and symplectic operators of integrable systems in 1 1 dimensions using these two results we prove under some natural assumptions the maltsev novikov conjecture stating that higher hamiltonian symplectic and recursion operators of integrable systems in 1 1 dimensions are weakly nonlocal i e the coefficients of these operators are local and these operators contain at most one integration operator in each term

on the local systems hamiltonian in the weakly non local poission brackets we study in this work the important class of non local poisson brackets pb which we call weakly non local they appeared recently in some investigations in soliton theory however there was no theory of such brackets except for very special first order case even in this case the theory was not developed enough in particular we introduce the physical forms and find casimirs momentum and canonical forms for the most important hydrodynamic type pb of that kind and their dependence on the boundary conditions

hydrodynamic chains and the classification of their poisson brackets infinite component poisson brackets of the dubrovin novikov type sov math dokl 27 665 669 1983 are considered the corresponding jacobi identity is significantly simplified in the liouville coordinates since the skew symmetry condition is automatically satisfied the concept of m poisson bracket connected with hydrodynamic chains is introduced then the jacobi identity is a nonlinear system of equations in partial derivatives which can be completely integrated in such a case a classification of infinite component poisson brackets of the dubrovin novikov type can be obtained two simplest examples m 0 and m 1 are considered also infinite component poisson brackets of the ferapontov am math soc transl 170 38 58 1995 type can be simplified in the liouville coordinates the jacobi identity for infinite component poisson brackets of the ferapontov mokhov type russ math surveys 45 218 219 1990 is presented in the liouville coordinates

a bi hamiltonian structure for the integrable discrete non linear schrodinger system abstract this paper shows that the al ablowitz ladik hierarchy of integrable equations can be explicitly viewed as a hierarchy of commuting flows which a are hamiltonian with respect to both a standard local poisson operator j and a new non local skew almost poisson operator k on the appropriate space b can be recursively generated from a recursion operator r k j 1 in addition the proof of these facts relies upon two new pivotal resolvent identities which suggest a general method for uncovering bi hamiltonian structures for other families of discrete integrable equations

generalized strichartz estimates and scattering for 3d zakharov system we obtain scattering for the 3d zakharov system with non radial small data in the energy space with angular regularity of degree one the main ingredient is a generalized strichartz estimate for the schrodinger equation in the space of l2 angular integrability

reciprocal transformations of hamiltonian operators of hydrodynamic type nonlocal hamiltonian formalism for linearly degenerate systems reciprocal transformations of hamiltonian operators of hydrodynamic type are investigated the transformed operators are generally nonlocal possessing a number of remarkable algebraic and differential geometric properties we apply our results to linearly degenerate semi hamiltonian systems in riemann invariants a typical example being rti m 1nrm ri rxi i 1 2 n since all such systems are linearizable by appropriate generalized reciprocal transformations our formulas provide an infinity of mutually compatible nonlocal hamiltonian structures explicitly parametrized by n arbitrary functions of one variable

global strichartz estimates for solutions to the wave equation exterior to a convex obstacle hhin this paper we show that certain local strichartz estimates for solutions of the wave equation exterior to a convex obstacle can be extended to estimates that are global in both space and time this extends the work that was done previously by h smith and c sogge in odd spatial dimensions in order to prove the global estimates we explore weighted strichartz estimates for solutions of the wave equation when the cauchy data and forcing term are compactly supported

an elementary proof of strauss conjecture abstract in this paper we consider the cauchy problem for semilinear wave equation in high dimensions first we use a morawetz energy estimate which is obtained by integrating on the light cone to get a weighted l 2 l 2 estimate of the solution and then give an elementary proof of the weighted strichartz estimate in georgiev et al 3 hence the strauss conjecture we also obtain a variant of the weighted strichartz estimates and give the sharp estimate of the lifespan for the semilinear wave equation with subcritical nonlinearity

lenard scheme for two dimensional periodic volterra chain we prove that for compatible weakly nonlocal hamiltonian and symplectic operators hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating nijenhuis operators are weakly nonlocal or not we construct a recursion operator of the two dimensional periodic volterra chain from its lax representation and prove that it is a nijenhuis operator furthermore we show that this system is a generalized bi hamiltonian system rather surprisingly the product of its weakly nonlocal hamiltonian and symplectic operators gives rise to the square of the recursion operator

whitham systems and deformations we consider the deformations of whitham systems including the dispersion terms and having the form of dubrovin zhang deformations of frobenius manifolds the procedure is connected with the b a dubrovin problem of deformations of frobenius manifolds corresponding to the whitham systems of integrable hierarchies under some nondegeneracy requirements we suggest a general scheme of the deformation of the hyperbolic whitham systems using the initial nonlinear system the general form of the deformed whitham system coincides with the form of the low dispersion asymptotic expansions used by b a dubrovin and y zhang in the theory of deformations of frobenius manifolds

on complete integrability of the mikhailov novikov wang system in the present paper we consider a new two component fifth order integrable system recently found by mikhailov novikov and wang and show that this system possesses a hereditary recursion operator and infinitely many commuting symmetries and conservation laws as well as infinitely many compatible hamiltonian and symplectic structures and is therefore completely integrable the system in question admits a reduction to the kaup kupershmidt equation

global convergence of a class of trust region algorithms for optimization with simple bounds this paper extends the known excellent global convergence properties of trust region algorithms for unconstrained optimization to the case where bounds on the variables are present weak conditions on the accuracy of the hessian approximations are considered it is also shown that when the strict complementarily condition holds the proposed algorithms reduce to an unconstrained calculation after finitely many iterations allowing a fast asymptotic rate of convergence

some remarks on strichartz estimates for homogeneous wave equation abstract we give several remarks on strichartz estimates for homogeneous wave equation with special attention to the cases of l x estimates radial solutions and initial data from the inhomogeneous sobolev spaces in particular we give the failure of the endpoint estimate l t 4 l x for n 2

combined effects of two nonlinearities in lifespan of small solutions to semi linear wave equations this paper investigates the combined effects of two distinctive power type nonlinear terms with parameters p q 1 in the lifespan of small solutions to semi linear wave equations we determine the full region of p q to admit global existence of small solutions at least for spatial dimensions n 2 3 moreover for many p q when there is no global existence we obtain sharp lower bound of the lifespan which is of the same order as the upper bound of the lifespan

on abstract strichartz estimates and the strauss conjecture for nontrapping obstacles we establish the strauss conjecture concerning small data global existence for nonlinear wave equations in the setting of exterior domains to compact obstacles for space dimensions n 3 and 4 the obstacle is assumed to be nontrapping and the solution is assumed to satisfy either dirichlet or neumann conditions along the boundary of the obstacle the key step in the proof is establishing certain abstract strichartz estimates for the linear wave equation on exterior domains

reciprocal transformations and flat metrics on hurwitz spaces we consider hydrodynamic systems which possess a local hamiltonian structure to such a system there are also associated an infinite number of nonlocal hamiltonian structures we give the necessary and sufficient conditions so that after a nonlinear transformation of the independent variables the reciprocal system still possesses a local hamiltonian structure we show that under our hypotheses bi hamiltonicity is preserved by the reciprocal transformation finally we apply such results to the reciprocal systems of genus g whitham kdv modulation equations

global and asymptotic convergence rate estimates for a class of projected gradient processes projected gradient processes of the goldstein levitin polyak type are considered for constrained minimization problems min omega f with omega a convex set in a hilbert space x and f x to mathbb r 1 a differentiable functional global and local convergence theorems are established for a large class of these processes including those generated with implicit step length rules proposed by bertsekas and goldstein in this analysis traditional uniform strong positivity conditions on the hessian nabla 2 f are replaced by weaker pseudoconvexity conditions and growth conditions on f when f has a unique minimizes in omega convergence rates are shown to depend on how rapidly the function gamma sigma inf r f x f xi mid x in omega x xi geqq sigma grows with increasing sigma 0 if gamma sigma geqq b sigma nu for some b 0 the processes f n in question converge to inf f like o n nu nu 2 linear

jacobi structures of evolutionary partial differential equations abstract in this paper we introduce the notion of infinite dimensional jacobi structure to describe the geometrical structure of a class of nonlocal hamiltonian systems which appear naturally when applying reciprocal transformations to hamiltonian evolutionary pdes we prove that our class of infinite dimensional jacobi structures is invariant under the action of reciprocal transformations that only change the spatial variable the main technical tool is in a suitable generalization of the classical schouten nijenhuis bracket to the space of the so called quasi local multi vectors and a simple realization of this structure in the framework of supermanifolds these constructions are used to compute the lichnerowicz jacobi cohomologies and to prove a darboux theorem for jacobi structures with hydrodynamic leading terms we also introduce the notion of bi jacobi structures and consider the integrability of a system of evolutionary pdes that possesses a bi jacobi structure

on nonlinear elastic waves in 2 d abstract for the cauchy problem of 2 d nonlinear elastic waves of isotropic homogeneous and hyperelastic materials satisfying null conditions assuming the smallness of h 3 r 2 h 2 r 2 norm and the boundedness of h 4 r 2 h 3 r 2 norm for radially symmetric initial data we show the global existence of classical solutions similar results for exterior domain problems are also proved

finite time blow up to critical semilinear wave equation outside the ball in 3 d abstract in this paper we study the initial boundary value problem of critical semilinear wave equation with small initial data in 3 d blow up result will be established no matter how small the data are we find a radial and positive solution of the corresponding linear problem then our blow up result follows from the idea of zhou and han 2014 what is more we get the upper bound of the lifespan estimate

on semilinear tricomi equations with critical exponents or in two space dimensions abstract this paper is a complement of our recent works on the semilinear tricomi equations in 9 and 10 for the semilinear tricomi equation t 2 u t u u p with initial data u 0 t u 0 u 0 u 1 where t 0 x r n n 3 p 1 and u i c 0 r n i 0 1 we have shown in 9 and 10 that there exists a critical exponent p crit n 1 such that the solution u in general blows up in finite time when 1 p p crit n and there is a global small solution for p p crit n in the present paper firstly we prove that the solution of t 2 u t u u p will generally blow up for the critical exponent p p crit n and n 2 secondly we establish the global existence of small data solution to t 2 u t u u p for p p c r i t n and n 2 thus we have given a systematic study on the blowup or global existence of small data solution u to the equation t 2 u t u u p for n 2

an active set newton algorithm for large scale nonlinear programs with box constraints a new algorithm for large scale nonlinear programs with box constraints is introduced the algorithm is based on an efficient identification technique of the active set at the solution and on a nonmonotone stabilization technique it possesses global and superlinear convergence properties under standard assumptions a new technique for generating test problems with known characteristics is also introduced the implementation of the method is described along with computational results for large scale problems

algorithms for bound constrained quadratic programming problems we present an algorithm which combines standard active set strategies with the gradient projection method for the solution of quadratic programming problems subject to bounds we show in particular that if the quadratic is bounded below on the feasible set then termination occurs at a stationary point in a finite number of iterations moreover if all stationary points are nondegenerate termination occurs at a local minimizer a numerical comparison of the algorithm based on the gradient projection algorithm with a standard active set strategy shows that on mildly degenerate problems the gradient projection algorithm requires considerable less iterations and time than the active set strategy on nondegenerate problems the number of iterations typically decreases by at least a factor of 10 for strongly degenerate problems the performance of the gradient projection algorithm deteriorates but it still performs better than the active set method

maximal functions associated to filtrations abstract let t be a bounded linear or sublinear operator from lp y to lq x a maximal operator t f x supj t f yj x is associated to any sequence of subsets yj of y under the hypotheses that q p and the sets yj are nested we prove that t is also bounded classical theorems of menshov and zygmund are obtained as corollaries multilinear generalizations of this theorem are also established these results are motivated by applications to the spectral analysis of schrodinger operators

a class of nonmonotone stabilization methods in unconstrained optimization this paper deals with the solution of smooth unconstrained minimization problems by newton type methods whose global convergence is enforced by means of a nonmonotone stabilization strategy in particular a stabilization scheme is analyzed which includes different kinds of relaxation of the descent requirements an extensive numerical experimentation is reported

an active set quasi newton method with projected search for bound constrained minimization we analyze an active set quasi newton method for large scale bound constrained problems our approach combines the accurate active set identification function and the projected search both of these strategies permit fast change in the working set the limited memory method is employed to update the inactive variables while the active variables are updated by simple rules a further division of the active set enables the global convergence of the new algorithm numerical tests demonstrate the efficiency and performance of the present strategy and its comparison with some existing active set strategies

the multi dimensional hamiltonian structures in the whitham method we consider the averaging of local field theoretic poisson brackets in the multi dimensional case as a result we construct a local poisson bracket for the regular whitham system in the multidimensional situation the procedure is based on the procedure of averaging of local conservation laws and follows the dubrovin novikov scheme of the bracket averaging suggested in one dimensional case however the features of the phase space of modulated parameters in higher dimensions lead to a different natural class of the averaged brackets in comparison with the one dimensional situation here we suggest a direct procedure of construction of the bracket for the whitham system for d 1 and discuss the conditions of applicability of the corresponding scheme at the end we discuss canonical forms of the averaged poisson bracket in the multidimensional case

the strauss conjecture on kerr black hole backgrounds we examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the strauss conjecture on the schwarzschild and kerr with small angular momentum black hole backgrounds the key estimates are a class of weighted strichartz estimates which are used near infinity where the metrics can be viewed as small perturbations of the minkowski metric and a localized energy estimate on the black hole background which handles the behavior in the remaining compact set

on the boundary strichartz estimates for wave and schrodinger equations abstract we consider the l t 2 l x r estimates for the solutions to the wave and schrodinger equations in high dimensions for the homogeneous estimates we show l t 2 l x estimates fail at the critical regularity in high dimensions by using stable levy process in r d moreover we show that some spherically averaged l t 2 l x estimate holds at the critical regularity as a by product we obtain strichartz estimates with angular smoothing effect for the inhomogeneous estimates we prove double l t 2 type estimates

extension of integrable equations in this communication we show that there is general construction to produce non evolutionary integrable equations from a given integrable evolutionary equation to support the main theorem a few examples are explicitly given

critical criteria of fujita type for a system of inhomogeneous wave inequalities in exterior domains abstract we consider blow up results for a system of inhomogeneous wave inequalities in exterior domains we will handle three type boundary conditions dirichlet type neumann type and mixed boundary conditions we use a unified approach to show the optimal criteria of fujita type for each case our study yields naturally optimal nonexistence results for the corresponding stationary wave system and equation we provide many new results and close some open questions

weakly nonlocal symplectic structures whitham method and weakly nonlocal symplectic structures of hydrodynamic type we consider the special type of field theoretical symplectic structures called weakly nonlocal the structures of this type are in particular very common for integrable systems such as kdv or nls we introduce here the special class of weakly nonlocal symplectic structures which we call weakly nonlocal symplectic structures of hydrodynamic type we investigate then the connection of such structures with the whitham averaging method and propose the procedure of averaging the weakly nonlocal symplectic structures the averaging procedure gives the weakly nonlocal symplectic structure of hydrodynamic type for the corresponding whitham system the procedure also gives action variables corresponding to the wave numbers of m phase solutions of the initial system which give the additional conservation laws for the whitham system

strichartz estimates for dirichlet wave equations in two dimensions with applications we establish the strauss conjecture for nontrapping obstacles when the spatial dimension n is two as pointed out in 7 this case is more subtle than n 3 or 4 due to the fact that the arguments of the first two authors 11 burq 1 and metcalfe 9 showing that local strichartz estimates for obstactles imply global ones require that the sobolev index equal 1 2 when n 2 we overcome this difficulty by interpolating between energy estimates 0 and ones for 1 2 that are generalizations of minkowski space estimates of fang and the third author 4 5 the second author 12 and sterbenz 14

blow up of solutions to critical semilinear wave equations with variable coefficients abstract we verify the critical case p p 0 n of strauss conjecture 30 concerning the blow up of solutions to semilinear wave equations with variable coefficients in r n where n 2 the perturbations of laplace operator are assumed to be smooth and decay exponentially fast at infinity we also obtain a sharp lifespan upper bound for solutions with compactly supported data when p p 0 n the unified approach to blow up problems in all dimensions combines several classical ideas in order to generalize and simplify the method of zhou 43 and zhou han 45 exponential eigenfunctions of the laplacian 37 are used to construct the test function q for linear wave equation with variable coefficients and john s method of iterations 13 is augmented with the slicing method of agemi kurokawa and takamura 1 for lower bounds in the critical case

projected newton methods for optimization problems with simple constraints we consider the problem min f x x geqq 0 and propose algorithms of the form x k 1 x k alpha k d k nabla f x k where cdot denotes projection on the positive orthant alpha k is a stepsize chosen by an armijo like rule and d k is a positive definite symmetric matrix which is partly diagonal we show that d k can be calculated simply on the basis of second derivatives of f so that the resulting newton like algorithm has a typically superlinear rate of convergence with other choices of d k convergence at a typically linear rate is obtained the algorithms are almost as simple as their unconstrained counterparts they are well suited for problems of large dimension such as those arising in optimal control while being competitive with existing methods for low dimensional problems the effectiveness of the newton like algorithm is demonstrated via computational examples involving as many as 10 000 variables extensions to general linearly constrained pr

universal lie algebra extensions and their casimir invariants via commutative structures abstract we consider a special kind of lie algebra extension called universal extension introduced recently we show that there is one to one correspondence with commutative associative algebras we also investigate the casimir invariants for the universal extensions and consider in detail the so called coextension structures arising in the calculation of the casimir functions

a direct active set algorithm for large sparse quadratic programs with simple bounds we show how a direct active set method for solving definite and indefinite quadratic programs with simple bounds can be efficiently implemented for large sparse problems all of the necessary factorizations can be carried out in a static data structure that is set up before the numeric computation begins the space required for these factorizations is no larger than that required for a single sparse cholesky factorization of a matrix with the same sparsity structure as the hessian of the quadratic we propose several improvements to this basic algorithm a new way to find a search direction in the indefinite case that allows us to free more than one variable at a time and a new heuristic method for finding a starting point these ideas are motivated by the two norm trust region problem additionally we also show how projection techniques can be used to add several constraints to the active set at each iteration our experimental results show that an algorithm with these improvements runs much faster than the basic algorithm for positive definite problems and finds local minima with lower function values for indefinite problems

reciprocal transformations and local hamiltonian structures of hydrodynamic type systems we start from a hyperbolic dubrovin and novikov dn hydrodynamic type system of dimension n which possesses riemann invariants and we settle the necessary conditions on the conservation laws in the reciprocal transformation so that after such a transformation of the independent variables one of the metrics associated with the initial system is flat we prove the following statement let n 3 in the case of reciprocal transformations of a single independent variable or n 5 in the case of transformations of both the independent variables then the reciprocal metric may be flat only if the conservation laws in the transformation are linear combinations of the canonical densities of conservation laws i e the casimirs the momentum and the hamiltonian densities associated with the hamiltonian operator for the initial metric then we restrict ourselves to the case in which the initial metric is either flat or of constant curvature and we classify the reciprocal transformations of one or both the independent variables so that the reciprocal metric is flat such characterization has an interesting geometric interpretation the hypersurfaces of two diagonalizable dn systems of dimension n 5 are lie equivalent if and only if the corresponding local hamiltonian structures are related by a canonical reciprocal transformation

almost global existence for nonlinear wave equations in an exterior domain in two space dimensions in this paper we deal with the exterior problem for a system of nonlinear wave equations in two space dimensions assuming that the initial data is small and smooth we establish the same type of lower bound of the lifespan for the problem as that for the cauchy problem despite of the weak decay property of the solution in two space dimensions