stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

approximation order of bivariate spline interpolation for arbitrary smoothness by using the algorithm of nurnberger and riessinger 1995 we construct hermite interpolation sets for spaces of bivariate splines sqr 1 of arbitrary smoothness defined on the uniform type triangulations it is shown that our hermite interpolation method yields optimal approximation order for q 3 5r 1 in order to prove this we use the concept of weak interpolation and arguments of birkhoff interpolation

swapping edges of arbitrary triangulations to achieve the optimal order of approximation in the representation of scattered data by smooth pp piecewise polynomial functions perhaps the most important problem is to find an optimal triangulation of the given sample sites called vertices of course the notion of optimality depends on the desirable properties in the approximation or modeling problems in this paper we are concerned with optimal approximation order with respect to the given order r of smoothness and degree k of the polynomial pieces of the smooth pp functions we will only consider c1 pp approximation with r 1 and k 4 the main result in this paper is an efficient method for triangulating any finitely many arbitrarily scattered sample sites such that these sample sites are the only vertices of the triangulation and that for any discrete data given at these sample sites there is a c1 piecewise quartic polynomial on this triangulation that interpolates the given data with the fifth order of approximation

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

persistence properties and unique continuation of solutions of the camassa holm equation it is shown that a strong solution of the camassa holm equation initially decaying exponentially together with its spacial derivative must be identically equal to zero if it also decays exponentially at a later time in particular a strong solution of the cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution

optimal order approximation by mixed three directional spline elements this paper is concerned with a study of approximation order and construction of locally supported elements for the space s14 of c1 quartic pp piecewise polynomial functions on a triangulation of a connected polygonal domain in r2 it is well known that when is a three directional mesh 1 the order of approximation of s14 1 is only 4 not 5 though a local clough tocher refinement procedure of an arbitrary triangulation yields the optimal fifth order of approximation from the space s14 see 1 it needs more data points in addition to the vertex set of the triangulation in this paper we will introduce a particular mixed three directional mesh 3 and construct so called mixed three directional elements we prove that the space s14 3 achieves its optimal order of approximation by constructing an interpolation scheme using mixed three directional elements

macro elements and stable local bases for splines on powell sabin triangulations macro elements of arbitrary smoothness are constructed on powell sabin triangle splits these elements are useful for solving boundary value problems and for interpolation of hermite data it is shown that they are optimal with respect to spline degree and we believe they are also optimal with respect to the number of degrees of freedom the construction provides local bases for certain superspline spaces defined over powell sabin refinements these bases are shown to be stable as a function of the smallest angle in the triangulation which in turn implies that the associated spline spaces have optimal order approximation power

a robust solver for a second order mixed finite element method for the cahn hilliard equation abstract we develop a robust solver for a second order mixed finite element splitting scheme for the cahn hilliard equation this work is an extension of our previous work in which we developed a robust solver for a first order mixed finite element splitting scheme for the cahn hilliard equation the key ingredient of the solver is a preconditioned minimal residual algorithm with a multigrid preconditioner whose performance is independent of the spatial mesh size and the time step size for a given interfacial width parameter the dependence on the interfacial width parameter is also mild

optimal lagrange interpolation by quartic c1 splines on triangulations we develop a local lagrange interpolation scheme for quartic c 1 splines on triangulations given an arbitrary triangulation d we decompose d into pairs of neighboring triangles and add diagonals to some of these pairs only in exceptional cases a few triangles are split based on this simple refinement of d we describe an algorithm for constructing lagrange interpolation points such that the interpolation method is local stable and has optimal approximation order the complexity for computing the interpolating splines is linear in the number of triangles for the local lagrange interpolation methods known in the literature about half of the triangles have to be split

developments in bivariate spline interpolation the aim of this survey is to describe developments in the field of interpolation by bivariate splines we summarize results on the dimension and the approximation order of bivariate spline spaces and describe interpolation methods for these spaces moreover numerical examples are given

stability of optimal order approximation by bivariate splines over arbitrary triangulations let a be a triangulation of some polygonal domain in r2 and skr a the space of all bivariate cr piecewise polynomials of total degree 3r 2 that can be used to provide the highest order of approximation with the property that the approximation constant of this order is independent of the geometry of a with the exception of the smallest angle in the partition this result is obtained by means of a careful choice of locally supported basis functions which however require a very technical proof to justify their stability in optimal order approximation a new formulation of smoothness conditions for piecewise polynomials in terms of their b net representations is derived for this purpose

conservation laws of the camassa holm equation we use the bi hamiltonian structure of the camassa holm equation to show that its conservation laws h n m are homogeneous with respect to the scaling m lambdam moreover a direct argument is presented proving that h 1 h 2 are of local character finally simple representations of the conservation laws in terms of their variational derivatives are derived and used to obtain a constructive scheme for computation of the h n s

local well posedness and stability of peakons for a generalized dullin gottwald holm equation abstract we establish the local well posedness for a generalized dullin gottwald holm equation by using kato s theory furthermore the orbital stability of the peaked solitary waves is also proved

persistence properties for a generalized camassa holm equation in weighted lp spaces abstract in present paper we study the cauchy problem for a generalized camassa holm equation which was discovered by hakkaev and kirchev 2005 12 our purpose here is to establish persistence properties and some unique continuation properties of the solutions of this equation in weighted l p spaces l p l p r p d x our results extends the work of brandolese 2012 2 on persistence properties to more general equation with higher order nonlinearities

well posedness and analyticity for the cauchy problem for the generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem for the generalized camassa holm equation which was proposed by hakkaev and kirchev s hakkaev k kirchev local well posedness and orbital stability of solitary wave solutions for the generalized camassa holm equation comm partial differential equations 30 2005 761 781 we establish the local well posedness for a range of besov spaces moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time which extends some results of danchin r danchin a few remarks on the camassa holm equation differential integral equations 14 2001 953 988 and himonas and misiolek a himonas g misiolek analyticity of the cauchy problem for an integrable evolution equation math ann 327 2003 575 584 to more general equations

continuity and asymptotic behaviors for a shallow water wave model with moderate amplitude abstract in this paper we study continuity and persistence for a nonlinear evolution equation describing the free surface of shallow water wave with a moderate amplitude which was proposed by constantin and lannes 7 by the approach for approximate solutions and well posedness estimates we obtain two sequences of solution for constantin lannes equation which are bounded in the sobolev space h s r with s 3 2 and the distance between the two sequences is lower bounded by a positive constant for any time t but converges to zero at the initial time this implies that the solution map is not uniformly continuous furthermore the solution map for constantin lannes equation is shown holder continuous in h r topology for all 0 r s with exponent depending on s and r in addition we also investigate the asymptotic behaviors of the strong solutions to constantin lannes equation at infinity within its lifespan provided the initial data in weighted l p l p r p d x spaces

the local well posedness and existence of weak solutions for a generalized camassa holm equation abstract a generalization of the camassa holm equation a model for shallow water waves is investigated using the pseudoparabolic regularization technique its local well posedness in sobolev space h s r with s 3 2 is established via a limiting procedure in addition a sufficient condition for the existence of weak solutions of the equation in lower order sobolev space h s with 1 s 3 2 is developed

macro elements and stable local bases for splines on clough tocher triangulations macro elements of arbitrary smoothness are constructed on clough tocher triangle splits these elements can be used for solving boundary value problems or for interpolation of hermite data and are shown to be optimal with respect to spline degree we conjecture they are also optimal with respect to the number of degrees of freedom the construction provides local bases for certain superspline spaces defined over clough tocher refinements of arbitrary triangulations these bases are shown to be stable as a function of the smallest angle in the triangulation which in turn implies that the associated spline spaces have optimal order approximation power

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

the ivp for the benjamin ono equation in weighted sobolev spaces abstract in this work we continue our study initiated in fonseca and ponce 2011 11 on the uniqueness properties of real solutions to the ivp associated to the benjamin ono bo equation in particular we shall show that the uniqueness results established in fonseca and ponce 2011 11 do not extend to any pair of non vanishing solutions of the bo equation also we shall prove that the uniqueness result established in fonseca and ponce 2011 11 under a hypothesis involving information of the solution at three different times cannot be relaxed to two different times

error estimates in w 2 semi norms for discrete interpolating d 2 splines we derive error estimates in w 2 semi norms for multivariate discrete d 2 splines that interpolate an unknown function at the vertices of given triangulations these results are widely based on the construction of approximation operators and linear projectors onto piecewise polynomial spaces having weakly stable local bases