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

vertexwise criteria for admissibility of alcoves we give a new description of the set rm adm mu of admissible alcoves as an intersection of certain obtuse cones of alcoves and we show this description may be given by imposing conditions vertexwise we use this to prove the vertexwise admissibility conjecture of pappas rapoport smithling the same idea gives simple proofs of two ingredients used in the proof of the kottwitz rapoport conjecture on existence of crystals with additional structure

the periodic cauchy problem for a two component non isospectral cubic camassa holm system abstract in this paper we study the periodic cauchy problem for a two component non isospectral cubic camassa holm system which includes the fokas olver rosenau qiao forq or modified camassa holm mch equation and the two component mch system as two special cases the system is integrable in the sense of possessing a non isospectral lax pair with the spectrum depending on time t and admits multi peakon solutions in an explicit form furthermore we establish the local well posedness for the system in the besov space b 2 r s t with s 3 2 1 r where the key ingredients include the friedrichs regularization method the littlewood paley decomposition theory and the transport theory in besov spaces then we derive a precise blow up criteria which is dependent of the parameters t and t moreover by the intrinsic structure of the system we obtain a new blow up result for strong solutions with sufficient conditions on the initial data and parameters the entire proof procedure relies upon a newly derived transport equation which is involved in nonlocal velocity term along the characteristic curves

a note on well posedness for camassa holm equation in this note we investigate the problem of well posedness for a shallow water equation with data having critical regularity our results are based on the use of besov spaces b2 rs which generalize the sobolev spaces hs with critical index s 3 2

stability in the energy space of the sum of n peakons for a modified camassa holm equation with higher order nonlinearity considered herein is the orbital stability in the energy space h1 r of a decoupled sum of n peakons for a modified camassa holm equation with higher order nonlinearity the equation studied here admits single peakon and multi peakons based on our obtained result of the stability of a single peakon and then combining modulation argument with monotonicity of the local energy h1 norm we get the stability of the sum of n peakons considered herein is the orbital stability in the energy space h1 r of a decoupled sum of n peakons for a modified camassa holm equation with higher order nonlinearity the equation studied here admits single peakon and multi peakons based on our obtained result of the stability of a single peakon and then combining modulation argument with monotonicity of the local energy h1 norm we get the stability of the sum of n peakons

restriction of fourier transforms to curves an endpoint estimate with affine arclength measure consider the fourier restriction operators associated to curves in r d 3 we prove for various classes of curves the endpoint restricted strong type estimate with respect to affine arclength measure on the curve an essential ingredient is an interpolation result for multilinear operators with symmetries acting on sequences of vector valued functions

the camassa holm equation as a geodesic flow on the diffeomorphism group misiolek j geom phys 24 203 208 1998 has shown that the camassa holm equation is a geodesic flow on the bott virasoro group in this paper it is shown that the camassa holm equation for the case 0 is the geodesic spray of the weak riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the h1 inner product over the entire group this paper uses the right trivialization technique to rigorously verify that the euler poincare theory for lie groups can be applied to diffeomorphism groups the observation made in this paper has led to physically meaningful generalizations of the ch equation to higher dimensional manifolds

non uniform dependence on initial data for the two component novikov system considered herein is the initial value problem for the two component novikov system it is shown that the solution map of this problem is not uniformly continuous in sobolev spaces hs hs 1 for s 5 2 on the line based on the well posedness result and the lifespan for this problem the method of approximate solutions is utilized

propagation of ultra short optical pulses in cubic nonlinear media abstract we derive a partial differential equation that approximates solutions of maxwell s equations describing the propagation of ultra short optical pulses in nonlinear media and which extends the prior analysis of alterman and rauch phys lett a 264 2000 390 diffractive nonlinear geometric optics for short pulses preprint 2002 we discuss non rigorously conditions under which this approximation should be valid but the main contributions of this paper are 1 an emphasis on the fact that the model equation for short pulse propagation may depend on the details of the optical susceptibility in the wavelength regime under consideration 2 a numerical comparison of solutions of this model equation with solutions of the full nonlinear partial differential equation 3 a local well posedness result for the model equation and 4 a proof that in contrast to the nonlinear schrodinger equation which models slowly varying wavetrains this equation has no smooth pulse solutions which propagate with fixed shape and speed

appendix on parahoric subgroups we give the proofs of some simple facts on parahoric subgroups and on iwahori weyl groups used in t haines the base change fundamental lemma for central elements in parahoric hecke algebras preprint 2008 g pappas m rapoport twisted loop groups and their affine flag varieties adv math 219 2008 118 198 m rapoport a guide to the reduction modulo p of shimura varieties asterisque 298 2005 271 318

twisted loop groups and their affine flag varieties we develop a theory of affine flag varieties and of their schubert varieties for reductive groups over a laurent power series local field k t with k a perfect field this can be viewed as a generalization of the theory of affine flag varieties for loop groups to a twisted case a consequence of our results is that our construction also includes the flag varieties for kac moody lie algebras of affine type we also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k t and some applications to local models of shimura varieties

on the global existence and wave breaking criteria for the two component camassa holm system considered herein is a two component camassa holm system modeling shallow water waves moving over a linear shear flow a wave breaking criterion for strong solutions is determined in the lowest sobolev space hs s 32 by using the localization analysis in the transport equation theory moreover an improved result of global solutions with only a nonzero initial profile of the free surface component of the system is established in this sobolev space hs

on the flatness of local models for the symplectic group abstract we investigate the bad reduction of certain shimura varieties associated to the symplectic group more precisely we look at a model of the shimura variety at a prime p with parahoric level structure at p we show that this model is flat as conjectured by rapoport and zink ann of math stud 141 1996 and that its special fibre is reduced a crucial ingredient is faltings theorem on the normality of schubert varieties in the affine flag variety

on the blow up rate and the blow up set of breaking waves for a shallow water equation we consider the problem of the development of singularities for classical solutions to a new periodic shallow water equation blow up can occur only in the form of wave breaking i e the solution remains bounded but its slope becomes unbounded in finite time a quite detailed description of the wave breaking phenomenon is given there is at least a point in general depending on time where the slope becomes infinite exactly at breaking time the precise blow up rate is established and for a large class of initial data we also determine the blow up set

lipschitz metric for the novikov equation we consider the lipschitz continuous dependence of solutions for the novikov equation with respect to the initial data in particular we construct a finsler type optimal transport metric which renders the solution map lipschitz continuous on bounded sets of h 1 mathbb r cap w 1 4 mathbb r although it is not lipschitz continuous under the natural sobolev metric from an energy law due to the finite time gradient blowup by an application of thom s transversality theorem we also prove that when the initial data is in an open dense subset of h 1 mathbb r cap w 1 4 mathbb r the solution is piecewise smooth this generic regularity result helps us extend the lipschitz continuous metric to the general weak solutions our method of constructing the metric can be used to treat other kinds of quasi linear equations provided a good knowledge about the energy concentration

generalized hunter saxton equation and the geometry of the group of circle diffeomorphisms we study an equation lying mid way between the periodic hunter saxton and camassa holm equations and which describes evolution of rotators in liquid crystals with external magnetic field and self interaction we prove that it is an euler equation on the diffeomorphism group of the circle corresponding to a natural right invariant sobolev metric we show that the equation is bihamiltonian and admits both cusped and smooth traveling wave solutions which are natural candidates for solitons we also prove that it is locally well posed and establish results on the lifespan of its solutions throughout the paper we argue that despite similarities to the kdv ch and hs equations the new equation manifests several distinctive features that set it apart from the other three

on the flatness of models of certain shimura varieties of pel type consider a pel shimura variety associated to a unitary group that splits over an unramified extension of q p rapoport and zink have defined a model of the shimura variety over the ring of integers of the completion of the reflex field at a place lying over p with parahoric level structures at p we show that this model is flat as conjectured by rapoport and zink and that its special fibre is reduced

integrable peakon equations with cubic nonlinearity we present a new integrable partial differential equation found by vladimir novikov like the camassa holm and degasperis procesi equations this new equation admits peaked soliton peakon solutions but it has nonlinear terms that are cubic rather than quadratic we give a matrix lax pair for v novikov s equation and show how it is related by a reciprocal transformation to a negative flow in the sawada kotera hierarchy infinitely many conserved quantities are found as well as a bi hamiltonian structure the latter is used to obtain the hamiltonian form of the finite dimensional system for the interaction of n peakons and the two body dynamics n 2 is explicitly integrated finally all of this is compared with some analogous results for another cubic peakon equation derived by zhijun qiao

topological flatness of local models in the ramified case local models are schemes defined in terms of linear algebra which can be used to study the local structure of integral models of certain shimura varieties with parahoric level structure we investigate the local models for groups of the form open image in new window is a totally ramified extension as defined by pappas and rapoport and show that they are topologically flat in the linear case flatness can be deduced from this

optimal frames for erasures we study frames from the viewpoint of coding theory we introduce a numerical measure of how well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

stability of periodic peakons for the modified μ camassa holm equation abstract considered herein is the dynamical stability of periodic peaked solitons for the modified camassa holm equation with the cubic nonlinearity the equation is a version of the modified camassa holm equation and is integrable with the lax pair and bi hamiltonian structure the equation admits the periodic peakons it is shown that the periodic peakons are dynamically stable under small perturbations in the energy space by finding inequalities related to the first three conservation laws with global maximum and minimum of the solution

dynamics of director fields an asymptotic equation for weakly nonlinear hyperbolic waves governed by variational principles is derived and analyzed the equation is used to study a nonlinear instability in the director field of a nematic liquid crystal it is shown that smooth solutions of the asymptotic equation break down in finite time also constructed are weak solutions of the equation that are continuous despite the fact that their spatial derivative blows up

topological flatness of orthogonal local models in the split even case i local models are schemes defined in terms of linear algebra that were introduced by rapoport and zink to study the etale local structure of integral models of certain pel shimura varieties over p adic fields a basic requirement for the integral models or equivalently for the local models is that they be flat in the case of local models for even orthogonal groups genestier observed that the original definition of the local model does not yield a flat scheme in a recent article pappas and rapoport introduced a new condition to the moduli problem defining the local model the so called spin condition and conjectured that the resulting spin local model is flat we prove a preliminary form of their conjecture in the split iwahori case namely that the spin local model is topologically flat an essential combinatorial ingredient is the equivalence of admissibility and permissibility for two minuscule cocharacters in root systems of type d

the size of lp improving measures abstract a measure on the locally compact abelian group g is lp improving if for some 1 p p g l r g l space properties of such are discussed if is lp improving and non negative and f is a bounded borel function then f is lp improving the non negativity of and boundedness of f are essential as is shown in several ways in particular there exists on each infinite compact abelian group an lp improving such that is not lp improving the distribution function of every lp improving measure on the circle group is shown to satisfy a lipschitz condition but that is not sufficient for a measure to be lp improving except in restricted cases the gelfand transform on of an lp improving measure is shown to be small in a quantitative sense from that the strong continuity of lp improving measures follows the spectrum of an lp improving measure as an operator on lp is shown to be precisely the closure of related results and open problems are included

uniform estimates for fourier restriction to polynomial curves in ℝ d we prove uniform l p to l q bounds for fourier restriction to polynomial curves in bbb r d with affine arclength measure in the conjectured range

schubert varieties in twisted affine flag varieties and local models abstract our concern in this paper is the dimension and inclusion relations of schubert varieties in twisted partial affine flag varieties in the end we apply our results to some local models of certain shimura varieties

a generalized nonisospectral camassa holm equation and its multipeakon solutions abstract motivated by the paper of beals sattinger and szmigielski 2000 3 we propose an extension of the camassa holm equation which also admits the multipeakon solutions the novel aspect is that our approach is mainly based on classic determinant technique furthermore the proposed equation is shown to possess a nonisospectral lax pair

global weak solutions for a new periodic integrable equation with peakon solutions abstract we prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

some new integrable systems constructed from the bi hamiltonian systems with pure differential hamiltonian operators abstract when both hamiltonian operators of a bi hamiltonian system are pure differential operators we show that the generalized kupershmidt deformation gkd developed from the kupershmidt deformation in 10 offers an useful way to construct new integrable system starting from the bi hamiltonian system we construct some new integrable systems by means of the generalized kupershmidt deformation in the cases of harry dym hierarchy classical boussinesq hierarchy and coupled kdv hierarchy we show that the gkd of harry dym equation gkd of classical boussinesq equation and gkd of coupled kdv equation are equivalent to the new integrable rosochatius deformations of these soliton equations with self consistent sources we present the lax pair for these new systems therefore the generalized kupershmidt deformation provides a new way to construct new integrable systems from bi hamiltonian systems and also offers a new approach to obtain the rosochatius deformation of soliton equation with self consistent sources

well posedness wave breaking and peakons for a modified μ camassa holm equation considered herein is a modified periodic camassa holm equation with cubic nonlinearity which is called the modified camassa holm equation the proposed equation is shown to be formally integrable with the lax pair and bi hamiltonian structure local well posedness of the initial value problem to the modified camassa holm equation in the besov space is established existence of peaked traveling wave solutions and formation of singularities of solutions for the equation are then investigated it is shown that the equation admits a single peaked soliton and multi peakon solutions with a similar character of the camassa holm equation singularities of the solutions can occur only in the form of wave breaking and several wave breaking mechanisms for solutions with certain initial profiles are described in detail

on the blow up structure for the generalized periodic camassa holm and degasperis procesi equations considered herein are the generalized camassa holm and degasperis procesi equations in the spatially periodic setting the precise blow up scenarios of strong solutions are derived for both of equations several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established the exact blow up rates are also determined finally geometric descriptions of these two integrable equations from non stretching invariant curve flows in centro equiaffine geometries pseudo spherical surfaces and affine surfaces are given

universal l p improving for averages along polynomial curves in low dimensions we prove sharp lp lq estimates for averaging operators along general polynomial curves in two and three dimensions these operators are translation invariant given by convolution with the so called affine arclength measure of the curve and we obtain universal bounds over the class of curves given by polynomials of bounded degree our method relies on a geometric inequality for general vector polynomials together with a combinatorial argument due to m christ almost sharp lorentz space estimates are obtained as well

global solutions to a special case of the generalized weakly dissipative periodic two component camassa holm system abstract we establish global well posedness to the generalized weakly dissipative periodic two component camassa holm system in some special case

continuous properties of the data to solution map for a generalized μ camassa holm integrable equation this work studies a generalized type integrable equation with both quadratic and cubic nonlinearities the camassa holm and modified camassa holm equations are members of this family of equations it has been shown that the cauchy problem for this generalized camassa holm integrable equation is locally well posed for initial data u0 hs s 5 2 in this work we further investigate the continuity properties to this equation it is proved in this work that the data to solution map of the proposed equation is not uniformly continuous it is also found that the solution map is holder continuous in the hr topology when 0 r 5 2 in this work we further investigate the continuity properties to this equation it is proved in this work that the data to solution map of the proposed equation is not uniformly continuous it is also found that the solution map is holder continuous in the hr topology when 0 r s with holder exponent depending on both s and r

the pythagorean theorem i the finite case the pythagorean theorem and variants of it are studied the variations evolve to a formulation in terms of noncommutative conditional expectations on von neumann algebras that displays the theorem as the basic result of noncommutative metric euclidean geometry the emphasis in the present article is finite dimensionality both discrete and continuous

non uniform dependence on initial data for the modified μ camassa holm equation considered herein is the initial value problem for the modified camassa holm equation it is shown that the solution map of this problem is not uniformly continuous in sobolev spaces hs s for s 5 2 on the circle based on the well posedness result and the lifespan for this problem the method of approximate solutions is utilized

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

the camassa holm equation a loop group approach abstract a map is presented that associates with each element of a loop group a solution of an equation related by a simple change of coordinates to the camassa holm ch equation certain simple automorphisms of the loop group give rise to backlund transformations of the equation these are used to find 2 soliton solutions of the ch equation as well as some novel singular solutions

on the solvability of the painleve vi equation a rigorous method was introduced by fokas and zhou 1992 for studying the riemann hilbert problem associated with the painleve ii and iv equations the same methodology has been applied to the painleve i iii and v equations in this paper we will apply the same methodology to the painleve vi equation we will show that the cauchy problem for the painleve vi equation admits in general a global meromorphic solution in t furthermore the special solution which can be written in terms of a hypergeometric function is obtained via solving the special case of the riemann hilbert problem

the korteweg de vries hierarchy as dynamics of closed curves in the plane the korteweg de vries modified korteweg de vries and harry dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane these purely local dynamics conserve an infinite number of global geometric properties of the curves such as perimeter and enclosed area several techniques used to study these integrable systems are shown to have simple differential geometric interpretations connections with incompressible inviscid fluid flow in two dimensions are suggested

blow up solutions and peakons to a generalized μ camassa holm integrable equation considered here is a generalized type integrable equation which can be regarded as a generalization to both the camassa holm and modified camassa holm equations it is shown that the proposed equation is formally integrable with the lax pair and the bi hamiltonian structure and its scale limit is an integrable model of hydrodynamical systems describing short capillary gravity waves local well posedness of the cauchy problem in the suitable sobolev space is established by the viscosity method existence of peaked traveling wave solutions and formation of singularities of solutions for the equation are investigated it is found that the equation admits single and multi peaked traveling wave solutions the effects of varying camassa holm and modified camassa holm nonlocal nonlinearities on blow up criteria and wave breaking are illustrated in detail our analysis relies on the method of characteristics and conserved quantities and is proceeded with a priori differential estimates

restriction of the fourier transform to some complex curves abstract the purpose of this paper is to prove a fourier restriction estimate for certain 2 dimensional surfaces in r 2 d d 3 these surfaces are defined by a complex curve z of simple type which is given by a mapping of the form z z z z 2 z d 1 z where z is an analytic function on a domain c this is regarded as a real mapping z x y x y from r 2 to r 2 d our results cover the case z z n for any nonnegative integer n in all dimensions d 3 the main result is a uniform estimate valid when d 3 where z may be taken to be an arbitrary polynomial of degree at most n it is uniform in the sense that the operator norm is independent of the coefficients of the polynomial these results are analogues of the uniform restricted strong type estimates in 5 valid for polynomial curves of simple type and some other classes of curves in r d d 3

uniform estimates for the x ray transform restricted to polynomial curves abstract we establish near optimal mixed norm estimates for the x ray transform restricted to polynomial curves with a weight that is a power of the affine arclength the bounds that we establish depend only on the spatial dimension and the degree of the polynomial some of our results are new even in the well curved case

fourier restriction to polynomial curves i a geometric inequality we prove a fourier restriction result for general polynomial curves in bbb r d measuring the fourier restriction with respect to the affine arclength measure of the curve we obtain a universal estimate for the class of all polynomial curves of bounded degree our method relies on establishing a geometric inequality for general polynomial curves which is of interest in its own right applications of this geometric inequality to other problems in euclidean harmonic analysis have recently been established

on inverse recursion operator and tri hamiltonian formulation for a kaup newell system of dnls equations an inverse of the recursion operator is computed and a clear presentation of a tri hamiltonian formulation is provided for a kaup newell system of derivative nonlinear schrodinger dnls equations therefore all kaup newell systems in the whole hierarchy of dnls equations are tri hamiltonian and have an inverse hierarchy of common commuting symmetries

global existence and blow up phenomena for the degasperis procesi equation this paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the degasperis procesi equation on the line global strong solutions to the equation are determined for a class of initial profiles on the other hand it is shown that the first blow up can occur only in the form of wave breaking a new wave breaking mechanism for solutions is described in detail and two results of blow up solutions with certain initial profiles are established

alcoves associated to special fibers of local models the special fiber of the local model of a pel shimura variety with iwahori type level structure admits a cellular decomposition the set of strata is in a natural way a finite subset of the affine weyl group determined by the shimura data in this paper we generalize and give a new proof of a theorem of kottwitz and rapoport concerning the description of this set for the case of linear and symplectic groups for higher rank groups not of type an we produce counterexamples showing the situation is not as nice in general we also recover a theorem of deodhar characterizing the bruhat order on sn the local geometry can be approached using the rapoport zink local model m j 13 this protective op scheme is a local model for the singularities in the special fiber of sh g x9 k 9 but it is defined in terms of the pair g p using linear algebra and is somewhat easier to deal with than the shimura variety itself in this paper we address certain combinatorial questions which arise in the study of the special fiber m f of m to fix ideas consider the case where g is split it turns out that m f can be regarded as a finite dimensional union of iwahori orbits in the affine flag variety for g fp t see 5 and 8 it follows that m f has a stratification indexed by a finite set perm p of the extended affine weyl group w g for g equivalently perm p can be regarded as a finite set of alcoves in the affine coxeter complex determined by a splitting for g let wo denote the finite weyl group of g for each translation in the wo orbit

topological flatness of local models for ramified unitary groups i the odd dimensional case abstract local models are certain schemes defined in terms of linear algebraic moduli problems which give etale local neighborhoods of integral models of certain p adic pel shimura varieties defined by rapoport and zink when the group defining the shimura variety ramifies at p the local models and hence the shimura models as originally defined can fail to be flat and it becomes desirable to modify their definition so as to obtain a flat scheme in the case of unitary similitude groups whose localizations at q p are ramified quasi split gu n pappas and rapoport have added new conditions the so called wedge and spin conditions to the moduli problem defining the original local models and conjectured that their new local models are flat we prove a preliminary form of their conjecture namely that their new models are topologically flat in the case n is odd

extended affine surface area during the twentieth century many of the notions of classical differential geometry have been extended so as to be defined on the boundaries of arbitrary convex bodies without smoothness requirements for example the projection measures quermassintegrals are the natural extensions of the integrals of mean curvature see santa16 27 pp 215 2321 for a discussion the surface area measures of aleksandrov fenchel jessen are extensions of the indefinite integrals of the elementary symmetric functions of the principal radii of curvature while federer s curvature measures when restricted to convex hypersurfaces are extensions of the indefinite integrals of the elementary symmetric functions of the principal curvatures see schneider 30 pp 27 331 one of the important concepts of classical differential geometry which has been difficult to extend is afline surface area this has often led to difficulties which when circumvented have been done so only by considerable effort and ingenuity for example suppose we are given a measure on the unit sphere which satisfies the hypothesis of the minkowski problem the solution of the minkowski problem guarantees the existence of a convex body whose surface area measure is the given measure from the given measure we can find the surface area of the body the minkowski mixed volume inequality can be used to obtain various upper bounds for the volume of the body however we cannot even inquire about the affine surface area of the body unless we can somehow convince ourselves that the convex body is sufficiently smooth results regarding aftine surface area have a surprising number of applications for example a number of geometric inequalities which appear to have no connection with affine surface area are in fact consequences of inequalities involving afline surface area see for example petty 25 and also 151 since polytopes have zero afftne surface area it seems quite surprising that the notion of afline surface area arises frequently in the area of polytope approximation see the survey of gruber 9 recently leichtweiss 11 121 presented a definition of extended afftne

an integrable semi discretization of the camassa holm equation and its determinant solution an integrable semi discretization of the camassa holm ch equation is presented the keys of its construction are bilinear forms and determinant structure of solutions of the ch equation determinant formulas of n soliton solutions of the continuous and semi discrete camassa holm equations are presented based on determinant formulas we can generate multi soliton multi cuspon and multi soliton cuspon solutions numerical computations using the integrable semi discrete camassa holm equation are performed it is shown that the integrable semi discrete camassa holm equation gives very accurate numerical results even in the cases of cuspon cuspon and soliton cuspon interactions the numerical computation for an initial value condition which is not an exact solution is also presented

k symplectic structures the basic properties of k symplectic structures are introduced and developed in analogy with the well known symplectic differential geometry examples of such structures related to k symplectic lie algebras are given

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

generalizations of the camassa holm equation we classify generalized camassa holm type equations which possess infinite hierarchies of higher symmetries we show that the obtained equations can be treated as negative flows of integrable quasi linear scalar evolution equations of orders 2 3 and 5 we present the corresponding lax representations or linearization transformations for these equations some of the obtained equations seem to be new

global weak solutions and blow up structure for the degasperis procesi equation in this paper we study several qualitative properties of the degasperis procesi equation we first established the precise blow up rate and then determine the blow up set of blow up strong solutions to this equation for a large class of initial data we finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions

the multiscale expansions of difference equations in the small lattice spacing regime and a vicinity and integrability test i we propose an algorithmic procedure i to study the distance between an integrable pde and any discretization of it in the small lattice spacing regime and at the same time ii to test the asymptotic integrability properties of such discretization this method should provide in particular useful and concrete information on how good is any numerical scheme used to integrate a given integrable pde the procedure illustrated on a fairly general ten parameter family of discretizations of the nonlinear schr dinger equation consists of the following three steps i the construction of the continuous multiscale expansion of a generic solution of the discrete system at all orders in following degasperis et al 1997 physica d 100 187 211 ii the application to such an expansion of the degasperis procesi dp integrability test degasperis a and procesi m 1999 asymptotic integrability symmetry and perturbation theory spt98 ed a degasperis and g gaeta singapore world scientific pp 23 37 degasperis a 2001 multiscale expansion and integrability of dispersive wave equations lectures given at the euro summer school what is integrability isaac newton institute cambridge uk 13 24 august integrability lecture notes in physics vol 767 ed a mikhailov berlin springer to test the asymptotic integrability properties of the discrete system and its distance from its continuous limit iii the use of the main output of the dp test to construct infinitely many approximate symmetries and constants of motion of the discrete system through novel and simple formulas

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

orbital stability of periodic peakons to a generalized μ camassa holm equation in this paper we study the orbital stability of the periodic peaked solitons of the generalized camassa holm equation with nonlocal cubic and quadratic nonlinearities the equation is a version of a linear combination of the camassa holm equation and the modified camassa holm equation it is also integrable with the lax pair and bi hamiltonian structure and admits the single peakons and multi peakons by constructing an inequality related to the maximum and minimum of solutions with the conservation laws we prove that even in the case that the camassa holm energy counteracts in part the modified camassa holm energy the shapes of periodic peakons are still orbitally stable under small perturbations in the energy space

wave breaking and peakons for a modified camassa holm equation in this paper we investigate the formation of singularities and the existence of peaked traveling wave solutions for a modified camassa holm equation with cubic nonlinearity the equation is known to be integrable and is shown to admit a single peaked soliton and multi peakon solutions of a different character than those of the camassa holm equation singularities of the solutions can occur only in the form of wave breaking and a new wave breaking mechanism for solutions with certain initial profiles is described in detail

invariant submanifold flows given a lie group acting on a manifold our aim is to analyze the evolution of differential invariants under invariant submanifold flows the constructions are based on the equivariant method of moving frames and the induced invariant variational bicomplex applications to integrable soliton dynamics and to the evolution of differential invariant signatures used in equivalence problems and object recognition and symmetry detection in images are discussed

finite propagation speed for the camassa holm equation we prove that any classical solution of the camassa holm equation will have compact support if its initial data has this property

single and multi solitary wave solutions to a class of nonlinear evolution equations abstract in this paper an effective discrimination algorithm is presented to deal with equations arising from physical problems the aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations many examples are given to illustrate the algorithm at the same time some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations such as camassa holm equation kolmogorov petrovskii piskunov equation and so on then a direct constructive method called multi auxiliary equations expansion method is described to derive the multi solitary wave solutions of nonlinear evolution equations finally a class of novel multi solitary wave solutions of the 2 1 dimensional asymmetric version of the nizhnik novikov veselov equation are given by three direct methods the algorithm proposed in this paper can be steadily applied to some other nonlinear problems

integrable equations arising from motions of plane curves the motion of plane curves in klein geometry is studied it is shown that the kdv harry dym sawada kotera burgers the defocusing mkdv hierarchies the camassa holm and the kaup kupershmidt equation naturally arise from the motions of plane curves in sl 2 sim 2 sa 2 and a 2 geometries these local and nonlocal dynamics conserve global geometric quantities of curves such as perimeter and enclosed area motions of curves in euclidean special linear and similarity geometries corresponding to the traveling wave solutions of the mkdv kdv and burgers equations are discussed

some tricks from the symmetry toolbox for nonlinear equations generalizations of the camassa holm equation abstract the main subject of the paper is to give a survey and to present new methods on how integrability results i e results for symmetry groups inverse scattering formulations action angle transformations and the like can be transferred from one equation to others in case the equations are not related by backlund transformations as a main example the so called camassa holm equation is chosen for which the relevant results are obtained by having a look on the korteweg de vries kdv equation the camassa holm equation turns out to be a different factorization equation of the kdv it describes shallow water waves and reconciles the properties which were known for different orders of shallow water wave approximations we follow here an old method already marginally mentioned in fuchssteiner and fokas 1981 and fuchssteiner 1983 and recently applied by others olver and rosenau 1995 the method allows an immediate recovery of the recursion operator for the camassa holm equation from the invariance structure of the kdv although both equations are not related by backlund transformations however in addition and different from other approaches from there by use of the squared eigenfunction relation for the kdv equation the lax pair formulation for the different factorization equation is derived for the example under consideration it is of course the one obtained in camassa and holm 1993 since the methods proposed here can be transferred to any compatible factorization of recursion operators its application even in the special case which was chosen for illustration leads to a large class of integrable equations among which the camassa holm equation can be found as well as a three parameter family of generalizations of the equation the advantage of the general approach to the lax pair presentation is that direct transformations between action and angle variables are obtained so using this lax pair formulation as a novel result a direct transformation between action and angle variables for the camassa holm equation is derived further novel results in the paper are a hodograph link back from a backlund transformation of the camassa holm equation to a particular member of the kdv hierarchy additional symmetries and the construction of the conformal algebra for the hierarchy of the camassa holm equation the methods involved include hereditary operators bi hamiltonian formulations nilpotent flows and scaling symmetries

l p improving bounds for averages along curves we establish local l p l q mapping properties for averages on curves the exponents are sharp except for endpoints let n 2 and let m1 and m2 be two smooth n 1 dimensional manifolds 1 each containing a preferred origin 0m1 and 0m2 we shall abuse notation and write 0 for both 0m1 and 0m2 for the purposes of integration we shall place a smooth riemannian metric on m1 and m2 although the exact choice of this metric will not be relevant all our considerations shall be local to the origin 0 we are interested in the local l p improving properties of averaging operators on curves before we give the rigorous description of these operators let us first give an informal discussion informally we assume that we have a smooth assignment x2 7 x2 taking points in m2 to curves in m1 with a corresponding dual assignment x1 x1 taking points in m1 to curves in m2 such that x1 x2 x2 1 we then form the operator r taking functions 2 on m1 to functions on m2 defined

well posedness of a class of solutions to an integrable two component camassa holm system abstract in this paper we study a class of solutions to the cauchy problem for an integrable two component camassa holm system with the initial data u 0 0 1 h 1 r w 1 r l 2 r l r based on characteristics we study a corresponding ode and obtain a unique local solution by applying the contraction mapping principle we prove local existence and uniqueness of the solution to the camassa holm system by constructing a solution obtained from the ode and studying the regularity of the solution finally we show continuous dependence of the solution on the initial data in some weak sense

on adapted coordinate systems the notion of an adapted coordinate system for a given real analytic function introduced by v i arnol d plays an important role for instance in the study of asymptotic expansions of oscillatory integrals in two dimensions a n varchenko gave sufficient conditions for the adaptness of a given coordinate system and proved the existence of an adapted coordinate system for analytic functions without multiple components varchenko s proof is based on a two dimensional resolution of singularities result in this article we present a more elementary approach to these results which is based on the puiseux series expansion of roots of the given function this approach is inspired by the work of d h phong and e m stein on the newton polyhedron and oscillatory integral operators it applies to arbitrary real analytic functions and even to arbitrary smooth functions of finite type in particular we show that varchenko s conditions are in fact necessary and sufficient for the adaptedness of a given coordinate system and that adapted coordinates always exist in two dimensions even in the smooth finite type setting for analytic functions a construction of adapted coordinates by means of puiseux series expansions of roots has already been carried out in work by d h phong e m stein and j a sturm on the growth and stability of real analytic function as we learned after the completion of this paper in contrast to their work however our proof more closely follows varchenko s algorithm for the construction of an adapted coordinate system which turns out to be useful for the extension to the smooth setting

integrable evolution equations on spaces of tensor densities and their peakon solutions we study a family of equations defined on the space of tensor densities of weight on the circle and introduce two integrable pde one of the equations turns out to be closely related to the inviscid burgers equation while the other has not been identified in any form before we present their lax pair formulations and describe their bihamiltonian structures we prove local wellposedness of the corresponding cauchy problem and include results on blow up as well as global existence of solutions moreover we construct peakon and multi peakon solutions for all 0 1 and shock peakons for 3 we argue that there is a natural geometric framework for these equations that includes other well known integrable equations and which is based on v arnold s approach to euler equations on lie groups

computing the alternating trace of frobenius on the sheaves of nearby cycles on local models for gl4 and gl5 abstract consider a shimura variety defined over some number field and assume we have a model over the ring of integers at some prime of bad reduction it is then interesting to know the alternating trace of frobenius on the invariants under the inertia group of the sheaf of nearby cycles since these traces are related to the local factor of the hasse weil zeta function in this article we compute the semi simple alternating trace of frobenius for shimura varieties associated to the groups gu 2 2 and gu 3 2 with level structure of iwahori type by investigating the equations of the local model defined by rapoport and zink and by performing explicit blowing ups