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

vector bundles over configuration spaces of nonidentical particles topological potentials and internal degrees of freedom we consider configuration spaces of nonidentical pointlike particles the physically motivated assumption that any two particles cannot be located at the same point in space time leads to nontrivial topological structure of the configuration space for a quantum mechanical description of such a system we classify complex vector bundles over the configuration space and obtain potentials of topological origin similar to those that occur in the fiber bundle approach to dirac s magnetic monopole or in yang mills theory

non abelian quantum statistics on graphs we show that non abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces in particular we formulate a general framework for describing quantum statistics of particles constrained to move in a topological space x the framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of x which can be achieved by determining its homology groups we apply this methodology for configuration spaces of graphs as a conclusion we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non abelian anyons on networks that are used in quantum computing these conclusions are based on our solution of the so called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs

blow up blow up rate and decay of the solution of the weakly dissipative camassa holm equation in this paper we mainly study several problems on the weakly dissipative periodic camassa holm equation at first the local well posedness of the equation is obtained by kato s theorem a necessary and sufficient condition of the blow up of the solution and some criteria guaranteeing the blow up of the solution are established then the blow up rate of the solution is discussed moreover we prove that the equation has global solutions and these global solutions decay to zero as time goes to infinite provided the potentials associated to their initial date are of one sign

wave breaking for the periodic weakly dissipative dullin gottwald holm equation abstract in this paper we consider the periodic weakly dissipative dullin gottwald holm equation the present work is mainly concerned with blow up phenomena for the cauchy problem for this new kind of equation we apply the optimal constant to give sufficient conditions via an appropriate integral form of the initial data which guarantee the finite time singularity formation for the corresponding solution

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

wave breaking for a periodic shallow water equation the focus of this paper is on the blow up of a recently derived one dimensional shallow water equation which is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equations in the shallow water regime some new criteria guaranteeing the development of singularities in finite time for strong solutions with regular initial data are obtained for the periodic case

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

global existence of solution to the camassa holm equation abstract in this paper we prove the result of global existence of solution to camassa holm equation under the assumption that v 0 u 0 u 0 has unique zero point and that v 0 is negative at in this situation v 0 clearly changes sign thus our result completes the picture described in ann scuola norm sup pisa cl sci 4 xxvi 1998 303 where the global existence of solutions is shown under the assumption that v 0 does not change sign we also give the sufficient and necessary condition of global existence of the solution when v 0 is odd with unique zero point

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

a covariant approach to the quantization of a rigid body this paper concerns the quantization of a rigid body in the framework of covariant quantum mechanics on a curved spacetime with absolute time we consider the configuration space of n classical particles as the n fold product of the configuration space of one particle then we impose a rigid constraint and the resulting space is dealt with as a configuration space of a single abstract particle this classical framework turns out to be suitable for the formulation of covariant quantum mechanics according to this scheme thus we quantize such a particle accordingly this scheme can model e g the quantum dynamics of extremely cold molecules we provide a new mathematical interpretation of two valued wavefunctions on so 3 in terms of single valued sections of a new non trivial quantum bundle these results have clear analogies with spin

global existence and blow up phenomena for a weakly dissipative periodic 2 component camassa holm system we first establish local well posedness for a weakly dissipative periodic 2 component camassa holm system we then present two global existence results for strong solutions to the equation we finally obtain several blow up results and the blow up rate of strong solutions to the equation

weakly damped forced korteweg de vries equations behave as a finite dimensional dynamical system in the long time abstract it is shown that a weak dissipation of order zero is sufficient to impose a finite dimensional behavior for korteweg de vries equations with periodic boundary conditions a universal attractor which captures all the trajectories is constructed its fractal dimension is proved to be finite

on the weak solutions to a shallow water equation we obtain the existence of global in time weak solutions to the cauchy problem for a one dimensional shallow water equation that is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equation in the shallow water regime the solution is obtained as a limit of viscous approximation the key elements in our analysis are some new a priori one sided supernorm and space time higher norm estimates on the first order derivatives 2000 john wiley sons inc

on a dissipative form of camassa holm equation in this paper we consider several problems on the camassa holm equation with a weak dissipation term first we establish the local well posedness result and then we present several new blow up results and discuss the global existence of the solution for the periodic case finally we examine the persistence of decay properties for the weakly dissipative camassa holm equation and we prove that any nontrivial classical solution of the equation will not have compact support if its initial data have this property

blow up phenomena for the weakly dissipative dullin gottwald holm equation revisited in this paper we investigate the cauchy problem for the weakly dissipative dullin gottwald holm equation we first establish the local well posedness result by kato s semigroup theory then we obtain the precise blow up scenario and present some blow up results for strong solutions to the equation finally we discuss the blow up rate of the wave breaking solutions this result complements the early one in the literature such as novruzov j math phys 54 092703 2013

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

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

blowup and blowup rate of solutions to a weakly dissipative periodic rod equation in this paper we first establish the local well posedness for a weakly dissipative periodic rod equation we then present a precise blowup scenario and several blowup results of strong solutions to the equation we finally give the blowup rate of strong solutions to the equation when blowup occurs our blowup results for the weakly dissipative periodic rod equation improve considerably earlier results for the periodic rod equation

averaging techniques for the a posteriori bem error control for a hypersingular integral equation in two dimensions averaging techniques or gradient recovery techniques are frequently employed tools for the a posteriori finite element error analysis their very recent mathematical justification for partial differential equations allows unstructured meshes and nonsmooth exact solutions this paper establishes an averaging technique for the hypersingular integral equation on a one dimensional boundary and presents numerical examples that show averaging techniques can be employed for an effective mesh refining algorithm for the discussed test examples the provided estimator estimates the in general unknown error very accurately in the sense that the quotient error estimator stays bounded with a value close to 1