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

analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

on hotelling s formula for the volume of tubes and naiman s inequality abstract motivation by the question of testing for a nonlinear parameter in a regression model with independent homoscedastic normal residuals hotelling 1939 was led to consider the geometric problem of computing the volume of a tube of given radius around a curve in s to the n 1 power the unit sphere in ir to the nth power the answer involves only the arc length of the curve and not its curvature providing the radius of the tube is sufficiently small that there is no self overlap in the tube starting from a somewhat different statistical setting naiman 1986 arrived at the same geometric problem and showed that hotelling s result properly interpreted is an upper bound for the volume of a tube of arbitrary radius the purpose of this note is to give two new derivations of the hotelling naiman results the first involves differential inequalities the second is probabilistic using the concept of upcrossing borrowed from the theory of gaussian processes in the context of gaussian processes knowles 1987 has observed that approximations obtained from hotelling s result and bounds derived via upcrossings are related

a note on parameter effects curvature abstract the parameter effects curvature measure proposed by bates and watts 1980 is examined for a growth model and the fieller creasy problem exact confidence regions are constructed and compared to linear approximation regions for the growth model the agreement between the regions is good despite high curvature in the fieller creasy problem it is shown that the agreement can be quite poor despite low curvature

confidence regions in linear functional relationships a unified approach to deriving confidence regions in linear functional relationship models is presented based on the conditional likelihood ratio method of knowles siegmund and zhang in the case of a single latent predictor the confidence region for the slope produced by this approach is the familiar one of fieller and creasy however here it is shown how to derive a confidence region for the slope when it is known that the slope is positive that improves on merely intersecting the region for an unrestricted slope with 0 oo a geometric interpretation is given for fieller creasy confidence region for the ratio of population means fieller creasy problem regions are also derived for simultaneous estimation of the slope and intercept in the model with a single latent predictor and for the slopes in a model with two latent predictors

blow up phenomena and persistence property for the modified b family of equations abstract in this paper we study the blow up mechanism and persistence property of solutions to the modified b family of equations the dynamics of the blow up quantity along the characteristics is established by the riccati type differential inequality with various parameters the key feature of the method is to refine the analysis on the growth rate of the relative ratio between solution and its gradient by performing a vertical shift furthermore the persistence results for the solution are established in weighted spaces

on leibniz series defined by convex functions abstract it is shown that for every 0 we have k n 1 1 k 1 k 1 2 n n for some strictly decreasing sequence n n 1 such that 1 2 n 1 2 1 1 2 n 1 1 hence with lim n n 1 2 this is only a particular case of more general new results on leibniz series defined by convex functions

local in space criteria for blowup in shallow water and dispersive rod equations we unify a few of the best known results on wave breaking for the camassa holm equation by r camassa a constantin j escher l holm j hyman and others in a single theorem a sufficient condition for the breakdown is that u 0 u 0 is strictly negative in at least one point x 0 in mathbb r such blowup criterion looks more natural than the previous ones as the condition on the initial data is purely local in the space variable our method relies on the introduction of two families of lyapunov functions contrary to mckean s necessary and sufficient condition for blowup our approach applies to other equations that are not integrable we illustrate this fact by establishing new local in space blowup criteria for an equation modeling nonlinear dispersive waves in elastic rods

the nonexistence of 100 1 alpha confidence sets of finite expected diameter in errors in variables and related models on montre que pour certains modeles il est impossible de construire des intervalles de confiance pour les parametres cles qui aient a la fois une confiance positive et une longueur esperee finie

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

blow up issues for a two component system modelling water waves with constant vorticity abstract considered herein is the blow up mechanism to the two component system which models two dimensional shallow water waves with constant vorticity we first derive the precise blow up scenario for strong solutions to the system then we establish the new local in space blow up results simplifying and extending the earlier blow up criterion for this system finally the propagation speed of strong solutions has been investigated

on a class of physically important integrable equations abstract a methodology introduced by fuchssteiner and the author is used to derive a class of physically important integrable evolution equations among these equations are integrable generalizations of the korteweg devries kdv of the modified kdv of the nonlinear schrodinger nls and of the sine gordon equations the modeling of water waves as well as general asymptotic considerations are used to illustrate the occurrence of the generalized modified kdv and nls equations respectively

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

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

cesaro s theorems for complex sequences abstract the classical cesaro s theorems apply to ratios of real valued sequences like n g n where g n monotonically or n 0 and g n 0 monotonically here such results are generalized to complex valued sequences and at the same time the restriction of monotonicity on g n is removed in the real case such a condition is replaced with a more general property that does not require the ordering of the range an application to the asymptotics of linear difference equations is presented

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

analysis on the blow up of solutions to a class of integrable peakon equations abstract we investigate the blow up mechanism of solutions to a class of quasilinear integrable equations which could possess peakons the dynamics of the blow up quantity along the characteristics is established by the riccati type differential inequality which involves the interaction among three parts a local nonlinearity a nonlocal term and a term stemming from the weak linear dispersion to analyse the interplay among these quantities we provide two different approaches the first one is designed for the case when the equations do not exhibit a weak linear dispersion and hence focuses on the interplay between the first two parts the method is based on a refined analysis on either evolution of the solution u and its gradient u x that is c u u x or the growth rate of the relative ratio u x u the second one handles the general situation when all of three parts are present the idea is to extract the truly blow up component from the riccati type differential inequality and utilizes the morawetz type identity or higher order conservation laws to show that such a component blows up in finite time before the other component degenerates

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

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

blow up solutions for the modified b family of equations abstract in this paper we firstly derive the blow up criterion and the lower bound of the maximal time of existence to the modified b family of equations the new blow up mechanism for the solutions with certain initial profiles and various parameters is then described in detail

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

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

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

generic case complexity decision problems in group theory and random walks abstract we give a precise definition of generic case complexity and show that for a very large class of finitely generated groups the classical decision problems of group theory the word conjugacy and membership problems all have linear time generic case complexity we prove such theorems by using the theory of random walks on regular graphs

a new integrable equation with cuspons and w m shape peaks solitons in this paper we propose a new completely integrable wave equation mt mx u2 ux2 2m2ux 0 m u uxx the equation is derived from the two dimensional euler equation and is proven to have lax pair and bi hamiltonian structures this equation possesses new cusp solitons cuspons instead of regular peakons ce x ct with speed c through investigating the equation we develop a new kind of soliton solutions w m shape peaks solitons there exist no smooth solitons for this integrable water wave equation