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 a mixture autoregressive model we generalize the gaussian mixture transition distribution gmtd model introduced by le and co workers to the mixture autoregressive mar model for the modelling of non linear time series the models consist of a mixture of k stationary or non stationary ar components the advantages of the mar model over the gmtd model include a more full range of shape changing predictive distributions and the ability to handle cycles and conditional heteroscedasticity in the time series the stationarity conditions and autocorrelation function are derived the estimation is easily done via a simple em algorithm and the model selection problem is addressed the shape changing feature of the conditional distributions makes these models capable of modelling time series with multimodal conditional distributions and with heteroscedasticity the models are applied to two real data sets and compared with other competing models the mar models appear to capture features of the data better than other competing models do

on the cauchy problem for the integrable modified camassa holm equation with cubic nonlinearity abstract considered in this paper is the modified camassa holm equation with cubic nonlinearity which is integrable and admits the single peaked solitons and multi peakon solutions the short wave limit of this equation is known as the short pulse equation the main investigation is the cauchy problem of the modified camassa holm equation with qualitative properties of its solutions it is firstly shown that the equation is locally well posed in a range of the besov spaces the blow up scenario and the lower bound of the maximal time of existence are then determined a blow up mechanism for solutions with certain initial profiles is described in detail and nonexistence of the smooth traveling wave solutions is also demonstrated

two point taylor expansions and one dimensional boundary value problems we consider second order linear differential equations x y f x y g x y h x in the interval 1 1 with dirichlet neumann or mixed dirichlet neumann boundary conditions we consider x f x g x and h x analytic in a cassini disk with foci at x 1 containing the interval 1 1 the two point taylor expansion of the solution y x at the extreme points 1 is used to give a criterion for the existence and uniqueness of solution of the boundary value problem this method is constructive and provides the two point taylor approximation of the solution s when it exists

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

blow up phenomena and local well posedness for a generalized camassa holm equation with cubic nonlinearity abstract in this paper we consider the cauchy problem for a generalized camassa holm equation with cubic nonlinearity in besov spaces we first establish the local well posedness of the equation in the besov space b p r s by using the littlewood paley theory then under a sign condition we reach the sign preserved property and a precise blow up criterion applying this precise criterion we present a blow up result and the precise blow up rate for strong solutions to the equation

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

global wellposedness of cubic camassa holm equations abstract in this paper we study the cauchy problem of cubic camassa holm equation this equation is derived from a model of shallow water dynamics we prove the global existence of entropy weak solution to this problem in space h 1 with its derivative in b v the stability and uniqueness of entropy weak solution are obtained in w 1 1

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

ill posedness of the camassa holm and related equations in the critical space abstract we prove norm inflation and hence ill posedness for a class of shallow water wave equations such as the camassa holm equation degasperis procesi equation and novikov equation etc in the critical sobolev space h 3 2 and even in the besov space b p r 1 1 p for p 1 r 1 our results cover both real line and torus cases only real line case for novikov solving an open problem left in the previous works 5 14 16

a student t mixture autoregressive model with applications to heavy tailed financial data we introduce the class of student t mixture autoregressive models which is promising for financial time series modelling the model is able to capture serial correlations time varying means and volatilities and the shape of the conditional distributions can be time varied from short tailed to long tailed or from unimodal to multimodal the use of t distributed errors in each component of the model allows conditional leptokurtic distributions that account for the commonly observed excess unconditional kurtosis in financial data methods of parameter estimation and model selection are given finally the proposed modelling procedure is illustrated through a real example

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

hyperasymptotics and hyperterminants exceptional cases a new method is introduced for the computation of hyperterminants it is based on recurrence relations and can also be used to compute the parameter derivatives of the hyperterminants these parameter derivatives are needed in hyperasymptotic expansions in exceptional cases numerical illustrations and an application are included

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

remarks on the well posedness of camassa holm type equations in besov spaces abstract in this paper we prove the solution map of the cauchy problem of camassa holm type equations depends continuously on the initial data in nonhomogeneous besov spaces in the sense of hadamard by using the littlewood paley theory and the method introduced by kato 37 and danchin 21

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

exponentially improved asymptotic solutions of ordinary differential equations ii irregular singularities of rank one re expansions are found for the optimal remainder terms in the well known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one the re expansions are in terms of generalized exponential integrals and have greater regions of validity than the original expansions as well being considerably more accurate and providing a smooth interpretation of the stokes phenomenon they are also of strikingly simple form in addition explicit asymptotic expansions for the higher coefficients of the original asymptotic solutions are obtained

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

weak well posedness for the integrable modified camassa holm equation with the cubic nonlinearity abstract the goal of this paper is supposed to investigate the weak well posedness for the modified camassa holm equation with cubic nonlinearity after given an initial data u 0 h s r w 2 r 1 s 2 firstly we deduce an ode system whose equivalence to problem 1 2 with a spatial initial condition will be verified later next by studying the existence of the unique solution to the ode system we conclude the corresponding results of the original problem finally we end up with the weak continuity of solutions to the pde associated to the initial datum

on the geometric approach to the motion of inertial mechanical systems according to the principle of least action the spatially periodic motions of one dimensional mechanical systems with no external forces are described in the lagrangian formalism by geodesics on a manifold configuration space the group d of smooth orientation preserving diffeomorphisms of the circle the periodic inviscid burgers equation is the geodesic equation on d with the l 2 right invariant metric however the exponential map for this right invariant metric is not a c 1 local diffeomorphism and the geometric structure is therefore deficient on the other hand the geodesic equation on d for the h 1 rightinvariant metric is also a re expression of a model in mathematical physics we show that in this case the exponential map is a c 1 local diffeomorphism and that if two diffeomorphisms are sufficiently close on d t heycan be joined by a unique length minimizing geodesic a state of the system is transformed to another nearby state by going through a uniquely determined flow that minimizes the energy we also analyse for both metrics the breakdown of the geodesic flow

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

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

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

hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one a sequence of re expansions is developed for the remainder terms in the well known poincare series expansions of the solutions of homogeneous linear differential equations of higher order in the neighbourhood of an irregular singularity of rank one these re expansions are a series whose terms are a product of stokes multipliers coefficients of the original poincare series expansions and certain multiple integrals the so called hyperterminants each step of the process reduces the estimate of the error term by an exponentially small factor the method of this paper is based on the borel laplace transform which makes it applicable to other problems at the end of the paper the method is applied to integrals with saddles also a powerful new method is presented to compute the stokes multipliers a numerical example is included

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

on the asymptotic and numerical solution of linear ordinary differential equations an investigation is made of the nature of asymptotic solutions of homogeneous linear differential equations of arbitrary order in the neighborhood of a singularity of unit rank we introduce a classification of the solutions into two types explicit and implicit for the former there exists a sector on which the solution is dominated by all independent solutions as the singularity is approached no such sector exists for implicit solutions in consequence the two types of solution have different uniqueness properties another difference is that error bounds for the asymptotic expansions of explicit solutions are generally stronger than those for implicit solutions we also investigate the computation of the solutions by numerical integration of the differential equation it is shown that it is the exception rather than the rule for the integration process to be stable particularly so for implicit solutions to overcome these instabilities we develop boundary value methods complete with error analysis numerical examples illustrate the computation of both explicit and implicit solutions and also the associated stokes multipliers

multi point taylor approximations in one dimensional linear boundary value problems abstract we consider second order linear differential equations in a real interval i with mixed dirichlet and neumann boundary data we consider a representation of its solution by a multi point taylor expansion the number and location of the base points of that expansion are conveniently chosen to guarantee that the expansion is uniformly convergent x i we propose several algorithms to approximate the multi point taylor polynomials of the solution based on the power series method for initial value problems

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

finding the observed information matrix when using the em algorithm a procedure is derived for extracting the observed information matrix when the em algorithm is used to find maximum likelihood estimates in incomplete data problems the technique requires computation of a complete data gradient vector or second derivative matrix but not those associated with the incomplete data likelihood in addition a method useful in speeding up the convergence of the em algorithm is developed two examples are presented

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