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

an asymptotic property of the camassa holm equation abstract asymptotic densities are used to study an asymptotic property of the dispersionless camassa holm equation an asymptotic density of a global solution is a weak limit of its scaled momentum density along a sequence of time increasing to infinity for a global solution with non negative compactly supported initial momentum density we show that if the asymptotic density is unique then it is a positive combination of dirac measures supported in a bounded interval in the non negative axis with zero as the only possible accumulation point in other words if the scaled momentum density does not oscillate as time goes to infinity the solution behaves as a combination of peakons moving to the right at different speeds in contrast to many investigations on the topic our approach is not spectral theoretic and hence is independent of the structure of the isospectral problem associated with the equation

convexity of stokes waves of extreme form existence is established of a piecewise convex periodic planar curve s below which is defined a harmonic function which simultaneously satisfies prescribed dirichlet and neumann boundary conditions on s in hydrodynamics this corresponds to the existence of a periodic stokes wave of extreme form which has a convex profile between consecutive stagnation points where there is a corner with a contained angle of 120

global weak solutions for a two component camassa holm shallow water system abstract in this paper we prove the existence of global weak solution for an integrable two component camassa holm shallow water system provided the initial data satisfying some certain conditions

instability criteria for the flow of an inviscid incompressible fluid we present a geometric estimate from below on the growth rate of a small perturbation of a three dimensional steady flow of an ideal fluid and thus we obtain effective criteria for local instability for euler s equations we use these criteria to demonstrate the instability of several simple flows and to show that any flow with a hyperbolic stagnation point is unstable

blow up phenomena and global existence for a two component camassa holm system with an arbitrary smooth function abstract in the paper we mainly investigate blow up phenomena and global existence of strong solutions to a two component camassa holm systems with an arbitrary smooth function h for three types of smooth functions h by using a conservation law and the sign preserving property of strong solutions we obtain two new blow up results and a new global existence result for the two component system our obtained results generalize and cover the recent results in yan et al 2015 zhang and yin 2015 2016

local well posedness and blow up phenomena of the generalized short pulse equation in this paper we investigate the cauchy problem of the generalized short pulse equation in periodic domain we first establish the local well posedness of the equation in hs s s 2 by taking advantage of the kato method and present some useful conservation laws then we obtain the boundedness of the solution by virtue of the conservation laws finally we give a precise blow up criterion for 0 or 0 and get several blow up results provided the initial data satisfies some certain conditions

particle trajectories beneath small amplitude shallow water waves in constant vorticity flows abstract we investigate the particle trajectories in a constant vorticity shallow water flow over a flat bed as periodic waves propagate on the water s free surface within the framework of small amplitude waves we find the solutions of the nonlinear differential equations system which describes the particle motion in the considered case and we describe the possible particle trajectories depending on the relation between the initial data and the constant vorticity some particle trajectories are undulating curves to the right or to the left others are loops with forward drift or with backward drift others can follow some peculiar shapes

well posedness and blow up phenomena for a higher order shallow water equation this paper deals with the cauchy problem for a higher order shallow water equation yt auxy buyx 0 where y 2ku i x2 ku and k 2 the local well posedness of solutions for the cauchy problem in sobolev space hs r with s 7 2 is obtained under some assumptions the existence and uniqueness of the global solutions to the equation are shown and conditions that lead to the development of singularities in finite time for the solutions are also acquired finally the weak solution for the equation is considered

local in space blow up criteria for a class of nonlinear dispersive wave equations abstract this paper is concerned with blow up phenomena for the nonlinear dispersive wave equation on the real line u t u x x t f u x f u x x x g u f u 2 u x 2 x 0 that includes the camassa holm equation as well as the hyperelastic rod wave equation f u k u 2 2 and g u 3 k u 2 2 as special cases we establish some a local in space blow up criterion i e a criterion involving only the properties of the data u 0 in a neighborhood of a single point simplifying and precising earlier blow up criteria for this equation

a spectral multidomain penalty method model for the simulation of high reynolds number localized incompressible stratified turbulence a spectral multidomain penalty method model has been developed for the simulation of high reynolds number localized stratified turbulence this is the first time that a penalty method with a particular focus on subdomain interface treatment has been used with a multidomain scheme to simulate incompressible flows the temporal discretization ensures maximum temporal accuracy by combining third order stiffly stable and backward differentiation schemes with a high order boundary condition for the pressure in the non periodic vertical direction a spectral multidomain discretization is used and its stability for under resolved simulations at high reynolds numbers is ensured through use of penalty techniques spectral filtering and strong adaptive interfacial averaging the penalty method is implemented in different formulations for both the explicit non linear term advancement and the implicit treatment of the viscous terms the multidomain model is validated by comparing results of simulations of the mid to late time stratified turbulent wake with non zero net momentum to the corresponding laboratory data for a towed sphere the model replicates correctly the characteristic vorticity and internal wave structure of the stratified wake and exhibits robust agreement with experiments in terms of the temporal power laws in the evolution of mean profile characteristic velocity and lengthscales

the analysis of multiple endpoints in clinical trials treatment comparisons in randomized clinical trials usually involve several endpoints such that conventional significance testing can seriously inflate the overall type i error rate one option is to select a single primary endpoint for formal statistical inference but this is not always feasible another approach is to apply bonferroni correction i e multiply each p value by the total number of endpoints its conservatism for correlated endpoints is examined for multivariate normal data a third approach is to derive an appropriate global test statistic and this paper explores one such test applicable to any set of asymptotically normal test statistics quantitative binary and survival endpoints are all considered within this general framework two examples are presented and the relative merits of the proposed strategies are discussed

exact free surface equatorial flows with general stratification in spherical coordinates this paper is concerned with the construction of a new exact solution to the geophysical fluid dynamics governing equations for inviscid and incompressible fluid in the equatorial region this solution represents a steady purely azimuthal flow with a free surface the novel aspect of the solution we derive is that the flow it prescribes accommodates a general fluid stratification the density may vary both with depth and with latitude the solution is presented in the terms of spherical coordinates hence at no stage do we invoke approximations by way of simplifying the geometry in the governing equations following the construction of our explicit solution we employ functional analytic considerations to prove that the pressure at the free surface defines implicitly the shape of the free surface distortion in a unique way exhibiting also the expected monotonicity properties finally using a short wavelength stability analysis we prove that certain flows defined by our exact solution are stable for a specific choice of the density distribution

free surface purely azimuthal equatorial flows in spherical coordinates with stratification abstract in this paper we derive an exact solution to the governing equations for geophysical fluid dynamics in spherical coordinates which incorporates fluid stratification this solution represents a steady purely azimuthal equatorial flow with an associated free surface following the derivation of the solution we demonstrate that there is a well defined relationship between the imposed pressure at the free surface and the resulting distortion of the surface s shape finally the solution for stratified fluid flow is subjected to a short wavelength stability analysis

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 cauchy problem for a generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem of the generalized camassa holm equation using a galerkin type approximation scheme it is shown that this equation is well posed in sobolev spaces h s s 3 2 for both the periodic and the nonperiodic case in the sense of hadamard that is the data to solution map is continuous furthermore it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous the nonuniform dependence is proved using the method of approximate solutions and well posedness estimates moreover it is shown that the solution map for the generalized camassa holm equation is holder continuous in h r topology finally with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time

on the existence of solutions and the pressure function related to the antarctic circumpolar current abstract this paper is concerned with the analysis of an incompressible inviscid fluid moving only in the azimuthal direction with no variations in this direction but allowing for a depth dependent variation as such this permits the consideration of the antarctic circumpolar current the most significant current in earth s oceans and the only current that encircles the polar axis more precisely working in spherical coordinates accounting thus for the geometry of the earth we derive an implicit equation for the function describing the free surface which is related to the pressure function we use this equation to prove by means of the implicit function theorem that any small deviation from the pressure required to maintain the free surface undisturbed gives rise to a unique veritable wave solution our approach comprises also a body force which is shown to have significant influence on the monotonicity properties of the pressure function and the height function describing the free surface

on the cauchy problem for a generalized degasperis procesi equation we mainly study the cauchy problem of the generalized degasperis procesi equation first we establish the local well posedness for the equation second we give the precise blow up scenario a conservation law and prove the equation has smooth solutions which blow up in finite time finally we investigate the blow up rate for the blow up solutions

applications of global statistics in analysing quality of life data quality of life qol instruments usually consist of a number of components each of which deals specifically with a particular functionally related dysfunction in a clinical trial whose primary aim is the evaluation of the treatment by means of qol instruments analysis of each of the components usually consists of either univariate analysis of variance anova or some non parametric methods this multiple testing approach can produce an increase in false positive findings one attempt to correct for this is the bonferroni adjustment another approach is to apply global statistics parametric or non parametric for the null hypothesis of no treatment difference versus the alternative hypothesis that one treatment is uniformly better than the other for qol instruments as a whole data from a randomized double blind trial of 111 congestive heart failure patients which involved four qol instruments were analysed with univariate anova bonferroni adjustment parametric and non parametric global statistics the global statistics complemented the univariate methods and made the presentation of qol data very effective i recommend the general use of global statistics in analysis of qol data

exponential profiles producing genuine three dimensional nonlinear flows relevant for equatorial ocean dynamics abstract a nonlinear three dimensional model for equatorial oceanic flows that are moving slowly in the azimuthal direction is investigated the consequences of including exponential terms in the azimuthal flow profile are determined some examples show that the model is able to capture simultaneously depth dependent currents with flow reversal poleward flow upwelling downwelling processes

a spectral numerical method for the navier stokes equations with applications to taylor couette flow a new spectral method for solving the incompressible navier stokes equations in a plane channel and between concentric cylinders is presented the method uses spectral expansions which inherently satisfy the boundary conditions and the continuity equation and yield banded matrices which are efficiently solved at each time step in addition the number of dependent variables is reduced resulting in a reduction in computer memory requirements several test problems have been computed for the channel flow and for flow between concentric cylinders including taylor couette flow with axisymmetric taylor vortices and wavy vortices in all cases agreement with available experimental and theoretical results is very good

global diffeomorphism of the lagrangian flow map defining equatorially trapped water waves abstract the aim of this paper is to prove that a three dimensional lagrangian flow which defines equatorially trapped water waves is dynamically possible this is achieved by applying a mixture of analytical and topological methods to prove that the nonlinear exact solution to the geophysical governing equations derived by constantin 2012 is a global diffeomorphism from the lagrangian labelling variables to the fluid domain beneath the free surface

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

instability of internal equatorial water waves abstract in the following paper we present criteria for the hydrodynamical instability of internal equatorial water waves we show by way of the short wavelength perturbation approach that certain geophysical waves propagating above the equatorial thermocline are linearly unstable when the wave steepness exceeds a given threshold

procedures for comparing samples with multiple endpoints summary five procedures are considered for the comparison of two or more multivariate samples these procedures include a newly proposed nonparametric rank sum test and a generalized least squares test also considered are the following tests ordinary least squares hotelling s p and a bonferroni per experiment error rate approach applications are envisaged in which each variable represents a qualitatively different measure of response to treatment the null hypothesis of no treatment difference is tested with power directed towards alternatives in which at least one treatment is uniformly better than the others in all simulations the nonparametric procedure provided relatively good power and accurate control over the size of the test and is recommended for general use alternatively the generalized least squares procedure may also be useful with normally distributed data in moderate or large samples a convenient expression for this procedure is obtained and its asymptotic relative efficiency with respect to the ordinary least squares test is evaluated clinical trials are often conducted for the purpose of evaluating the relative efficacy of two or more modes of therapy when the therapies are administered to independent groups and efficacy is measured on the basis of a single response variable appropriate parametric and nonparametric one way analysis of variance anova procedures and their properties are well known often however efficacy is measured by more than one variable although univariate methods for assessing each characteristic individually are useful in this setting there is often the additional need for a single overall objective probability statement that addresses the question of whether or not the experimental therapy is efficacious this need is particularly acute when the medical question is controversial and the sample size is small although a large literature exists on the comparison of multivariate samples the procedures tend to be infrequently used in practice as pointed out by meier 1975 p 523 the standard analysis for the comparison of two multivariate samples which is based on hotelling s t2 statistic addresses somewhat the wrong question and consequently has very poor power for the alternatives of primary interest specifically the t2 procedure addresses the question are one or more of the treatments different making no distinction between variables that change favorably and variables that change unfavorably a second approach would be to assign per experiment error rates to each of the univariate tests by using bonferroni s inequality i e by multiplying each univariate p value by the number of variables studied the desired overall probability statement is then given by the minimum of the per experiment univariate p values although this approach may be useful in some settings it may lack power for alternatives in which most or all measures of efficacy are improved this will be of particular concern when the number k of endpoints studied is large relative to sample size in fact statistical significance will not be possible in some instances this problem will be exacerbated when the measures of efficacy are highly

a three dimensional autonomous nonlinear dynamical system modelling equatorial ocean flows abstract we investigate a nonlinear three dimensional model for equatorial flows finding exact solutions that capture the most relevant geophysical features depth dependent currents poleward or equatorial surface drift and a vertical mixture of upward and downward motions

solitary wave solutions of the generalized two component hunter saxton system abstract the existence of solitary wave solutions of the generalized two component hunter saxton system is determined it is also shown that there are peaked and cusped solitary waves with singularities among those smooth solitary wave solutions

exponential decay and symmetry of solitary waves to degasperis procesi equation abstract we improve the decay argument by bona and li 1997 5 for solitary waves of general dispersive equations and illustrate it in the proof for the exponential decay of solitary waves to steady degasperis procesi equation in the nonlocal formulation in addition we give a method which confirms the symmetry of solitary waves including those of the maximum height finally we discover how the symmetric structure is connected to the steady structure of solutions to the degasperis procesi equation and give a more intuitive proof for symmetric solutions to be traveling waves the improved argument and new method above can be used for the decay rate of solitary waves to many other dispersive equations and will give new perspectives on symmetric solutions for general evolution equations

exact steady periodic water waves with vorticity we consider the classical water wave problem described by the euler equations with a free surface under the influence of gravity over a flat bottom we construct two dimensional inviscid periodic traveling waves with vorticity they are symmetric waves whose profiles are monotone between each crest and trough we use bifurcation and degree theory to construct a global connected set of such solutions c 2004 wiley periodicals inc

non uniform continuity of periodic holm staley b family of equations abstract we consider a family of non evolutionary partial differential equations known as the holm staley b family which includes the integrable camassa holm and degasperis procesi equations we show that the solution map is not uniformly continuous the proof relies on a construction of smooth periodic traveling waves with small amplitude

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