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

a fully supersymmetric akns theory we construct a fully supersymmetric bihamiltonian theory in four superfields admitting zero curvature and lax formulation this theory is an extension of the classical akns which can be recovered as a reduction other supersymmetric theories are obtained as reductions of the susy akns namely a nonlinear schrodinger a modified kdv and the manin radul kdv the susy nonlinear schrodinger hierarchy is related to the one of roelofs and kersten we determine its bihamiltonian and lax formulation finally we show that the susy kdv s mentioned before are related through a susy miura map

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

testing reliability in a stress strength model when x and y are normally distributed we consider the stress strength problem in which a unit of strength x is subjected to environmental stress y an important problem in stress strength reliability concerns testing hypotheses about the reliability parameter r p x yl in this article we consider situations in which x and y are independent and have normal distributions or can be transformed to normality we do not require the two population variances to be equal our approach leads to test statistics which are exact p values that are represented as one dimensional integrals on the basis of the p value one can also construct approximate confidence intervals for the parameter of interest we also present an extension of the testing procedure to the case in which both strength and stress depend on covariates for comparative purposes the bayesian solution to the problem is also presented we use data from a rocket motor experiment to illustrate the procedure

confidence intervals for steady state availability of a system with exponential operating time and lognormal repair time long run availability of a system assuming lognormal repair times and exponential failure times is considered with many repairable systems the time to repair is well characterized by a lognormal distribution however due to the difficulties associated with the lognormal distribution repair times are often modeled using the lognormal distribution assuming that the variance of the lognormal distribution is known or by using some other distribution such as the exponential distribution instead of the lognormal distribution in this paper using the generalized p value approach we propose confidence intervals and exact tests for steady state availability of systems using the two parameter lognormal distribution for repair time and exponential distribution for operating time here both parameters of the lognormal distribution are assumed to be unknown a couple of examples are given to illustrate the proposed procedures a simulation study is given to demonstrate the performance of the proposed procedure results are extended for the long run availability of a system consisting of several independent parallel sub systems

exact inference for growth curves with intraclass correlation structure summary we consider repeated observations taken over time for each of several subjects for example one might consider the growth curve of a cohort of babies over time we assume a simple linear growth curve model exact results based on sufficient statistics exact tests of the null hypothesis that a coefficient is zero or exact confidence intervals for coefficients are not available to make inference on regression coefficients when an intraclass correlation structure is assumed this paper will demonstrate that such exact inference is possible using generalized inference

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

on steady state availability of a system with lognormal repair time assuming two parameter lognormal distribution for repair times statistical inference for the steady state availability of a system is considered for the failure time distribution weibull gamma and lognormal distributions were considered using the generalized p value approach we propose confidence intervals and exact tests for the steady state availability of a system a couple of examples are given to illustrate the proposed procedures

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

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

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

biased diffusion in percolation systems indication of multifractal behaviour the authors study diffusion in percolation systems at criticality in the presence of a constant bias field e using the exact enumeration method they show that the mean displacement of a random walker varies as r t approximately log t a e where a e in 1 e 1 e for small e more generally diffusion on a given configuration is characterised by the probability p r t that the random walker is on site r at time t they find that the corresponding configurational average shows simple scaling behaviour and is described by a single exponent in contrast their numerical results indicate that the averaged moments pq t sigma rpq r t are described by an infinite hierarchy of exponents for zero bias field however all moments are determined by a single gap exponent

a convergent finite difference scheme for the camassa holm equation with general h 1 initial data we suggest a finite dfference scheme for the camassa holm equation that can handle general h 1 initial data the form of the difference scheme is judiciously chosen to ensure that it satisfies a total energy inequality we prove that the difference scheme converges strongly in h 1 towards a dissipative weak solution of the camassa holm equation

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

on traveling wave solutions of the θ equation of dispersive type abstract traveling wave solutions to a class of dispersive models u t u t x x u u x u u x x x 1 u x u x x are investigated in terms of the parameter including two integrable equations the camassa holm equation 1 3 and the degasperis procesi equation 1 4 as special models it was proved in h liu and z yin 2011 39 that when 1 2 1 smooth solutions persist for all time and when 0 1 2 strong solutions of the equation may blow up in finite time yielding rich traveling wave patterns this work therefore restricts to only the range 0 1 2 it is shown that when 0 only periodic travel wave is permissible and when 1 2 traveling waves may be solitary periodic or kink like waves for 0 1 2 traveling waves such as periodic solitary peakon peaked periodic cusped periodic or cusped soliton are all permissible

interval estimation and hypothesis testing of intraclass correlation coefficients the generalized variable approach in this paper we propose a novel approach using the concept of generalized variable gv for the confidence interval estimation of the difference of two intraclass correlation coefficients under unequal family sizes this approach can also easily provide p values for hypothesis testing simulation results show that the gv approach can provide confidence intervals with good coverage properties and perform hypothesis testing with satisfactory type i error control furthermore the confidence intervals and p values by gv approach can be easily obtained by simulation therefore the gv approach is a suitable candidate for making inference concerning two intraclass correlation coefficients

conservative weak solutions of the periodic cauchy problem for μhs equation we prove the existence and uniqueness of conservative weak solutions of the periodic cauchy problem for an integrable evolution equation from mathematical physics our method is to first prove the existence and uniqueness of solutions for all time in lagrangian coordinates and construct the maps to carry this result to the eulerian coordinates

inferences on the mean of zero inflated lognormal data the generalized variable approach this paper considers the problems of confidence interval estimation and hypothesis testing for the mean of zero inflated lognormal data we present an unified novel approach based on the concepts of generalized test variables and generalized pivotal quantities simulation results demonstrate that the coverage accuracy of proposed confidence intervals and the type i error control of the proposed exact tests are satisfactory

dissipative prolongations of the multipeakon solutions to the camassa holm equation abstract multipeakons are special solutions to the camassa holm equation they are described by an integrable geodesic flow on a riemannian manifold we present a bi hamiltonian formulation of the system explicitly and write down formulae for the associated first integrals then we exploit the first integrals and present a novel approach to the problem of the dissipative prolongations of multipeakons after the collision time we prove that an n peakon after a collision becomes an n 1 peakon for which the momentum is preserved

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

normal based methods for a gamma distribution prediction and tolerance intervals and stress strength reliability in this article we propose inferential procedures for a gamma distribution using the wilson hilferty wh normal approximation specifically using the result that the cube root of a gamma random variable is approximately normally distributed we propose normal based approaches for a gamma distribution for a constructing prediction limits one sided tolerance limits and tolerance intervals b for obtaining upper prediction limits for at least l of m observations from a gamma distribution at each of r locations and c assessing the reliability of a stress strength model involving two independent gamma random variables for each problem a normal based approximate procedure is outlined and its applicability and validity for a gamma distribution are studied using monte carlo simulation our investigation shows that the approximate procedures are very satisfactory for all of these problems for each problem considered the results are illustrated using practical examples our overall conclusion is tha

inferences on the common coefficient of variation the coefficient of variation is often used as a measure of precision and reproducibility of data in medical and biological science this paper considers the problem of making inference about the common population coefficient of variation when it is a priori suspected that several independent samples are from populations with a common coefficient of variation the procedures for confidence interval estimation and hypothesis testing are developed based on the concepts of generalized variables the coverage properties of the proposed confidence intervals and type i errors of the proposed tests are evaluated by simulation the proposed methods are illustrated by a real life example

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

a model containing both the camassa holm and degasperis procesi equations abstract a nonlinear dispersive partial differential equation which includes the famous camassa holm and degasperis procesi equations as special cases is investigated although the h 1 norm of the solutions to the nonlinear model does not remain constants the existence of its weak solutions in lower order sobolev space h s with 1 s 3 2 is established under the assumptions u 0 h s and u 0 x l the local well posedness of solutions for the equation in the sobolev space h s r with s 3 2 is also developed

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

long time asymptotic solution structure of camassa holm equation subject to an initial condition with non zero reflection coefficient of the scattering data in this article we numerically revisit the long time solution behavior of the camassa holm equation ut uxxt 2ux 3uux 2uxuxx uuxxx the finite difference solution of this integrable equation is sought subject to the newly derived initial condition with delta function potential our underlying strategy of deriving a numerical phase accurate finite difference scheme in time domain is to reduce the numerical dispersion error through minimization of the derived discrepancy between the numerical and exact modified wavenumbers additionally to achieve the goal of conserving hamiltonians in the completely integrable equation of current interest a symplecticity preserving time stepping scheme is developed based on the solutions computed from the temporally symplecticity preserving and the spatially wavenumber preserving schemes the long time asymptotic ch solution characters can be accurately depicted in distinct regions of the space time domain featuring with their own quantitatively very different solution behaviors we also aim to numerically confirm that in the two transition zones their long time asymptotics can indeed be described in terms of the theoretically derived painleve transcendents another attempt of this study is to numerically exhibit a close connection between the presently predicted finite difference solution and the solution of the painleve ordinary differential equation of type ii in two different transition zones

a self adaptive moving mesh method for the camassa holm equation a self adaptive moving mesh method is proposed for the numerical simulations of the camassa holm equation it is an integrable scheme in the sense that it possesses the exact n soliton solution it is named a self adaptive moving mesh method because the non uniform mesh is driven and adapted automatically by the solution once the non uniform mesh is evolved the solution is determined by solving a tridiagonal linear system due to these two superior features of the method several test problems give very satisfactory results even if by using a small number of grid points

quasiperiodic waves and asymptotic behavior for bogoyavlenskii s breaking soliton equation in 2 1 dimensions based on a multidimensional riemann theta function the hirota bilinear method is extended to explicitly construct multiperiodic quasiperiodic wave solutions for the 2 1 dimensional bogoyavlenskii breaking soliton equation among these periodic waves the one periodic waves are well known cnoidal waves their surface pattern is one dimensional and often they are used as one dimensional models of periodic waves in shallow water the two periodic biperiodic waves are a direct generalization of one periodic waves their surface pattern is two dimensional that is they have two independent spatial periods in two independent horizontal directions the two periodic waves may be considered to represent periodic waves in shallow water without the assumption of one dimensionality a limiting procedure is presented to analyze asymptotic behavior of the one and two periodic waves in details the exact relations between the periodic wave solutions and the well known soliton solutions are established it is rigorously shown that the periodic wave solutions tend to the soliton solutions under a small amplitude limit

the integral bifurcation method and its application for solving a family of third order dispersive pdes abstract in this paper an improved method named the integral bifurcation method is introduced in order to demonstrate its effectiveness for obtaining travelling wave solutions of the nonlinear wave equations a family of third order dispersive partial differential equations which were given by a degasperis d holm and a hone are studied many integral bifurcations are obtained for different parameter conditions by using these integral bifurcations many travelling wave solutions such as loop soliton solutions solitary wave solutions cusp soliton solutions and periodic wave solutions are obtained in particular under the conditions c 1 0 c 2 c 3 1 a very peculiar periodic wave solution is obtained

conditions for the existence of higher symmetries of evolutionary equations on the lattice in this paper we construct a set of five conditions necessary for the existence of generalized symmetries for a class of differential difference equations depending only on nearest neighboring interaction these conditions are applied to prove the existence of new integrable equations belonging to this class

a strange recursion operator demystified we show that a new integrable two component system of kdv type studied by karasu kalkanli et al 2004 acta appl math 83 85 94 is bi hamiltonian and its recursion operator which has a highly unusual structure of nonlocal terms can be written as a ratio of two compatible hamiltonian operators found by us using this we prove that the system in question possesses an infinite hierarchy of local commuting generalized symmetries and conserved quantities in involution and the evolution systems corresponding to these symmetries are bi hamiltonian as well we also show that upon introduction of suitable nonlocal variables the nonlocal terms of the recursion operator under study can be written in the usual form with the integration operator d 1x appearing in each term at most once

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

confidence intervals for p y1 y2 with normal outcomes in linear models recently there is an emerging interest in the inference of p y1 y2 where y1 and y2 stand for two independent continuous random variables so far most of the research in this field focuses on simply comparing two outcomes without adjusting for covariates this paper mainly presents a large sample approach based on a noncentral t distribution for the confidence interval estimation of p y1 y2 with normal outcomes in linear models furthermore the performance of the proposed large sample approach is compared with that of a generalized variable approach and a bootstrap approach simulation studies demonstrate that for small to medium sample sizes both the large sample approach and the generalized variable approach provide confidence intervals with satisfying coverage probabilities whereas the bootstrap approach can be slightly liberal for certain scenarios the proposed approaches are applied to three real life data sets copyright 2008 john wiley sons ltd

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

confidence limits for stress strength models with explanatory variables a lower confidence bound is obtained for pr y x z 1 z 2 where x and y are independent normal variables with explanatory variables z 1 and z 2 respectively for equal residual variances an exact solution is obtained but for the unequal variance case an approximate lower confidence bound is developed examples of the use of these procedures are given

peakon solutions of the novikov equation and properties of the data to solution map abstract we consider both the periodic and the non periodic cauchy problem for the novikov equation and discuss continuity results for the data to solution map in sobolev spaces in particular we show that the data to solution map is not globally uniformly continuous in sobolev spaces with exponent less than 3 2 to accomplish this we construct sequences of peakon solutions whose distance initially goes to zero but later becomes large

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

generalized confidence intervals abstract the definition of a confidence interval is generalized so that problems such as constructing exact confidence regions for the difference in two normal means can be tackled without the assumption of equal variances under certain conditions the extended definition is shown to preserve a repeated sampling property that a practitioner expects from exact confidence intervals the proposed procedure is also applied to the problem of constructing confidence intervals for the difference in two exponential means and for variance components in mixed models a repeated sampling property of generalized p values is also given with this characterization one can carry out fixed level tests of parameters of continuous distributions on the basis of generalized p values finally pratt s paradox is revisited and a procedure that resolves the paradox is given

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

anova under unequal error variances by taking a generalized approach to findingp values the classical f test of the one way anova is extended to the case of unequal error variances the relationship of this result to other solutions in the literature is discussed an exact test for comparing variances of a number of populations is also developed scheff6 s procedure of multiple comparison is extended to the case of unequal variances the possibility and the approach that one can take to extend the results to simple designs involving more than one factor are briefly discussed

random walk in a percolation cluster external field dependence random walk in a percolation cluster was studied in the simultaneous application of both dc bias and ac driving fields by monte carlo calculations amplitude and phase shift of the random walk response to the external field of b0 b1 sin omega t were calculated as a function of b1 omega and b0 some new observations are made which cannot be expected simply from the linear superposition of the two separate results for a constant bias field b0 and an ac driving field b1 sin omega t respectively

integrable symplectic maps abstract in this paper we first give a terse survey of symplectic maps their canonical formulation and integrability then we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices a number of illustrative examples are provided

a comparison of several exact and approximate tests for structural shift under heteroscedasticity abstract it is known that the chow test for equality of regression coefficients is not robust to heteroscedasticity much recent attention has been paid to this problem by way of new tests the bulk of the new procedures are justified only asymptotically and little has been done to examine finite sample properties even less attention has been focused on comparisons of tests under either the null or alternative hypotheses in this paper we redress these shortcomings and also contribute by suggesting additional tests the results indicate that there are tests available with good sampling characteristics some of which are computationally easy to carry out from both the standpoint of the researcher s time and computation time

the positive and negative camassa holm γ hierarchies zero curvature representations bi hamiltonian structures and algebro geometric solutions in this paper we extend the camassa holm ch equation to two different hierarchies of nonlinear evolution equations by resorting to two lenard recursion sequences one is of positive order ch hierarchy including a harry dym type equation while the other is of negative order ch hierarchy of integrodifferential equations with nonzero integration constants including the ch equation we provide the zero curvature representations for both hierarchies by solving a key matrix equation moreover the lenard sequences are used to construct a lax matrix that satisfies a stationary zero curvature equation which enables us to establish bi hamiltonian structures for both hierarchies by applying trace identity in addition the lenard sequence and lax pair are used to systematically derive the its matveev trace formula and the dubrovin type equations associated with the ch equation the hyperelliptic curve and abel jacobi coordinates are then introduced to linearize the associated flow from which the algebro geometric solutions to the ch equation are constructed by using standard jacobi inversion technique in this paper we extend the camassa holm ch equation to two different hierarchies of nonlinear evolution equations by resorting to two lenard recursion sequences one is of positive order ch hierarchy including a harry dym type equation while the other is of negative order ch hierarchy of integrodifferential equations with nonzero integration constants including the ch equation we provide the zero curvature representations for both hierarchies by solving a key matrix equation moreover the lenard sequences are used to construct a lax matrix that satisfies a stationary zero curvature equation which enables us to establish bi hamiltonian structures for both hierarchies by applying trace identity in addition the lenard sequence and lax pair are used to systematically derive the its matveev trace formula and the dubrovin type equations associated with the ch equation the hyperelliptic curve and abel jacobi coordinates are then introduced to linearize the associated flow from which the a

a local discontinuous galerkin method for the camassa holm equation in this paper we develop analyze and test a local discontinuous galerkin ldg method for solving the camassa holm equation which contains nonlinear high order derivatives the ldg method has the flexibility for arbitrary h and p adaptivity we prove the l 2 stability for general solutions and give a detailed error estimate for smooth solutions and provide numerical simulation results for different types of solutions of the nonlinear camassa holm equation to illustrate the accuracy and capability of the ldg method

coupled camassa holm equations n peakons and infinitely many conservation laws abstract three coupled camassa holm equations are proposed which are associated with a 4 4 matrix spectral problem with two potentials with the aid of the zero curvature equation we derive a hierarchy of new nonlinear evolution equations and establish their bi hamiltonian structures the three coupled camassa holm equations are exactly negative flows in the hierarchy and exact solutions with n peakons and infinite sequences of conserved quantities are discussed

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

field induced drift and trapping in percolation networks the authors study the effects of a bias producing external field on a random walk on the infinite cluster in the percolation problem there are two competing physical effects drift and trapping which result in a drift velocity nu which rises and then falls as the field increases they study these effects on a one dimensional lattice with random length branches and on the diluted bethe lattice they calculate nu in the first model with a maximum allowed length for each branch in the limit that this cutoff length becomes infinite they find that the velocity vanishes identically above a finite critical value of the field for the bethe lattice they derive an upper bound on the critical field which varies as p pc 1 2 as the percolation concentration pc is approached in the one dimensional model they also investigate the anomalous regime in which the velocity vanishes they discuss the distribution of steady state times required to traverse n sites and find that it can be described in terms of a stable distribution of index x with superimposed oscillations the index x of the stable distribution is given by l xi where xi is a characteristic branch length and l is a bias induced length which describes the exponential buildup of the steady state density of particles towards the end of a branch

periodic solutions of the camassa holm equation based on the bilinear form since the famous work of camassa and holm various analytic techniques have been used to find the exact solutions of camassa holm ch equation and many interesting results have been found by the different methods in this paper we first give an extended bilinear form of the ch equation which contains four more free parameters than those in the results proposed in parkers finding for soliton solutions thus we construct the smooth periodic solution of the ch equation in a parametric form based on the previous work of nakumura we also show that the periodic solution degenerates into the smooth solitary wave solution by limiting behavior of is the parameter appeared from the riemann theta function

monte carlo study of biased diffusion at the percolation threshold computer simulations confirm dhar s prediction ibid vol 17 p l257 1984 that the asymptotic exponent for the rms distance against time relation is zero right at the percolation threshold if a particle diffuses under the influence of a constant force in a random medium for intermediate times and weak force the exponent for the mean displacement as a function of time in the direction of the force confirms the prediction of ohtsuki and keyes phys rev lett vol 52 p 1177 1984

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

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

non linear response in percolation systems the authors study diffusion in d 2 percolation clusters at criticality under the influence of an external time dependent field e t e0sin omega t they find that the mean displacement x t of a random walker is a periodic function with a phase shift phi approximately pi 3 independent of omega and e0 the amplitude a e0 omega of x t is linear in e0 below a crossover field e 0 omega above e 0 omega a e0 omega is strongly non linear showing a maximum as a function of e0 they find that both the linear and the non linear regime can be described by a single scaling function a e0 omega approximately e0 omega 2d f e0 e 0 omega where d approximately 2 87 is the diffusion exponent and e 0 omega approximately omega beta with beta approximately 0 3 the linear response regime is reached for e0 omega beta 1

the multiple soliton solution of the camassa holm equation this paper refines johnso s implementation of constantin s method for solving the camassa holm equation for a multiple soliton solution an analytical formula for the q y and an explicit relation between x and y are found an algorithm of solving for u y is presented how to introduce time variable t into the solution is also clearly explained

estimating treatment effects in a two arm parallel trial of a continuous outcome for a continuous outcome in a two arm trial that satisfies normal distribution assumptions we can transform the standardized mean difference with the use of the cumulative distribution function to be the effect size measure p x

darboux transformation and multi soliton solutions of the camassa holm equation and modified camassa holm equation in this paper we propose a new approach to calculate multi soliton solutions of camassa holm ch equation and modified camassa holm mch equation with the aid of darboux transformation dt the new approach simplifies the approach presented by li and zhang proc r soc london ser a 460 2617 2627 2004 we first map the ch and mch equation to a negative order kdv nkdv equation by a reciprocal transformation then we proceed to apply the dt to solve the nkdv equation in the usual way finally we invert the reciprocal transformation to recover the solutions of the ch equation and mch equation

well posedness travelling waves and geometrical aspects of generalizations of the camassa holm equation abstract in this paper we consider a five parameter equation including the camassa holm and the dullin gottwald holm equations among others we prove the existence and uniqueness of solutions of the cauchy problem using kato s approach conservation laws of the equation up to second order are also investigated from these conservation laws we establish some properties for the solutions of the equation and we also find a quadrature for it the quadrature obtained is of capital importance in a classification of bounded travelling wave solutions we also find some explicit solutions given in terms of elliptic integrals finally we classify the members of the equation describing pseudo spherical surfaces

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

on the camassa holm equation and a direct method of solution iii n soliton solutions in the concluding study designated iii we modify the direct bilinear transformation method for solving the camassa holm ch equation that was set down in part ii of this work we demonstrate its efficacy for finding analytic multisoliton solutions of the equation and give explicit expressions for the first few solitons it is shown that at each order n the n soliton has a non standard representation that is characterized by an extra parameter the stucture of this parameter is investigated and a procedure for constructing the general n soliton solution of the ch equation is presented

on solutions of the camassa holm equation the techniques that have been developed for the application of the inverse scattering transform method to the solution of the camassa holm equation principally by constantin are implemented we use this approach first to represent the known solitary wave solution in a simple parametric form and then second to obtain the general two and three soliton solutions these latter two solutions require rather extensive use of mathematical packages mathematica and maple in order to complete the construction of solutions a number of examples are presented the phase shifts evident after an interaction are found and the special limit that recovers the peakon solutions is discussed

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

on some wave breaking for the nonlinear integrable shallow water wave equations mathematicians in this paper based on the conservation laws and the blow up scenario some new blow up phenomena are derived for the camassa holm equation and degasperis procesi equation which have attracted much attention due to their structure c 2015 elsevier ltd all rights reserved

generalized p values in significance testing of hypotheses in the presence of nuisance parameters abstract this article examines some problems of significance testing for one sided hypotheses of the form h 0 0 versus h 1 0 where is the parameter of interest in the usual setting let x be the observed data and let t x be a test statistic such that the family of distributions of t x is stochastically increasing in define cx as x t x t x 0 the p value is p x sup 0 pr x cx in the presence of a nuisance parameter there may not exist a nontrivial cx with a p value independent of we consider tests based on generalized extreme regions of the form cx x t x x t x x and conditions on t x x are given such that the p value p x sup 0 pr x cx is free of the nuisance parameter where t is stochastically increasing in we provide a solution to the problem of testing hypotheses about the differences in means of two independent exponential distributions a problem for which the fixed level testing approach

testing regression equality with unequal variances on obtient un test exact similaire a celui de chow 1960 qui comporte une simple integration unidimensionnelle

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

two super camassa holm equations reciprocal transformations and applications reciprocal transformations are introduced for two super camassa holm ch equations under these transformations and appropriate changes of dependent variables the super ch equation proposed by geng et al stud appl math 130 1 2013 is converted to a negative member of the super korteweg de vries kdv hierarchy studied by geng and wu in 2010 appl math lett 23 716 2010 while the other super ch equation due to zhang and zuo j math phys 52 073503 2011 is related to a new super kdv hierarchy in the latter case algebraic properties of this new super kdv hierarchy are established including hamiltonian operators a recursion operator and conserved quantities

diffusion on percolation clusters at criticality the concept of fractal dimensionality is used to study the problem of diffusion on percolation clusters the authors find from monte carlo simulations that the fractal dimensionality of a random walk on a critical percolation cluster in three dimensional space is d 3 3 or 0 1 where the size of the cluster is restricted to be larger than the span of the walk and is d 3 9 or 0 1 for a walk on clusters not subject to this restriction for two dimensional space they find d approximately d approximately 2 7 or p0 1 the exponent d and d is related to the scaling of the average length r of n steps via rd varies as n the fracton dimensionality which is related to the density of states was found to be 1 26 or 0 1 these results are in good agreement with the predictions of alexander and orbach 1982

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

the modified camassa holm equation backlund transformation and nonlinear superposition formula this paper is concerned with the modified camassa holm equation and its backlund transformation with the aid of reciprocal transformation and the associated modified camassa holm equation a backlund transformation which involves both dependent and independent variables is constructed for the modified camassa holm equation the related nonlinear superposition formula is worked out and applications of both backlund transformation and nonlinear superposition formula are considered

on solutions of the two component camassa holm system following the approach of chen et al lett math phys 75 1 15 2006 we continue the study of the two component camassa holm system by using its relation with the first negative flow of the akns hierarchy we obtain a more general class of particular solutions including the multisoliton ones

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

multi soliton and multi cuspon solutions of a camassa holm hierarchy and their interactions in this paper we study an integrable camassa holm hierarchy whose high frequency limit is the camassa holm equation by a method associated with the darboux transform we construct the explicit multi soliton and multi cuspon solutions then we study in detail the interactions of soliton cuspon and soliton soliton further an interesting phenomenon is found the soliton with smaller amplitude can travel faster than the one with larger amplitude when they interact at last we investigate in detail the head on collision of one soliton and one cuspon