hodograph transformations for a camassa holm hierarchy in 2 1 dimensions a generalization of the negative camassa holm hierarchy to 2 1 dimensions is presented under the name chh 2 1 several hodograph transformations are applied in order to transform the hierarchy into a system of coupled cbs calogero bogoyavlenskii schiff equations in 2 1 dimensions that pass the painleve test a non isospectral lax pair for chh 2 1 is obtained through the above mentioned relationship with the cbs spectral problem

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 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

periodic solutions of some infinite dimensional hamiltonian systems associated with non linear partial difference equations ii this is our second paper devoted to the study of some non linear schrodinger equations with random potential we study the non linear eigenvalue problems corresponding to these equations we exhibit a countable family of eigenfunctions corresponding to simple eigenvalues densely embedded in the band tails contrary to our results in the first paper the results established in the present paper hold for an arbitrary strength of the non linear cubic term in the non linear schrodinger equation

intraband discrete breathers in disordered nonlinear systems ii localization abstract we find spatially localized time periodic solutions discrete breathers or dbs in disordered nonlinear systems with frequency inside the linear phonon spectrum under conditions that strictly prohibit their existence in their periodic counterparts for that purpose we develop a new in situ method for the accurate calculation of these solutions which does not make use of any continuation from an anticontinuous limit using this method we demonstrate that intraband localized modes intraband discrete breathers or idbs at a given site with frequencies inside the discrete linear spectrum do exist provided these frequencies do not belong to forbidden resonance gaps since there is a dense set of resonant frequencies we illustrate numerically in agreement with a theorem by albanese and frohlich that the localized dbs exist provided that their frequencies belong to fat cantor sets i e with finite measure such a set contains as accumulation points the linear frequency of the normal mode at the occupied site we check that many of these solutions are linearly stable and conjecture that their frequency belongs to another smaller fat cantor set our numerical methods provide a much wider set of exact solutions which are multisite breathers and suggest conjectures extending the existing theorems the physical implications of the existence of idbs and possible applications for glasses and the persistent spectral hole burning are discussed

on face numbers of rational simplicial polytopes with symmetry tight lower bounds are obtained for the h vector and consequently for all the face numbers of a rational simplicial polytope which has a fixed point free linear symmetry of prime order this extends a result of richard stanley for centrally symmetric polytopes

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

localization length of stationary states in the nonlinear schrodinger equation for the nonlinear schrodinger equation nlse in the presence of disorder exponentially localized stationary states are found we demonstrate analytically that the localization length is typically independent of the strength of the nonlinearity and is identical to the one found for the corresponding linear equation the analysis makes use of the correspondence between the stationary nlse and the langevin equation as well as of the resulting fokker planck equation the calculations are performed for the white noise random potential and an exact expression for the exponential growth of the eigenstates is obtained analytically it is argued that the main conclusions are robust

factorization and painleve analysis of a class of nonlinear third order partial differential equations a class of fully nonlinear third order partial differential equations pdes is considered this class contains several examples which have recently appeared in the literature and for which rather unusual travelling wave solutions have been given these solutions consist essentially of sums of exponentials we trace the occurrence of these exponentials back to the existence of a linear subequation which appears as a factor in the travelling wave reduction in addition we consider the painleve analysis of this set of equations both for the original pde and also for reductions to ordinary differential equations odes no equation in the class considered survives the combination of pde and ode tests also an equation in the class considered which is known to be integrable is shown to possess only the weak painleve property our analysis therefore confirms the limitations of the painleve test as a test for complete integrability when applied to fully nonlinear pdes

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

integrable 1 1 dimensional hierarchies arising from reduction of a non isospectral problem in 2 1 dimensions this work presents a classical lie point symmetry analysis of a two component non isospectral lax pair of a hierarchy of partial differential equations in 2 1 dimensions which can be considered as a modified version of the camassa holm hierarchy in 2 1 dimensions a classification of reductions for this spectral problem is performed non isospectral reductions in 1 1 dimensions are considered of remarkable interest

anomalous diffusion in nonlinear oscillators with multiplicative noise the time asymptotic behavior of undamped nonlinear oscillators with a random frequency is investigated analytically and numerically we find that averaged quantities of physical interest such as the oscillator s mechanical energy root mean square position and velocity grow algebraically with time the scaling exponents and associated generalized diffusion constants are calculated when the oscillator s potential energy grows as a power of its position u x approximately x 2n for x infinity correlated noise yields anomalous diffusion exponents equal to half the value found for white noise

statistics of finite time lyapunov exponents in a random time dependent potential the sensitivity of trajectories over finite time intervals t to perturbations of the initial conditions can be associated with a finite time lyapunov exponent lambda obtained from the elements m ij of the stability matrix m for globally chaotic dynamics lambda tends to a unique value the usual lyapunov exponent lambda infinity for almost all trajectories as t is sent to infinity but for finite t it depends on the initial conditions of the trajectory and can be considered as a statistical quantity we compute for a particle moving in a randomly time dependent one dimensional potential how the distribution function p lambda t approaches the limiting distribution p lambda infinity delta lambda lambda infinity our method also applies to the tail of the distribution which determines the growth rates of moments of m ij the results are also applicable to the problem of wave function localization in a disordered one dimensional potential

subdiffusion in classical and quantum nonlinear schrdinger equations with disorder the review is concerned with the nonlinear schrdinger equation nlse in the presence of disorder disorder leads to localization in the form of the localized anderson modes am while nonlinearity is responsible for the interaction between the ams and transport the dynamics of an initially localized wave packets are concerned in both classical and quantum cases in both cases there is a subdiffusive spreading which is explained in the framework of a continuous time random walk ctrw and it is shown that subdiffusion is due to the transitions between those ams which are strongly overlapped this overlapping being a common feature of both classical and quantum dynamics leads to the clustering with an effective trapping of the wave packet inside each cluster therefore the dynamics of the wave packet corresponds to the ctrw where the basic mechanism of subdiffusion is an entrapping of the wave packet with delay or waiting times distributed by the power law w t 1 t1 where is the transport exponent it is obtained that 1 3 for the classical nlse and 1 2 for the quantum nlse

dynamics of wave packets for the nonlinear schrodinger equation with a random potential the dynamics of an initially localized anderson mode is studied in the framework of the nonlinear schrodinger equation in the presence of disorder it is shown that the dynamics can be described in the framework of the liouville operator an analytical expression for a wave function of the initial time dynamics is found by a perturbation approach as follows from a perturbative solution the initially localized wave function remains localized at asymptotically large times the dynamics can be described qualitatively in the framework of a phenomenological probabilistic approach by means of a probability distribution function it is shown that the probability distribution function may be governed by the fractional fokker planck equation and corresponds to subdiffusion

anderson localization for multi dimensional systems at large disorder or large energy we prove that discrete schrodinger operators on d with a random potential have almost surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy this is the first proof of localization for multi dimensional anderson models

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

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

multiscaling of noise induced parametric instability we describe the statistical properties of growth rates of a linear oscillator driven by a parametric noise we show that in general the fluctuations of local lyapunov exponents are non gaussian and demonstrate multiscaling analytical calculations of the generalized lyapunov exponents are complemented with approximative and numerical results this allows us to identify the parameter range where the deviations from the gaussian statistics become important

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

rational solutions for schwarzian integrable hierarchies we give an approach to finding rational solutions of completely intagrable hierarchies which makes use of the relationship between modifications and the schwarzian equations obtained via the singular manifold method this extends the recent work of kudryashov which allowed a simple derivation of the iteration used to construct sequences of such solutions we also give a closed form for the index polynomial of the schwarzian korteweg de vries hierarchy in addition we consider the representation of rational solutions using lower families of the hierarchy we give a simple representation under which the rational solutions remain solutions of every flow of the hierarchy this representation also allows the inclusion of arbitrary data corresponding to negative indices we use our method to derive an alternative form of the backlund transformation for the hierarchy of the second painleve equation as well as new solutions of a hierarchy of breaking soliton equations we also present here for the first time a schwarzian version of this breaking soliton hierarchy

moment modification multipeakons and nonisospectral generalizations abstract firstly a formal correspondence is established between the camassa holm ch equation and a two component modified ch or called sqq equation according to the method of moment modification for multipeakon formulae secondly based on the generalized nonisospectral ch equation in chang et al 2014 14 and the interlacing multipeakons of the two component modified ch equation in chang et al 2016 15 we propose a new generalized two component modified ch equation with two parameters which possesses a nonisospectral lax pair the proposed equation still admits multipeakon solutions of explicit and closed form sufficient conditions for global existence of solutions are given and two concrete examples with certain interesting phenomenon are presented last of all as a by product a generalized nonisospectral modified ch equation is deduced together with its lax pair

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 painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

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

perturbation theory for periodic orbits in a class of infinite dimensional hamiltonian systems we consider a class of hamiltonian systems describing an infinite array of coupled anharmonic oscillators and we study the bifurcation of periodic orbits off the equilibrium point the family of orbits we construct can be parametrized by their periods which belong to cantor sets of large measure containing certain periods of the linearized problem as accumulation points the infinitely many holes forming a dense set on which the existence of a periodic orbit cannot be proven originate from a dense set of resonances that are present in the system we also have a result concerning the existence of solutions of arbitrarily large amplitude

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

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

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 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

asymptotic localization of stationary states in the nonlinear schrodinger equation the mapping of the nonlinear schrodinger equation with a random potential on the fokker planck equation is used to calculate the localization length of its stationary states the asymptotic growth rates of the moments of the wave function and its derivative for the linear schrodinger equation in a random potential are computed analytically and resummation is used to obtain the corresponding growth rate for the nonlinear schrodinger equation and the localization length of the stationary states

anderson localization for one and quasi one dimensional systems we prove almost sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete schrodinger operators on one and quasi one dimensional lattices this paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential our results allow us to prove that e g for strong disorder the smallest positive lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity

smith type inequalities for a polytope with a solvable group of symmetries the object of this paper is to obtain a set of inequalities relating the face numbers of different orbit types of a simplicial polytope p with a finite solvable group g of linear symmetries it is assumed that 1 for each subgroup h of g the fixed point set ph is a subpolytope of p and 2 the toric variety x p associated to p is nonsingular the action of g on p induces an action on x p and we describe a set of smith type inequalities between the betti numbers of x p h where h ranges through the set of subgroups of g by relating each x p h with x ph we then express these inequalities in terms of the face numbers of the different orbit types of p and the rank of fixed point sets of certain compact tori this rank is determined explicitly when g is abelian moreover assumption 2 is removed for a polytope of dimension 2

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

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

on face numbers of manifolds with symmetry abstract necessary conditions on the face numbers of cohen macaulay simplicial complexes admitting a proper action of the cyclic group z p z of a prime order are given this result is extended further to necessary conditions on the face numbers and the betti numbers of buchsbaum simplicial complexes with a proper z p z action adin s upper bounds on the face numbers of cohen macaulay complexes with symmetry are shown to hold for all d 1 dimensional buchsbaum complexes with symmetry on n 3 d 2 vertices a generalization of kuhnel s conjecture on the euler characteristic of 2 k dimensional manifolds and sparla s analog of this conjecture for centrally symmetric 2 k manifolds are verified for all 2 k manifolds on n 6 k 3 vertices connections to the upper bound theorem are discussed and its new version for centrally symmetric manifolds is established

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

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

non isospectral problem in 2 1 dimensions in this paper the singular manifold method allows us to obtain a non isospectral lax pair darboux transformations and miura transformations for an equation in 2 1 dimensions and its modified version in this way we can iteratively build different kinds of solutions with solitonic behaviour

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

coupled harry dym equations with multi hamiltonian structures the authors consider the isospectral flows of delta 2 sigma 1n upsilon i lambda i psi alpha psi using an unusual form of the lax approach they derive in a particularly simple manner a the second half of the lax pair b the associated hamiltonian structures c an infinite hierarchy of poisson commuting hamiltonians in this way they show that these equations possess n 1 compatible purely differential hamiltonian structures the case n 1 is just the bi hamiltonian harry dym hierarchy they thus extend the recent results on multi hamiltonian coupled kdv equations