improved least squares identification an improved recursive least squares algorithm for parameter estimation is presented which includes on off criteria to prevent parameter drift during periods of low excitation a variable forgetting factor which maintains the trace of the covariance matrix at a user specified value data preprocessing and normalization to improve numerical accuracy scaling of the regressor vector to minimize the condition number of the covariance matrix plus independent estimation of the mean values of the input output data which can be used to eliminate errors due to d c bias or slowly drifting elements in the regressor vector the algorithm can also include parameter projection to constrain the estimates to a priori specified regions and retains the formal properties such as convergence of a true weighted least squares algorithm the proposed algorithm is compared with other modifications suggested in the literature and its advantages are demonstrated by a simulated example

bounded disturbance adaptive control with exponential and selective data weightings a method of combination of selective and exponential data weighting algorithms for parameter identification is proposed bounded disturbances in systems with unknown statistical properties are considered the proposed method ensures a stable control system in addition it does not need the condition of persistent excitation when the predicted output error is small

symmetries as integrability criteria for differential difference equations in this paper we review the results obtained by the generalized symmetry method in the case of differential difference equations during the last 20 years together with general theory of the method classification results are discussed for classes of equations which include the volterra toda and relativistic toda lattice equations

short note viscous and inviscid regularizations in a class of evolutionary partial differential equations we investigate solution properties of a class of evolutionary partial differential equations pdes with viscous and inviscid regularization an equation in this class of pdes can be written as an evolution equation involving only first order spatial derivatives coupled with the helmholtz equation a recently developed two step iterative method p h chiu l lee t w h sheu a dispersion relation preserving algorithm for a nonlinear shallow water wave equation j comput phys 228 2009 8034 8052 is employed to study this class of pdes the method is in principle superior for pde s in this class as it preserves their physical dispersive features in particular we focus on a leray type regularization h s bhat r c fetecau a hamiltonian regularization of the burgers equation j nonlinear sci 16 2006 615 638 of the hopf equation proposed in alternative to the classical burgers viscous term we show that the regularization effects induced by the alternative model can be vastly different from those induced by burgers viscosity depending on the smoothness of initial data in the limit of zero regularization we validate our numerical scheme by comparison with a particle method which admits closed form solutions further effects of the interplay between the dispersive terms comprising the leray regularization are illustrated by solutions of equations in this class resulting from regularized burgers equation by selective elimination of dispersive terms

dirichlet series and the integrability of multilinear differential equations we explore the relationship between the integrability of nonsemilinear partial differential equations and formal dirichlet series solutions to this end we consider a variety of integrable and nonintegrable differential equations we also demonstrate that such series solutions are a natural extension to the nonlinear case of those obtained by basic solution techniques for linear ordinary differential equations

a robust algorithm for random parameter tracking an algorithm for tracking time varying parameters is studied it is shown to be exponentially stable independently of the choice of algorithm parameters we derive some useful nonasymptotic estimates for the tracking error and develop the asymptotically optimal version of the method

brief papers independent tracking and regulation adaptive control with forgetting factor this paper considers a discrete time adaptive control algorithm with a forgetting factor applicable to minimum phase plants the tracking and regulation objectives are independently specified the relevance of the eigenvalues of the gain matrix f k used in the updating equation for the adaptive parameters gq k is shown it is proved that if the maximum eigenvalue of the inverse of the gain matrix f k has an upper bound and a non zero lower bound then the global convergence of the control algorithm is insured the result of the design is a simple control scheme using a linear constant feedforward controller and a nonlinear feedback controller the performance of the control structure in tracking and regulation are evaluated by simulations

radius of analyticity for the camassa holm equation on the line abstract using estimates in sobolev spaces we prove that the solution to the cauchy problem for the camassa holm equation on the line with analytic initial data u 0 x and satisfying the mckean condition that is the quantity m 0 x 1 x 2 u 0 x does not change sign is analytic in the spatial variable for all time furthermore we obtain explicit lower bounds for the radius of spatial analyticity r t given by r t a 1 1 c 1 b t 1 exp c 0 u 0 h 1 t where a b c 1 and c 0 are suitable positive constants

analytic solutions of the cauchy problem for two component shallow water systems this paper is mainly concerned with the cauchy problem for a modified two component camassa holm shallow water system with analytic initial data the analyticity of its solutions is proved in both variables globally in space and locally in time the obtained results can be also applied to the classical integrable two component camassa holm shallow water system and the b equation

degenerate form of a three component degasperis procesi equation abstract we discuss the short wave limit of a new three component degasperis procesi equation and construct infinitely many conserved quantities for the degenerate system we study the reductions for a spectral problem of the degenerate system which are shown to contain spectral problems of some famous or new systems found recently

on the structure of 2 1 dimensional commutative and noncommutative integrable equations we develop the symbolic representation method to derive the hierarchies of 2 1 dimensional integrable equations from the scalar lax operators and to study their properties globally the method applies to both commutative and noncommutative cases in the sense that the dependent variable takes its values in c or a noncommutative associative algebra we prove that these hierarchies are indeed quasi local in the commutative case as conjectured by mikhailov and yamilov j phys a 31 6707 1998 we propose a ring extension in the noncommutative case based on the symbolic representation as examples we give noncommutative versions of kadomtsev petviashvili kp modified kadomtsev petviashvili mkp and boussinesq equations

on a functional equation related to the intermediate long wave equation we resolve an open problem stated by ablowitz et al 1982 j phys a math gen 15 781 concerning the integral operator appearing in the intermediate long wave equation we explain how this is resolved using the perturbative symmetry approach introduced by one of us with mikhailov by solving a certain functional equation we prove that the intermediate long wave equation and the benjamin ono equation are the unique integrable cases within a particular class of integro differential equations furthermore we explain how the perturbative symmetry approach is naturally extended to treat equations on a periodic domain

non existence of truly solitary waves in water with small surface tension this paper considers permanent capillary gravity waves on the free surface of a two dimensional incompressible inviscid fluid of finite depth it is shown that there are no solitary wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity if the surface tension coefficient is less than its critical value and lies in some intervals the proof is based upon an estimate of a constant that is related to the approximation of the solution if it exists near its singularity the approximation satisfies a fourth order nonlinear ordinary differential equation when the solution is extended to the complex plane then the non existence of truly solitary waves is obtained by using a contradiction on this constant

non uniqueness for the fokas olver rosenau qiao equation abstract for the fokas olver rosenau qiao equation forq on both the line and the circle it is proved that there exist initial data in the sobolev spaces h s with s 3 2 which have nonunique solutions this phenomenon is demonstrated by constructing a 2 peakon solution with an asymmetric peakon antipeakon initial profile the peakon and antipeakon eventually collide and the solution at the collision time coincides with a second solitary peakon solution using the time reversibility of the equation this scenario proves nonuniqueness

the riemann problem for the leray burgers equation abstract for riemann data consisting of a single decreasing jump we find that the leray regularization captures the correct shock solution of the inviscid burgers equation however for riemann data consisting of a single increasing jump the leray regularization captures an unphysical shock this behavior can be remedied by considering the behavior of the leray regularization with initial data consisting of an arbitrary mollification of the riemann data as we show for this case the leray regularization captures the correct rarefaction solution of the inviscid burgers equation additionally we prove the existence and uniqueness of solutions of the leray regularized equation for a large class of discontinuous initial data all of our results make extensive use of a reformulation of the leray regularized equation in the lagrangian reference frame the results indicate that the regularization works by bending the characteristics of the inviscid burgers equation and thereby preventing their finite time crossing

towards classification of dimensional integrable equations integrability conditions i in this paper we attempt to extend the symmetry approach well developed in the case of 1 1 dimensional equations to the 2 1 dimensional case presence of nonlocal terms in symmetries and conservation laws is the main feature of integrable 2 1 dimensional equations we have introduced a concept of quasi local functions to characterize nonlocalities we have found a few first integrability conditions for a class of scalar equations in terms of quasi local functions and have demonstrated that they are suitable for testing integrability

analysis of a general recursive prediction error identification algorithm a general class of algorithms for recursive identification of stochastic dynamical systems is studied in this class the discrepancy between the measured output and the output predicted from previous data according to a candidate model the prediction error is minimized over the model set using a stochastic approximation approach it is proved that this class of methods has the same convergence properties as its off line counterparts under mild and general assumptions these assumptions do not for example include stationary conditions or conditions that the true system can be exactly represented within the model set the considered class of methods contains as special cases several well known algorithms other common recursive identification methods like the extended least squares method and the extended kalman filter can be interpreted as approximate prediction error methods with simplified gradient calculations therefore the approach taken here may also serve as a basis for unified description of many recursive identification methods

technical communique convergence analysis of recursive identification algorithms with forgetting factor a convergence analysis of a modified version of the least squares recursive identification algorithm with forgetting factor is given it is shown that the parametric distance converges to a zero mean random variable it is also shown that under persistent excitation condition on both system input and output the condition number of the adaptation gain matrix is bounded the variance of the parametric distance is bounded by the product of the noise variance times the upper bound of the condition number of the gain matrix this is done by normalizing the measurement vector entering in the identification algorithm and by using a forgetting factor verifying l t 0

on the integrability of homogeneous scalar evolution equations abstract we determine the existence of infinitely many symmetries for equations of the formut uk f u uk 1 when they are homogeneous with respect to the scalinguk k with 0 algorithms are given to determine whether a system has a symmetry also independent oftandx if it has one generalized symmetry we prove it has infinitely many and these can be found using recursion operators or master symmetries the method of proof uses the symbolic method and results from diophantine approximation theory we list the 10 integrable hierarchies the methods can in principle be applied to the 0 case as we illustrate for one example with 0 which seems to be new in principle they can also be used for systems of evolution equations but so far this has only been demonstrated for one class of examples

autonomous ovsyannikov theorem and applications to nonlocal evolution equations and systems abstract this work presents an ovsyannikov type theorem for an autonomous abstract cauchy problem in a scale of decreasing banach spaces which in addition to existence and uniqueness of solution provides an estimate about the analytic lifespan of the solution then using this theorem it studies the cauchy problem for camassa holm type equations and systems with initial data in spaces of analytic functions on both the circle and the line which is the main goal of this paper finally it studies the continuity of the data to solution map in spaces of analytic functions

on asymptotically equivalent shallow water wave equations abstract the integrable third order korteweg de vries kdv equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves however at quadratic order this asymptotic expansion produces an entire family of shallow water wave equations that are asymptotically equivalent to each other under a group of nonlinear non local normal form transformations introduced by kodama in combination with the application of the helmholtz operator these kodama helmholtz kh transformations are used to present connections between shallow water waves the integrable fifth order kdv equation and a generalization of the camassa holm ch equation that contains an additional integrable case the dispersion relation of the full water wave problem and any equation in this family agree to fifth order the travelling wave solutions of the ch equation are shown to agree to fifth order with the exact solution

integrable peakon equations with cubic nonlinearity we present a new integrable partial differential equation found by vladimir novikov like the camassa holm and degasperis procesi equations this new equation admits peaked soliton peakon solutions but it has nonlinear terms that are cubic rather than quadratic we give a matrix lax pair for v novikov s equation and show how it is related by a reciprocal transformation to a negative flow in the sawada kotera hierarchy infinitely many conserved quantities are found as well as a bi hamiltonian structure the latter is used to obtain the hamiltonian form of the finite dimensional system for the interaction of n peakons and the two body dynamics n 2 is explicitly integrated finally all of this is compared with some analogous results for another cubic peakon equation derived by zhijun qiao

a regularization of the burgers equation using a filtered convective velocity this paper examines the properties of a regularization of the burgers equation in one and multiple dimensions using a filtered convective velocity which we have dubbed as the convectively filtered burgers cfb equation a physical motivation behind the filtering technique is presented an existence and uniqueness theorem for multiple dimensions and a general class of filters is proven multiple invariants of motion are found for the cfb equation which are shown to be shared with the viscous and inviscid burgers equations traveling wave solutions are found for a general class of filters and are shown to converge to weak solutions of the inviscid burgers equation with the correct wave speed numerical simulations are conducted in 1d and 2d cases where the shock behavior shock thickness and kinetic energy decay are examined energy spectra are also examined and are shown to be related to the smoothness of the solutions this approach is presented with the hope of being extended to shock regularization of compressible euler equations

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 a regularization of the compressible euler equations for an isothermal gas abstract we consider a leray type regularization of the compressible euler equations for an isothermal gas the regularized system depends on a small parameter 0 using riemann invariants we prove the existence of smooth solutions for the regularized system for every 0 the regularization mechanism is a non linear bending of characteristics that prevents their finite time crossing we prove that in the 0 limit the regularized solutions converge strongly however based on our analysis and numerical simulations the limit is not the unique entropy solution of the euler equations the numerical method used to support this claim is derived from the riemann invariants for the regularized system this method is guaranteed to preserve the monotonicity of characteristics

global weak solutions for the novikov equation in the paper we mainly study the existence and uniqueness of global weak solutions for the novikov equation we first recall some results and definitions on strong solutions and weak solutions for the equation then we show that the equation has smooth solutions which exist globally in time provided the initial data satisfy certain sign conditions finally we prove the existence and uniqueness of global weak solutions to the equation with the initial data satisfying certain sign conditions

dispersive regularizations and numerical discretizations for the inviscid burgers equation we study centred second order in time and space discretizations of the inviscid burgers equation although this equation in its continuum formulation supports non smooth shock wave solutions the discrete equation generically supports smooth solitary wave solutions using backward error analysis we derive the modified equation associated with the numerical scheme we identify three different equations the korteweg de vries kdv equation the camassa holm ch equation and the b 0 member of the b family solutions of the first two equations are solitary waves and do not converge to the shock solutions of the burgers equation the third equation however supports solutions which strongly approximate weak solutions of the burgers equation we corroborate our analytical results with numerical simulations

a family of wave equations with some remarkable properties we consider a family of homogeneous nonlinear dispersive equations with two arbitrary parameters conservation laws are established from the point symmetries and imply that the whole family admits square integrable solutions recursion operators are found for two members of the family investigated for one of them a lax pair is also obtained proving its complete integrability from the lax pair we construct a miura type transformation relating the original equation to the korteweg de vries kdv equation this transformation on the other hand enables us to obtain solutions of the equation from the kernel of a schrodinger operator with potential parametrized by the solutions of the kdv equation in particular this allows us to exhibit a kink solution to the completely integrable equation from the 1 soliton solution of the kdv equation finally peakon type solutions are also found for a certain choice of the parameters although for this particular case the equation is reduced to a homogeneous second order nonlinear evolution equation

paper unification of discrete time explicit model reference adaptive control designs various discrete time explicit mrac designs based on stability considerations are presented from a unified point of view a general structure and a design procedure are given which contain as particular cases various possible designs some of these designs have been already given in the literature but some are new the general structure considered removes some of the restrictions encountered in previous designs and solves the important problem of independent specification of tracking and regulation objectives the proof of boundedness of the control variables is also given simulations are included in order to evaluate the various designs in a deterministic environment

on the cauchy problem for the modified novikov equation with peakon solutions abstract a new nonlinear dispersive partial differential equation with cubic nonlinearity which includes the famous novikov equation as special case is investigated we first establish the local well posedness in a range of the besov spaces b p r s p r 1 s max 3 2 1 1 p but s 2 1 p which generalize the sobolev spaces h s well posedness in h s with s 3 2 is also established by applying kato s semigroup theory then we give the precise blow up scenario moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time finally we prove that peakon solutions to the equation are global weak solutions

the power series method for nonlocal and nonlinear evolution equations abstract the initial value problem for a 4 parameter family of nonlocal and nonlinear evolution equations with data in a space of analytic functions is solved by using a power series method in abstract banach spaces in addition to determining the power series expansion of the solution this method also provides an estimate of the analytic lifespan expressed in terms of the norm of the initial data thus establishing an abstract cauchy kovalevsky type theorem for these equations

on the volume of sets bounded by refinable functions we present a method for the precise determination of the volume of subsets of r d which are bounded by a hypersurface parametrized by a set of refinable functions the derivation is based on the linear refinement equations rather than on closed form expressions of these functions which may not be available in particular our approach makes it possible to compute the area of planar domains bounded by subdivision curves or the volume of spatial domains bounded by subdivision surfaces

exponential convergence of adaptive identification and control algorithms output and equation error adaptive identification algorithms are shown to be exponentially convergent under a deterministic or stochastic persistently exciting or spanning condition on the system inputs together with several other standard conditions an adaptive control algorithm is shown to be exponentially convergent under a deterministic or stochastic persistently exciting condition on the reference trajectory together with some standard conditions

the degasperis procesi equation as a non metric euler equation in this paper we present a geometric interpretation of the degasperis procesi dp equation as the geodesic flow of a right invariant symmetric linear connection on the diffeomorphism group of the circle we also show that for any evolution in the family of b equations there is neither gain nor loss of the spatial regularity of solutions this in turn allows us to view the dp and the camassa holm equation as an ode on the frechet space of all smooth functions on the circle

study of the transient phase of the forgetting factor rls we investigate the convergence properties of the forgetting factor rls algorithm in a stationary data environment using the settling time as our performance measure we show that the algorithm exhibits a variable performance that depends on the particular combination of the initialization and noise level specifically when the observation noise level is low high snr rls when initialized with a matrix of small norm it has an exceptionally fast convergence convergence speed decreases as we increase the norm of the initialization matrix in a medium snr environment the optimum convergence speed of the algorithm is reduced as compared with the previous case however rls becomes more insensitive to initialization finally in a low snr environment we show that it is preferable to initialize the algorithm with a matrix of large norm

old problems and new challenges in subdivision abstract subdivision has been an active research topic for about 40 years and it has seen commercial exploitation in animated computer graphics the question has to be asked is there anything left to do research wise this note contains the authors responses yes much has been learnt and does not need to be relearnt but there are still results that we cannot prove properties of specific schemes that we cannot compute exactly and there are still ways in which we need to improve the technology to make it useful in the more demanding contexts of cad cam and cae we have approached the subject by looking first at curves then at surfaces and finally at the trivariate structures which are required for cae for each dimension we identify first what is already established and then proceed to suggest directions for further research three different labels indicate for each useful piece of work the relative difficulty and hence the prospects of success download download high res image 24kb download download full size image download download high res image 25kb download download full size image download download high res image 40kb download download full size image of course these ratings are just our personal estimates we will only know for sure how hard the questions were when the work has been done further we do not claim completeness of the list of topics discussed here also the list of references selects only a few samples from the vast literature

asymptotic stability of peakons for the novikov equation abstract the novikov equation is an integrable camassa holm type equation with a cubic nonlinearity one of the most important features of this equation is the existence of peaked traveling waves also called peakons this paper aims to prove the asymptotic stability of peakon solutions under h 1 r perturbations satisfying that their associated momentum density defines a non negative radon measure motivated by molinet s work 24 25 26 we shall first prove a liouville property for h 1 r global solutions belonging to a certain class of almost localized functions more precisely we show that such solutions have to be a peakon the main novelty in our analysis in comparison to the camassa holm equation comes from the fact that in our present case the momentum is not a conserved quantity and may be unbounded along the trajectory in this regard to prove the liouville property we used a new lyapunov functional not related to the not conserved momentum of the equation

c1 continuity of the generalized four point scheme abstract we show that the interpolatory four point scheme with tension parameter generates c 1 limit curves if and only if 0 where 0 19273 is the unique real solution of the cubic equation 32 3 4 1 0

classification of integrable one component systems on associative algebras this paper is devoted to the complete classification of integrable one component evolution equations whose field variable takes its values in an associative algebra the proof that the list of non commutative integrable homogeneous evolution equations is complete relies on the symbolic method each equation in the list has infinitely many local symmetries and these can be generated by its recursion recursive operator or master symmetry

partially integrable nonlinear equations with one higher symmetry in this letter we present a family of second order in time nonlinear partial differential equations which have only one higher symmetry these equations are not integrable but have a solution depending on one arbitrary function

decay properties of solutions to a 4 parameter family of wave equations in this paper persistence properties of solutions are investigated for a 4 parameter family k abc equation of evolution equations having k 1 degree nonlinearities and containing as its integrable members the camassa holm the degasperis procesi novikov and fokas olver rosenau qiao equations these properties will imply that strong solutions of the k abc equation will decay at infinity in the spatial variable provided that the initial data does furthermore it is shown that the equation exhibits unique continuation for appropriate values of the parameters k a b and c

generalizations of the camassa holm equation we classify generalized camassa holm type equations which possess infinite hierarchies of higher symmetries we show that the obtained equations can be treated as negative flows of integrable quasi linear scalar evolution equations of orders 2 3 and 5 we present the corresponding lax representations or linearization transformations for these equations some of the obtained equations seem to be new

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

a riemann hilbert approach to the modified camassa holm equation with nonzero boundary conditions this paper aims at developing the riemann hilbert problem approach to the modified camassa holm mch equation in the case when the solution is assumed to approach a non zero constant at both infinities of the space variable in this case the spectral problem for the associated lax pair equation has a continuous spectrum which allows formulating the inverse spectral problem as a riemann hilbert factorization problem with jump conditions across the real axis we obtain a representation for the solution of the cauchy problem for the mch equation and also a description of certain soliton type solutions both regular and non regular

on a regularization of a scalar conservation law with discontinuous coefficients this paper is devoted to a scalar conservation law with a linear flux function involving discontinuous coefficients it is clear that the delta standing wave should be introduced into the riemann solution in some nonclassical situation in order to study the formation of delta standing wave we consider a regularization of the discontinuous coefficient with the helmholtz mollifier and then obtain a regularized system which depends on a regularization parameter 0 the regularization mechanism is a nonlinear bending of characteristic curves that prevents their finite time intersection it is proved rigorously that the solutions of regularized system converge to the delta standing wave solution in the 0 limit compared with the classical method of vanishing viscosity here it is clear to see how the delta standing wave forms naturally along the characteristics

stability of stationary solutions for nonintegrable peakon equations abstract the camassa holm equation with linear dispersion was originally derived as an asymptotic equation in shallow water wave theory among its many interesting mathematical properties which include complete integrability perhaps the most striking is the fact that in the case where linear dispersion is absent it admits weak multi soliton solutions peakons with a peaked shape corresponding to a discontinuous first derivative there is a one parameter family of generalized camassa holm equations most of which are not integrable but which all admit peakon solutions numerical studies reported by holm and staley indicate changes in the stability of these and other solutions as the parameter varies through the family in this article we describe analytical results on one of these bifurcation phenomena showing that in a suitable parameter range there are stationary solutions leftons which are orbitally stable

weakly nonlocal solitary waves in a singularly perturbed korteweg de vries equation a fifth order korteweg de vries equation is considered where the fifth order derivative term is multiplied by a small parameter it is known that solitary wave solutions of this model equation are nonlocal in that the central core of the wave is accompanied by copropagating trailing oscillations here using the techniques of exponential asymptotics these solutions are reexamined and it is established that they form a one parameter family characterized by the phase shift of the trailing oscillations explicit asymptotic formula relating the oscillation amplitude to the phase shift are obtained

on dispersive difference schemes i etude du comportement oscillatoire et de la convergence faible des solutions des approximations aux differences dispersives de u t 1 2 u 2 x 0

the periodic b equation and euler equations on the circle in this note we show that the periodic b equation can only be realized as a euler equation on the lie group diff s1 of all smooth and orientation preserving diffeomorphisms on the circle if b 2 i e for the camassa holm equation in this case the inertia operator generating the metric on diff s1 is given by a 1 x2 in contrast the degasperis procesi equation for which b 3 is not a euler equation on diff s1 for any inertia operator our result generalizes a recent result of kolev some geometric investigations on the degasperis procesi shallow water equation wave motion 46 412 419 2009

a globally convergent adaptive predictor this paper establishes global convergence for an adaptive prediction algorithm key features of the algorithm are i a priori residuals are used in the regression vector ii the input signal need not be persistently exciting iii only an upper bound is required on the system order iv no monitoring or projection procedure is required to guarantee stability and v arbitrary feedback is allowed between output and input

on integrability of systems of evolution equations we prove the conjecture formulated in bsw98 that almost all systems in the family formula have at most finitely many symmetries by using number theory we list the nine exceptional cases when the systems do have infinitively many symmetries for such systems we give the recursive operators to generate their symmetries we treat both 1the commutative and the noncommutative or quantum cases this is the first example of a class of equations where such a classification has been possible

persistence properties and unique continuation for a generalized camassa holm equation in this paper persistence properties of solutions are investigated for a generalized camassa holm equation g k bch having k 1 degree nonlinearities and containing as its integrable members the camassa holm the degasperis procesi and the novikov equations the persistence properties will imply that strong solutions of the g k bch equation will decay at infinity in the spatial variable provided that the initial data does furthermore it is shown that the equation exhibits unique continuation for appropriate values of the parameters b and k finally existence of global solutions is established when b k 1

integrable equations in 2 1 dimensions deformations of dispersionless limits we classify integrable third order equations in 2 1 dimensions which generalize the examples of kadomtsev petviashvili veselov novikov and harry dym equations our approach is based on the observation that dispersionless limits of integrable systems in 2 1 dimensions possess infinitely many multi phase solutions coming from the so called hydrodynamic reductions in this paper we adopt a novel perturbative approach to the classification problem based on the method of hydrodynamic reductions we first classify integrable quasilinear systems which may potentially occur as dispersionless limits of soliton equations in 2 1 dimensions to reconstruct dispersive deformations we require that all hydrodynamic reductions of the dispersionless limit be inherited by the corresponding dispersive counterpart this procedure leads to a complete list of integrable third order equations some of which are apparently new

perturbative symmetry approach the aim of our paper is to formulate a perturbative version of the symmetry approach in the symbolic representation and to generalize it in order to make it suitable for the study of nonlocal and non evolution equations our formalism is the development and incorporation of the perturbative approach of zakharov and schulman the symbolic method of sanders and wang and the standard symmetry approach of shabat et al we apply our theory to describe integrable generalizations of the benjamin ono type equations and to isolate integrable cases of the camassa holm type equations

the cauchy problem for a 4 parameter family of equations with peakon traveling waves abstract the initial value problem for a novel 4 parameter family of evolution equations which are nonlinear and nonlocal and possess peakon traveling wave solutions is studied on both the line and the circle it is proved that this family of equations is well posed in the sense of hadamard when the initial data belong to the sobolev spaces h s with s 5 2 also it is shown that the data to solution map is not uniformly continuous however if h s s 5 2 is equipped with a weaker h r norm 0 r s then the solution map becomes holder continuous