stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

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

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

the camassa holm equation as a geodesic flow on the diffeomorphism group misiolek j geom phys 24 203 208 1998 has shown that the camassa holm equation is a geodesic flow on the bott virasoro group in this paper it is shown that the camassa holm equation for the case 0 is the geodesic spray of the weak riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the h1 inner product over the entire group this paper uses the right trivialization technique to rigorously verify that the euler poincare theory for lie groups can be applied to diffeomorphism groups the observation made in this paper has led to physically meaningful generalizations of the ch equation to higher dimensional manifolds

well posedness global existence and blow up phenomena for an integrable multi component camassa holm system abstract this paper is concerned with a new integrable multi component camassa holm system which has been proven to be integrable and has peakon solutions this system includes many one component and two component camassa holm type systems as special cases in this paper we first establish the local well posedness and a continuation criterion for the system then we present several global existence and blow up results for two important integrable two component subsystems our obtained results cover and improve considerably recent results in gui et al 2013 yan et al 2015

stabilizing receding horizon h sub spl infin controls for linear continuous time varying systems in this note new matrix inequality conditions on the terminal weighting matrices are proposed for linear continuous time varying systems under these conditions nonincreasing and nondecreasing monotonicities of the saddle point value of a dynamic game are shown to be guaranteed it is proved that the proposed terminal inequality conditions ensure the closed loop stability of the receding horizon h sub spl infin control rhhc the stabilizing rhhc guarantees the h sub spl infin norm bound of the closed loop system the proposed terminal inequality conditions for the monotonicity of the saddle point value and the closed loop stability include most well known existing terminal conditions as special cases the results for time invariant systems are obtained correspondingly from those in the time varying case

propagation of ultra short optical pulses in cubic nonlinear media abstract we derive a partial differential equation that approximates solutions of maxwell s equations describing the propagation of ultra short optical pulses in nonlinear media and which extends the prior analysis of alterman and rauch phys lett a 264 2000 390 diffractive nonlinear geometric optics for short pulses preprint 2002 we discuss non rigorously conditions under which this approximation should be valid but the main contributions of this paper are 1 an emphasis on the fact that the model equation for short pulse propagation may depend on the details of the optical susceptibility in the wavelength regime under consideration 2 a numerical comparison of solutions of this model equation with solutions of the full nonlinear partial differential equation 3 a local well posedness result for the model equation and 4 a proof that in contrast to the nonlinear schrodinger equation which models slowly varying wavetrains this equation has no smooth pulse solutions which propagate with fixed shape and speed

the cauchy problem for shallow water waves of large amplitude in besov space abstract in this paper we consider a nonlinear evolution equation modelling the propagation of surface waves in the shallow water regime of large amplitude which is characterised by some cubical nonlinearities first we establish the local well posedness in besov space b 2 1 3 2 then we give a blow up criterion finally with a given analytic initial data we establish the analyticity of the solutions in both variables globally in space and locally in time

the linear quadratic optimal control approach to feedback control design for systems with delay abstract a generalized linear quadratic optimal control problem for systems with delay is formulated the optimal solution is given as a state feedback form which requires a solution of coupled infinite dimensional riccati equations it is shown that the closed loop system formed by the optimal state feedback control has some desirable sensitivity and robustness properties the generalization exists in the state quadratic form of the cost functional which makes it possible to discuss a pole location problem within the framework of the linear quadratic optimal control problem it is also shown that the generalized cost functional contains a special class of cost functionals for which the optimal control can be realized by solving only a finite dimensional riccati equation based on these results about the generalized linear quadratic optimal control a design method of feedback controls is proposed and an illustrative example is then presented

the periodic cauchy problem for a combined ch mch integrable equation abstract this paper is concerned with the periodic cauchy problem for a generalized camassa holm integrable equation which can be viewed as a generalization to both the camassa holm ch and modified camassa holm mch equations we mainly make a detailed presentation on the effects of varying the ch and mch nonlocal nonlinearities on the non uniform dependence and holder continuity of the solution map using a galerkin type approximation method we first establish the local well posedness result in sobolev spaces h s s 5 2 with continuous dependence on the initial data then we prove that this dependence is sharp by showing that the data to solution map is not uniformly continuous which is based on well posedness estimates and the method of approximate solutions furthermore we demonstrate that the solution map is holder continuous in the h topology with 0 s

on the blow up rate and the blow up set of breaking waves for a shallow water equation we consider the problem of the development of singularities for classical solutions to a new periodic shallow water equation blow up can occur only in the form of wave breaking i e the solution remains bounded but its slope becomes unbounded in finite time a quite detailed description of the wave breaking phenomenon is given there is at least a point in general depending on time where the slope becomes infinite exactly at breaking time the precise blow up rate is established and for a large class of initial data we also determine the blow up set

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

on time dynamics of coagulation fragmentation processes we establish a characterization of coagulation fragmentation processes such that the induced birth and death processes depicting the total number of groups at time t 0 are time homogeneous based on this we provide a characterization of mean field gibbs coagulation fragmentation models which extends the one derived by hendriks et al as a by product of our results the class of solvable models is widened and a question posed by n berestycki and pitman is answered under restriction to mean field models

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

receding horizon controls for input delayed systems this paper presents a receding horizon control rhc for an unconstrained input delayed system to begin with we derive a finite horizon optimal control for a quadratic cost function including two final weighting terms the rhc is easily obtained by changing the initial and final times of the finite horizon optimal control a linear matrix inequality lmi condition on two final weighting matrices is proposed to meet the cost monotonicity under which the optimal cost on the horizon is monotonically nonincreasing with time and hence the asymptotical stability is guaranteed only if an observability condition is met it is shown through simulation that the proposed rhc stabilizes the input delayed system effectively and its performance can be tuned by adjusting weighting matrices with respect to the state and the input

blow up phenomena and peakons for the b family of forq mch equations abstract this paper is devoted to studying the modified b family of equations with cubic nonlinearity called the b family of forq mch equations which includes the cubic camassa holm equation also called fokas olver rosenau qiao equation as a special case we first study the local well posedness for the cauchy problem of the equation and then make good use of fine structure of the equation we derive the precise blow up scenario and a new blow up result with respect to initial data finally peakon solutions are derived

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

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

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

sample period effects in optimally designed digital control systems a systematic procedure for obtaining an optimal sampled data control system which treats the sample period as a free design parameter is developed the sensitivity of the closed loop performance with respect to the sample period is examined for various choices of the penalty weighting factors and the compensator structure an optimal nonzero sample period may exist depending on the penalty weighting assigned to the system states even when a boundary condition is imposed that requires the optimal sampled data control system to converge to an optimal continuous control system as the sample period approaches zero

general receding horizon control for linear time delay systems a general receding horizon control rhc or model predictive control mpc for time delay systems is proposed the proposed rhc is obtained by minimizing a new cost function that includes two terminal weighting terms which are closely related to the closed loop stability the general solution of the proposed rhc is derived using the generalized riccati method furthermore an explicit solution is obtained for the case where the horizon length is less than or equal to the delay size a linear matrix inequality lmi condition on the terminal weighting matrices is proposed under which the optimal cost is guaranteed to be monotonically non increasing it is shown that the monotonic condition of the optimal cost guarantees closed loop stability of the rhc simulations demonstrate that the proposed rhc effectively stabilizes time delay systems

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

associated stieltjes carlitz polynomials and a generalization of heun s differential equation abstract the generating function of stieltjes carlitz polynomials is a solution of heun s differential equation and using this relation carlitz was the first to get exact closed forms for some heun functions similarly the associated stieltjes carlitz polynomials lead to a new differential equation which we call associated heun thanks to the link with orthogonal polynomials we are able to deduce two integral relations connecting associated heun functions with different parameters and to exhibit the set of associated heun functions which generalize carlitz s part of these results were used by the author to derive the stieltjes transform of the measure of orthogonality for the associated stieltjes carlitz polynomials using asymptotic analysis here we present a new derivation of this result

dual birth and death processes and orthogonal polynomials we discuss connections between birth and death processes and orthogonal polynomials we survey certain spectral properties of orthogonal polynomials and spectra of birth and death processes birth and death processes with linear birth and death rates are studied in some detail we also mention some results concerning birth and death processes whose transition rates are rational functions

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

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

associated laguerre and hermite polynomials explicit orthogonality relations are found for the associated laguerre and hermite polynomials one consequence is the construction of the n 1 n pade approximation to a 1 b x a b x where a b x is the second solution to the confluent hypergeometric differential equation that does not grow rapidly at infinity

the cauchy problem and blow up phenomena of a new integrable two component camassa holm system abstract this paper considers the cauchy problem and blow up phenomena of a new integrable two component camassa holm system which is a natural extension of the fokas olver rosenau qiao equation firstly the local well posedness of the system in the critical besov space b 2 1 1 2 r b 2 1 1 2 r is investigated and it is shown that the data to solution mapping is holder continuous then a blow up criteria for the cauchy problem in the critical besov space is derived moreover with conditions on the initial data a new blow up criteria is obtained by virtue of the blow up criteria at hand and the conservative property of m and n along the characteristics finally a global existence result for the strong solution is established

smooth and singular multisoliton solutions of a modified camassa holm equation with cubic nonlinearity and linear dispersion we develop a direct method for solving a modified camassa holm equation with cubic nonlinearity and linear dispersion under the rapidly decreasing boundary condition we obtain a compact parametric representation for the multisoliton solutions and investigate their properties we show that the introduction of a linear dispersive term exhibits various new features in the structure of solutions in particular we find the smooth solitons whose characteristics are different from those of the camassa holm equation as well as the novel types of singular solitons a remarkable feature of the soliton solutions is that the underlying structure of the associated tau functions is the same as that of a model equation for shallow water waves introduced by ablowitz et al 1974 stud appl math 53 249 315 finally we demonstrate that the short wave limit of the soliton solutions recovers the soliton solutions of the short pulse equation which describes the propagation of ultra short optical pulses in nonlinear media

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

indeterminate moment problems and the theory of entire functions the set of solutions to an indeterminate hamburger moment problem is given by the nevanlinna parametrization involving four entire functions a b c d we review the properties of these functions and discuss recent progress in the still open problem of characterizing the class of entire functions which occur in this way they belong to the krein class and are of minimal exponential type we deduce the analogous parametrization of the set of solutions to an indeterminate stieltjes moment problem from the corresponding symmetric hamburger problem the survey ends with a discussion of three special cases where the functions a b c d are known explicitly

robust memory state feedback model predictive control for discrete time uncertain state delayed systems in this paper we propose a memory state feedback model predictive control mpc law for a discrete time uncertain state delayed system with input constraints the model uncertainty is assumed to be polytopic and the delay is assumed to be unknown but with a known upper bound we derive a sufficient condition for cost monotonicity in terms of lmi which can be easily solved by an efficient convex optimization algorithm a delayed state dependent quadratic function with an estimated delay index is considered for incorporating mpc problem formulation the mpc problem is formulated to minimize the upper bound of infinite horizon cost that satisfies the sufficient conditions therefore a less conservative sufficient conditions in terms of linear matrix inequality lmi can be derived to design a more robust mpc algorithm a numerical example is included to illustrate the effectiveness of the proposed method

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

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

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

nevanlinna extremal measures for some orthogonal polynomials related to birth and death processes abstract we consider two classes of birth and death processes which lead to indeterminate stieltjes moment problems and we calculate the entire functions c and d from the corresponding nevanlinna matrix this permits a direct computation of a nevanlinna extremal orthogonality measure for each of the processes

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

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

the 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

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

oscillation induced blow up to the modified camassa holm equation with linear dispersion abstract in this paper we provide a blow up mechanism to the modified camassa holm equation with varying linear dispersion we first consider the case when linear dispersion is absent and derive a finite time blow up result with an initial data having a region of mild oscillation a key feature of the analysis is the development of the burgers type inequalities with focusing property on characteristics which can be deduced from tracing the ratio between solution and its gradient using the continuity and monotonicity of the solutions we then extend this blow up criterion to the case of negative linear dispersion and determine that the finite time blow up can still occur if the initial momentum density is bounded below by the magnitude of the linear dispersion and the initial datum has a local mild oscillation region finally we demonstrate that in the case of non negative linear dispersion the formation of singularities can be induced by an initial datum with a sufficiently steep profile in contrast to the camassa holm equation with linear dispersion the effect of linear dispersion of the modified camassa holm equation on the blow up phenomena is rather delicate

wave breaking and peakons for a modified camassa holm equation in this paper we investigate the formation of singularities and the existence of peaked traveling wave solutions for a modified camassa holm equation with cubic nonlinearity the equation is known to be integrable and is shown to admit a single peaked soliton and multi peakon solutions of a different character than those of the camassa holm equation singularities of the solutions can occur only in the form of wave breaking and a new wave breaking mechanism for solutions with certain initial profiles is described in detail

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

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

orthogonal polynomials for a quartic birth and death process abstract a one parameter family of solutions for a quartic birth and death process has been obtained recently leading to new orthogonal polynomials the techniques which give these results are reviewed

well posedness and wave breaking of the degenerate novikov equation abstract in this paper we investigate the degenerate novikov equation which is proposed as a scaling limit of the novikov equation and is integrable since it admits bi hamiltonian structure and lax pair geometrically it arises from an intrinsic space curve flow in the centro equiaffine geometry in the periodic setting local well posedness of the initial value problem to the equation in the sobolev space is established via kato s theory it turns out that singularities of the solutions occur only in the form of wave breaking a sufficient condition on initial data is obtained to guarantee the formation of singularities finally a kind of singular solutions are also presented

exact solutions of some quadratic and quartic birth and death processes and related orthogonal polynomials we study a birth and death process with quartic transition rates for which the transition probabilities but for the honest ones are not unique for their related polynomials we exhibit infinitely many different orthogonality measures among which stems the honest one these results are obtained firstly by reducing the kolmogorov equations to linear partial differential equations and secondly by mapping them to partial differential equations with constant coefficients which can be solved analytically

some tricks from the symmetry toolbox for nonlinear equations generalizations of the camassa holm equation abstract the main subject of the paper is to give a survey and to present new methods on how integrability results i e results for symmetry groups inverse scattering formulations action angle transformations and the like can be transferred from one equation to others in case the equations are not related by backlund transformations as a main example the so called camassa holm equation is chosen for which the relevant results are obtained by having a look on the korteweg de vries kdv equation the camassa holm equation turns out to be a different factorization equation of the kdv it describes shallow water waves and reconciles the properties which were known for different orders of shallow water wave approximations we follow here an old method already marginally mentioned in fuchssteiner and fokas 1981 and fuchssteiner 1983 and recently applied by others olver and rosenau 1995 the method allows an immediate recovery of the recursion operator for the camassa holm equation from the invariance structure of the kdv although both equations are not related by backlund transformations however in addition and different from other approaches from there by use of the squared eigenfunction relation for the kdv equation the lax pair formulation for the different factorization equation is derived for the example under consideration it is of course the one obtained in camassa and holm 1993 since the methods proposed here can be transferred to any compatible factorization of recursion operators its application even in the special case which was chosen for illustration leads to a large class of integrable equations among which the camassa holm equation can be found as well as a three parameter family of generalizations of the equation the advantage of the general approach to the lax pair presentation is that direct transformations between action and angle variables are obtained so using this lax pair formulation as a novel result a direct transformation between action and angle variables for the camassa holm equation is derived further novel results in the paper are a hodograph link back from a backlund transformation of the camassa holm equation to a particular member of the kdv hierarchy additional symmetries and the construction of the conformal algebra for the hierarchy of the camassa holm equation the methods involved include hereditary operators bi hamiltonian formulations nilpotent flows and scaling symmetries

stability of peakons for the generalized modified camassa holm equation abstract in this paper we study orbital stability of peakons for the generalized modified camassa holm gmch equation which is a natural higher order generalization of the modified camassa holm mch equation and admits hamiltonian form and single peakons we first show that the single peakon is the usual weak solution of the pdes some sign invariant properties and conserved densities are presented next by constructing the corresponding auxiliary function h t x and establishing a delicate polynomial inequality relating to the two conserved densities with the maximal value of approximate solutions the orbital stability of single peakon of the gmch equation is verified we introduce a new approach to prove the key inequality which is different from that used for the mch equation this extends the result on the stability of peakons for the mch equation qu et al 2013 36 successfully to the higher order case and is helpful to understand how higher order nonlinearities affect the dispersion dynamics

receding horizon control for multiplicative noise stochastic systems with input delay abstract this paper is concerned with finite horizon stabilization control for a class of discrete time stochastic systems subject to multiplicative noise and input delay by constructing a new cost function a complete solution to the problem of finite horizon stabilization is given for the first time based on previous work zhang et al 2015 it is shown that the system can be stabilized in the mean square sense with the receding horizon control rhc if and only if two new inequalities on the terminal weighting matrices are satisfied moreover the two inequalities can be solved by using iterative algorithm the explicit stabilizing controller is derived by solving a finite horizon optimal control problem simulations demonstrate the effectiveness of the proposed method

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

stability of peakons for an integrable modified camassa holm equation with cubic nonlinearity considered herein is the dynamical stability of the single peaked soliton and periodic peaked soliton for an integrable modified camassa holm equation with cubic nonlinearity the equation is known to admit a single peaked soliton and multi peakon solutions and is shown here to possess a periodic peaked soliton by constructing certain lyapunov functionals it is demonstrated that the shapes of these waves are stable under small perturbations in the energy space

a new integrable equation with cuspons and w m shape peaks solitons in this paper we propose a new completely integrable wave equation mt mx u2 ux2 2m2ux 0 m u uxx the equation is derived from the two dimensional euler equation and is proven to have lax pair and bi hamiltonian structures this equation possesses new cusp solitons cuspons instead of regular peakons ce x ct with speed c through investigating the equation we develop a new kind of soliton solutions w m shape peaks solitons there exist no smooth solitons for this integrable water wave equation

integrable equations arising from motions of plane curves the motion of plane curves in klein geometry is studied it is shown that the kdv harry dym sawada kotera burgers the defocusing mkdv hierarchies the camassa holm and the kaup kupershmidt equation naturally arise from the motions of plane curves in sl 2 sim 2 sa 2 and a 2 geometries these local and nonlocal dynamics conserve global geometric quantities of curves such as perimeter and enclosed area motions of curves in euclidean special linear and similarity geometries corresponding to the traveling wave solutions of the mkdv kdv and burgers equations are discussed