the complex geometry of weak piecewise smooth solutions of integrable nonlinear pde s of shallow water and dym type an extension of the algebraic geometric method for nonlinear integrable pde s is shown to lead to new piecewise smooth weak solutions of a class of n component systems of nonlinear evolution equations this class includes among others equations from the dym and shallow water equation hierarchies the main goal of the paper is to give explicit theta functional expressions for piecewise smooth weak solutions of these nonlinear pde s which are associated to nonlinear subvarieties of hyperelliptic jacobians

a family of wave breaking equations generalizing the camassa holm and novikov equations a 4 parameter polynomial family of equations generalizing the camassa holm and novikov equations that describe breaking waves is introduced a classification of low order conservation laws peaked travelling wave solutions and lie symmetries is presented for this family these classifications pick out a 1 parameter equation that has several interesting features it reduces to the camassa holm and novikov equations when the polynomial has degree two and three it has a conserved h1 norm and it possesses n peakon solutions when the polynomial has any degree and it exhibits wave breaking for certain solutions describing collisions between peakons and anti peakons in the case n 2

the periodic cauchy problem for a two component non isospectral cubic camassa holm system abstract in this paper we study the periodic cauchy problem for a two component non isospectral cubic camassa holm system which includes the fokas olver rosenau qiao forq or modified camassa holm mch equation and the two component mch system as two special cases the system is integrable in the sense of possessing a non isospectral lax pair with the spectrum depending on time t and admits multi peakon solutions in an explicit form furthermore we establish the local well posedness for the system in the besov space b 2 r s t with s 3 2 1 r where the key ingredients include the friedrichs regularization method the littlewood paley decomposition theory and the transport theory in besov spaces then we derive a precise blow up criteria which is dependent of the parameters t and t moreover by the intrinsic structure of the system we obtain a new blow up result for strong solutions with sufficient conditions on the initial data and parameters the entire proof procedure relies upon a newly derived transport equation which is involved in nonlocal velocity term along the characteristic curves

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

nonisospectral scattering problems a key to integrable hierarchies we show that certain partial differential equations associated to nonisospectral scattering problems in 2 1 dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equations this is illustrated using an extension of a known second order and two new third order nonisospectral scattering problems these scattering problems allow us to derive new hierarchies of integrable partial differential equations in both 1 1 and 2 1 dimensions together with their underlying linear problems isospectral and nonisospectral and also new hierarchies of integrable ordinary differential equations again with their underlying linear problems

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

multipeakons and the classical moment problem abstract classical results of stieltjes are used to obtain explicit formulas for the peakon antipeakon solutions of the camassa holm equation the closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem it is shown that collisions occur only in peakon antipeakon pairs and the details of the collisions are analyzed using results from the moment problem a sharp result on the steepening of the slope at the time of collision is given asymptotic formulas are given and the scattering shifts are calculated explicitly

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

global weak solutions to the novikov equation abstract the existence of global weak solutions to the cauchy problem for the novikov equation is established in the space c 0 r l 0 h 1 r without the sign condition on the initial value the limit of viscous approximations for the equation is used to establish the existence of the global weak solution the key elements in our analysis include a new one sided super bound and a new higher norm estimate on the first order derivatives of the solution

the existence of global strong and weak solutions for the novikov equation abstract the well posedness of the global strong and weak solutions for the novikov equation is investigated provided that initial value u 0 h s s 3 2 and satisfying a sign condition the existence and uniqueness of global strong solutions for the equation are shown to be valid in sobolev space the estimates in h q r space with 0 q 1 2 which are derived from the equation itself are developed to prove the existence and uniqueness of the global weak solutions

an application of pfaffians to multipeakons of the novikov equation and the finite toda lattice of bkp type abstract the novikov equation is an integrable analogue of the camassa holm equation with a cubic rather than quadratic nonlinear term both these equations support a special family of weak solutions called multipeakon solutions in this paper an approach involving pfaffians is applied to study multipeakons of the novikov equation first we show that the novikov peakon odes describe an isospectral flow on the manifold cut out by certain pfaffian identities then a link between the novikov peakons and the finite toda lattice of bkp type b toda lattice is established based on the use of pfaffians finally certain generalizations of the novikov equation and the finite b toda lattice are proposed together with special solutions to our knowledge it is the first time that the peakon problem is interpreted in terms of pfaffians

local well posedness and blow up criteria for a two component novikov system in the critical besov space abstract in this paper we mainly investigate the cauchy problem of a two component novikov system we first prove the local well posedness of the system in besov spaces b p r s 1 b p r s with p r 1 s max 1 1 p 3 2 by using the littlewood paley theory and transport equations theory then by virtue of logarithmic interpolation inequalities and the osgood lemma we establish the local well posedness of the system in the critical besov space b 2 1 1 2 b 2 1 3 2 moreover we present two blow up criteria for the system by making use of the conservation laws

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

local well posedness and blow up criteria for a three component camassa holm type equation considered herein is a three component camassa holm type equation that admits a bi hamiltonian structure and infinitely many conserved quantities in this paper we establish the local well posedness of this system in bp rs we also deduce two blow up criteria for this system

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

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

global analyticity for a generalized camassa holm equation and decay of the radius of spatial analyticity abstract global analytic solution in both the time and the space variables is proved for the cauchy problem of a generalized ch equation which contains as its members two integrable equations namely the camassa holm and the novikov equations the main assumptions are that the initial datum u 0 x is analytic on the line it has uniform radius of analyticity r 0 0 and is such that the mckean quantity m 0 x 1 x 2 u 0 x does not change sign furthermore an explicit lower bound on the radius of space analyticity at later times is obtained which is of the form l 3 exp l 1 exp l 2 t where l 1 l 2 and l 3 are appropriate positive constants

a new integrable two component system with cubic nonlinearity in this paper a new integrable two component system mt m uxvx uv uvx uxv x nt n uxvx uv uvx uxv x where m u uxx and n v vxx is proposed our system is a generalized version of the integrable system mt m ux2 u2 x which was shown having cusped solution cuspon and w m shape soliton solutions by qiao j math phys 47 112701 2006 the new system is proven integrable not only in the sense of lax pair but also in the sense of geometry namely it describes pseudospherical surfaces accordingly infinitely many conservation laws are derived through recursion relations furthermore exact solutions such as cuspons and w m shape solitons are also obtained

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

qualitative analysis for the new shallow water model with cubic nonlinearity abstract considered in this paper is the new shallow water model with cubic nonlinearity which admits the single peaked solitons and multi peakon solutions and includes both the modified camassa holm equation also called fokas olver rosenau qiao equation and the novikov equation as two special cases the main investigation is the cauchy problem of the new shallow water model with qualitative properties of its solutions we first establish the local well posedness for the cauchy problem of the new shallow water model and then derive a precise blow up scenario and the blow up mechanisms for solutions with certain initial profiles moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time finally we investigate the persistence properties of the solution to the cauchy problem of the new shallow water model

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

analytic bethe ansatz related to a one parameter family of finite dimensional representations of the lie superalgebra as is well known the type 1 lie superalgebra admits a one parameter family of finite dimensional irreducible representations we have carried out an analytic bethe ansatz related to this family of representations we present formulae which are deformations of previously proposed determinant formulae labelled by a young superdiagram these formulae will provide a transfer matrix eigenvalue in a dressed vacuum form related to the solutions of a graded yang baxter equation which depend not only on the spectral parameter but also on a non additive continuous parameter a class of transfer matrix functional relations among these formulae is briefly mentioned

blow up phenomenon for the integrable novikov equation abstract in this paper we investigate a new integrable equation derived recently by v s novikov generalizations of the camassa holm equation j phys a 42 34 2009 342002 14 pp analogous to the camassa holm equation and the degasperis procesi equation this new equation possesses the blow up phenomenon under the special structure of this equation we establish sufficient conditions on the initial data to guarantee the formulation of singularities in finite time a global existence result is also found

holder continuity of the solution map for the novikov equation the novikov equation ne has been discovered recently as a new integrable equation with cubic nonlinearities that is similar to the camassa holm and degasperis procesi equations which have quadratic nonlinearities ne is well posed in sobolev spaces hs on both the line and the circle for s 3 2 in the sense of hadamard and its data to solution map is continuous but not uniformly continuous this work studies the continuity properties of ne further for initial data in hs s 3 2 it is shown that the solution map for ne is holder continuous in hr topology for all 0 r s with exponent depending on s and r

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

a generalized nonisospectral camassa holm equation and its multipeakon solutions abstract motivated by the paper of beals sattinger and szmigielski 2000 3 we propose an extension of the camassa holm equation which also admits the multipeakon solutions the novel aspect is that our approach is mainly based on classic determinant technique furthermore the proposed equation is shown to possess a nonisospectral lax pair

on the cauchy problem for a two component b family system with high order nonlinearity abstract this paper is devoted to the cauchy problem of two component b family system with high order nonlinearity which includes the well known camassa holm equation degasperis procesi equation novikov equation and their two component extension systems as special cases firstly we establish the local well posedness of the system in the non homogeneous besov space by using the littlewood paley decomposition and the transport equation theory secondly we investigate the blow up of strong solutions to the system thirdly we study the gevrey regularity and analyticity of the solutions to the system in the gevrey sobolev spaces by using the generalized ovsyannikov theorem moreover we find a lower bound of the lifespan and prove the continuity of the data to solution mapping

optimal distributed controls of a class of nonlinear dispersive equations with cubic nonlinearity abstract this paper is devoted to the optimal distributed control problem governed by a class of nonlinear dispersive equations with cubic nonlinearity which contains the famous novikov equation as special case we first investigate the existence and uniqueness of the weak solution for the controlled system and then find an optimal solution for the controlled system with the generalized cost functional moreover by means of the method suggested by a ya dubovitskii and a a milyutin we establish the first order necessary optimality condition of optimal control for the controlled system in the fixed final horizon case

riemann hilbert approach for the fqxl model a generalized camassa holm equation with cubic and quadratic nonlinearity in this paper the inverse scattering transform associated with a riemann hilbert problem is formulated for the fqxl model a generalized camassa holm equation mt 12k1 m u2 ux2 x 12k2 2mux mxu m u uxx which was originally included in the work of fokas physica d 87 145 1995 and was recently shown to be integrable in the sense of lax pair bi hamilton structure and conservation laws by qiao xia and li e print arxiv 1205 2028v2 2012 we have discussed the following properties direct scattering problems and jost solutions asymptotical and analytical behavior of jost solutions the scattering equations in a riemann hilbert problem and the multi soliton solutions of the fqxl model then one soliton and two soliton solutions are presented in a parametric form as a special case of multi soliton solutions

orbital stability of the train of peakons for an integrable modified camassa holm equation abstract in this paper we consider the modified camassa holm equation with cubic nonlinearity which is integrable and admits the single peakons and multi peakons using energy argument and combining the method of the orbital stability of a single peakon with monotonicity of the local energy norm we prove that the sum of n sufficiently decoupled peakons is orbitally stable in the energy space

well posedness wave breaking and peakons for a modified μ camassa holm equation considered herein is a modified periodic camassa holm equation with cubic nonlinearity which is called the modified camassa holm equation the proposed equation is shown to be formally integrable with the lax pair and bi hamiltonian structure local well posedness of the initial value problem to the modified camassa holm equation in the besov space is established existence of peaked traveling wave solutions and formation of singularities of solutions for the equation are then investigated it is shown that the equation admits a single peaked soliton and multi peakon solutions with a similar character of the camassa holm equation singularities of the solutions can occur only in the form of wave breaking and several wave breaking mechanisms for solutions with certain initial profiles are described in detail

negative order korteweg de vries equations in this paper based on the regular korteweg de vries kdv system we study negative order kdv nkdv equations particularly their hamiltonian structures lax pairs conservation laws and explicit multisoliton and multikink wave solutions thorough bilinear backlund transformations the nkdv equations studied in our paper are differential and actually derived from the first member in the negative order kdv hierarchy the nkdv equations are not only gauge equivalent to the camassa holm equation through reciprocal transformations but also closely related to the ermakov pinney systems and the kupershmidt deformation the bi hamiltonian structures and a darboux transformation of the nkdv equations are constructed with the aid of trace identity and their lax pairs respectively the single and double kink wave and bell soliton solutions are given in an explicit formula through the darboux transformation the one kink wave solution is expressed in the form of tanh while the one bell soliton is in the form of sech and both forms are very standard the collisions of two kink wave and two bell soliton solutions are analyzed in detail and this singular interaction differs from the regular kdv equation multidimensional binary bell polynomials are employed to find bilinear formulation and backlund transformations which produce n soliton solutions a direct and unifying scheme is proposed for explicitly building up quasiperiodic wave solutions of the nkdv equations furthermore the relations between quasiperiodic wave solutions and soliton solutions are clearly described finally we show the quasiperiodic wave solution convergent to the soliton solution under some limit conditions

symbolic analysis and exact travelling wave solutions to a new modified novikov equation in this paper a new modified novikov equation is introduced multiple exact travelling wave solutions are obtained by using the extended tanh function method furthermore based on the homogeneous balance method an auto backlund transformation is derived and some solitary wave solutions to this new equation are established

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

non uniform dependence on initial data for the modified μ camassa holm equation considered herein is the initial value problem for the modified camassa holm equation it is shown that the solution map of this problem is not uniformly continuous in sobolev spaces hs s for s 5 2 on the circle based on the well posedness result and the lifespan for this problem the method of approximate solutions is utilized

analysis on the blow up of solutions to a class of integrable peakon equations abstract we investigate the blow up mechanism of solutions to a class of quasilinear integrable equations which could possess peakons the dynamics of the blow up quantity along the characteristics is established by the riccati type differential inequality which involves the interaction among three parts a local nonlinearity a nonlocal term and a term stemming from the weak linear dispersion to analyse the interplay among these quantities we provide two different approaches the first one is designed for the case when the equations do not exhibit a weak linear dispersion and hence focuses on the interplay between the first two parts the method is based on a refined analysis on either evolution of the solution u and its gradient u x that is c u u x or the growth rate of the relative ratio u x u the second one handles the general situation when all of three parts are present the idea is to extract the truly blow up component from the riccati type differential inequality and utilizes the morawetz type identity or higher order conservation laws to show that such a component blows up in finite time before the other component degenerates

well posedness and persistence properties for the novikov equation abstract recently novikov found a new integrable equation we call it the novikov equation in this paper which has nonlinear terms that are cubic rather than quadratic and admits peaked soliton solutions peakons firstly we prove that the cauchy problem for the novikov equation is locally well posed in the besov spaces b 2 r s which generalize the sobolev spaces h s with the critical index s 3 2 then well posedness in h s with s 3 2 is also established by applying kato s semigroup theory finally we present two results on the persistence properties of the strong solution for the novikov equation

continuity properties of the data to solution map for the generalized camassa holm equation abstract this work studies a generalized camassa holm equation with higher order nonlinearities g kbch the camassa holm the degasperis procesi and the novikov equations are integrable members of this family of equations g kbch is well posed in sobolev spaces h s s 3 2 on both the line and the circle and its solution map is continuous but not uniformly continuous in this work it is shown that the solution map is holder continuous in h s equipped with the h r topology for 0 r s and the holder exponent is expressed in terms of s and r

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 direct algorithm of one dimensional optimal system for the group invariant solutions a direct and systematic algorithm is proposed to find one dimensional optimal system for the group invariant solutions which is attributed to the classification of its corresponding one dimensional lie algebra since the method is based on different values of all the invariants the process itself can both guarantee the comprehensiveness and demonstrate the inequivalence of the optimal system with no further proof to leave the algorithm clear we illustrate each stage with a couple of well known examples the korteweg de vries equation and the heat equation finally we apply our method to the novikov equation and use the found optimal system to investigate the corresponding invariant solutions

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

blow up phenomena and local well posedness for a generalized camassa holm equation in the critical besov space abstract in this paper we mainly study the cauchy problem for a generalized camassa holm equation in a critical besov space first by using the littlewood paley decomposition transport equations theory logarithmic interpolation inequalities and osgood s lemma we establish the local well posedness for the cauchy problem of the equation in the critical besov space b 2 1 1 2 then we give a new blow up criterion for the cauchy problem of the equation finally we present a new blow up result and the exact blow up rate of strong solutions to the equation by making use of the conservation law and the obtained blow up criterion

new integrable hierarchy its parametric solutions cuspons one peak solitons and m w shape peak solitons in this paper we propose a new completely integrable hierarchy particularly in the hierarchy we draw two new soliton equations 1 mt 12 1 m2 xxx 12 1 m2 x 2 mt mx u2 ux2 2m2ux 0 m u uxx the first one is the second positive member in the hierarchy while the second one is the second negative member in the hierarchy both equations can be derived from the two dimensional euler equation by using the approximation procedure all equations in the hierarchy are proven to have bi hamiltonian operators and lax pairs through solving a crucial matrix equation moreover we develop parametric solutions of the entire hierarchy through constructing two kinds of constraints one is for all negative members of the hierarchy on a symplectic submanifold and the other is for all positive members in the standard symplectic space the most interesting things are both equations possess new type of peaked solitons continuous and piecewise smooth w m shape peak soliton solutions in addition we find new cusp solit

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

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 dual modified korteweg de vries fokas qiao equation geometry and local analysis we study a bi hamiltonian equation with cubic nonlinearity shown to appear in the theory of water waves by fokas derived by qiao using the two dimensional euler equation and also known to arise as the dual of the modified korteweg de vries equation thanks to work by fokas fuchssteiner olver and rosenau we present a quadratic pseudo potential we compute infinite sequences of local and nonlocal conservation laws and we construct an infinite dimensional lie algebra of symmetries which contains a semi direct sum of the sl 2 r loop algebra and the centerless virasoro algebra as an application we prove a theorem on the existence of smooth solutions and we construct some explicit examples moreover we consider the cauchy problem and we prove existence and uniqueness of weak solutions in the sobolev space hq 2 r q 1 2

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

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