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

the su u v group path integration path integration over the su u v group manifolds in particular su n and su n 1 1 are calculated in an appropriate polar coordinate representation giving explicitly their set of harmonic functions of all irreducible representations and the corresponding energy spectrum a path integral identity for multidimensional generalizations of the poschl teller potential is also stated

wave breaking analysis for the fornberg whitham equation abstract in this paper the wave breaking of the fornberg whitham equation is studied we first investigate the local well posedness and show a blow up scenario by local in time a priori estimates next we present a sufficient condition on the initial data to ensure wave breaking moreover the estimates of life span and blow up rate are obtained

squared eigenfunction symmetries for soliton equations part i abstract nonlinear integrable evolution equations in 1 1 dimensions arise from constraints of the 2 1 dimensional hierarchies associated with the kadomtsev petviashvili kp equation the modified kp equation and the dym equation respectively links of backlund type and of reciprocal type are known to exist between the 2 1 dimensional systems the corresponding links between the constrained flows are discussed in a general framework in particular squared eigenfunction symmetries generated by solutions of the associated linear scattering problems are considered the links between the soliton hierarchies are extended to these symmetries

the canonical formalism and path integrals in curved spaces the authors discuss the canonical and path integral quantisation of a particle moving on a curved manifold they advance a simple prescription to quantise the system within the canonical formalism an effective hamiltonian heff is defined as the classical hamiltonian plus a quantum potential term of order h cross 2 the hamilton operator is obtained by weyl ordering the expression that results when one replaces coordinates and momenta by their corresponding hermitian operators in the effective hamiltonian they construct the phase space path integral representation for the propagator in terms of heff when the propagator is expressed as a lagrangian path integral with an invariant measure an additional correction is introduced

on the cauchy problem for a class of shallow water wave equations with k 1 order nonlinearities abstract this paper considers the cauchy problem for a class of shallow water wave equations with k 1 order nonlinearities in the besov spaces t u t x 2 u u k x 3 u b u k 1 x u x 2 u b 1 u k x u which involves the camassa holm the degasperis procesi and the novikov equations as special cases firstly by means of the transport equation and the littlewood paley theory we obtain the local well posedness of the equations in the nonhomogeneous besov space b p r s s max 1 1 p 3 2 and p r 1 secondly we consider the local well posedness in b 2 r s with the critical index s 3 2 and show that the solutions continuously depend on the initial data thirdly the blow up criteria and the conservative property for the strong solutions are derived finally with the help of a new ovsyannikov theorem we investigate the gevrey regularity and analyticity of the solutions moreover we get a lower bound of the lifespan and the continuity of the data to solution mapping

on the cauchy problem for a two component b family system abstract this paper is concerned with the cauchy problem for a two component b family system we prove that the solution map of the cauchy problem of the b family system is not uniformly continuous in h s r h s 1 r s 5 2

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

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

the action angle transformation for soliton equations abstract for third order completely integrable equations in 1 1 dimensions canonical transformations which map in the multi soliton case gradients of action variables to gradients of angle variables are constructed furthermore this action angle map is shown always to be an infinitesimal symmetry group generator of the interacting soliton equation

a formula relating infinitesimal backlund transformations to hierarchy generating operators let u b eta u and l be respectively the elementary backlund transformation and hierarchy generating operator for the akns equations it is shown that db d eta b eta 1 sigma 3 l eta a similar formula relating to the general n n matrix spectral problem is also derived

stochastic perturbations to conservative dynamical systems on the plane ii recurrency conditions on considere les conditions sous lesquelles il existe une mesure invariante finie unique pour le systeme perturbe et la compacite faible de ces mesures pour un parametre de perturbation assez petit

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

application of exp function method to dullin gottwald holm equation in this paper we adopt the exp function method and the traveling wave transformation to study the so called dgh equation as a result a number of exact solutions of this equation have been found the family of solution including some exact solutions such as solitary wave pattern periodic traveling wave solution kink wave solution and new bounded wave solutions and explained some of the solutions physical meaning

limit theorem for a dynamical system in the presence of resonances and homoclinic orbits abstract the classical averaging theory is extended to a case where the oscillator admits a homoclinic orbit and the perturbation interacts resonantly with the free oscillations in the present approach we apply a method of vanishing diffusivity and obtain the weak convergence of the perturbed solutions to the solutions of an appropriately defined averaged system in the averaged system the concept of probability of trapping appears naturally the analysis involves an application of basic tools from the ito calculus followed by a reduction to a priori estimates on a family of parabolic partial differential equations and boundary layer analysis

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

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

blow up solutions of the general b equationa in this short paper we consider the b equation a new precise blow up scenario is described and some new results are obtained for the blow up phenomena

path integral treatment of ring shaped topological defects explicit path integration is carried out in a space with a ring shaped topological defect using toroidal coordinates the toroidal aharonov bohm experiment is taken as an example

global well posedness for the averaged euler equations in two dimensions abstract we prove global well posedness of the two dimensional averaged euler or euler equations for initial potential vorticity of class l2 this model generalizes the one dimensional fokas fuchssteiner camassa holm equation which describes the propagation of unidirectional waves on the surface of shallow water as such it can be realized as a geodesic equation for the h1 metric on the lie algebra of vector fields moreover in two dimensions the model obeys an advection equation for the so called potential vorticity in close analogy to the vorticity form of the euler equations we construct solutions to the weak form of the potential vorticity equation by taking the inviscid limit of solutions to a system regularized by artificial viscosity since the streamfunction vorticity relation is of order four we can show uniqueness even for potential vorticities in l2

new exact periodic wave solutions for the dullin gottwald holm equation abstract in this paper the dullin gottwald holm equation is studied using semi inverse method and integral bifurcation method new periodic waves such as peakon like periodic wave compacton like periodic wave and singular periodic wave are found and their dynamical behaviors and certain strange phenomena are explained using the proposed criterion the exact parametric representations of these waves are also presented

the bi hamiltonian structure of some nonlinear fifth and seventh order differential equations and recursion formulas for their symmetries and conserved covariants using a bi hamiltonian formulation we give explicit formulas for the conserved quantities and infinitesimal generators of symmetries for some nonlinear fifth and seventh order nonlinear partial differential equations among them the caudrey dodd gibbon sawada kotera equation and the kupershmidt equation we show that the lie algebras of the symmetry groups of these equations are of a very special form among the c vector fields they are generated from two given commuting vector fields by a recursive application of a single operator furthermore for some higher order equations those multisoliton solutions which for t asymptotically decompose into traveling wave solutions are characterized as eigenvector decompositions of certain operators

optimal control of the viscous weakly dispersive degasperis procesi equation abstract in this paper we study the optimal control problem for the viscous weakly dispersive degasperis procesi equation we deduce the existence and uniqueness of a weak solution to this equation in a short interval by using the galerkin method then according to optimal control theories and distributed parameter system control theories the optimal control of the viscous weakly dispersive degasperis procesi equation under boundary conditions is given and the existence of an optimal solution to the viscous weakly dispersive degasperis procesi equation is proved

recursion operators for infinitesimal transformations and their inverses for certain nonlinear evolution equations recursion operators generating infinitesimal transformations have been obtained systematically for certain nonlinear evolution equations their inverses have been evaluated and a double infinity of infinitesimal transformations and nonlinear equations generated

supersymmetric reciprocal transformation and its applications the supersymmetric analog of the reciprocal transformation is introduced this is used to establish a transformation between one of the supersymmetric harry dym equations and the supersymmetric modified korteweg de vries equation the reciprocal transformation as a backlund type transformation between these two equations is adopted to construct a recursion operator for the supersymmetric harry dym equation by proper factorization of the recursion operator a bi hamiltonian structure is found for the supersymmetric harry dym equation furthermore a supersymmetric kawamoto equation is proposed and is associated with the supersymmetric sawada kotera equation the recursion operator and odd bi hamiltonian structure of the supersymmetric kawamoto equation are also constructed

symmetries conserved quantities and hierarchies for some lattice systems with soliton structure basic invariants such as conserved quantities symmetries mastersymmetries and recursion operators are explicitly constructed for the following nonlinear lattice systems the modified korteweg de vries lattice the ablowitz ladik lattice the brusci ragnisco lattice the ragnisco tu lattice and some cases of the class of integrable systems introduced by bogoyavlensky the algorithmic basis for obtaining these quantities is described and the interrelation between the underlying mastersymmetry approach and the lax pair analysis is discussed by explicit presentation of the higher order members of the corresponding hierarchies new completely integrable lattice flows are found for all systems multi hamiltonian formulations are given

global existence and blow up for a weakly dissipative μdp equation abstract in this paper we study a weakly dissipative variant of the periodic degasperis procesi equation we show the local well posedness of the associated cauchy problem in h s s s 3 2 and discuss the precise blow up scenario for s 3 we also present explicit examples for globally existing solutions and blow up

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

global solutions and blow up phenomena for a generalized degasperis procesi equation abstract in this paper we mainly investigate global solutions and blow up phenomena for the cauchy problem of a generalized degasperis procesi equation we develop a new approach to consider the equation for 1 x u instead of 1 x 2 u we first prove a new conserved quantity 1 x u l 1 and give the local characterization of this conserved quantity these properties then allow us to improve considerably a previous global existence result in 38 and establish two new blow up results for strong solutions to the equation in the end based on vanishing viscosity method and using the conserved quantity we deduce the existence of global weak solutions to the equation for any initial data belonging to l 1 b v

a novel backlund invariance of a nonlinear differential equation abstract here the nonlinear ordinary differential equation yy s x is investigated the interest of the proposed study is twofold indeed the high nonlinearity exhibited by the considered equation does not allow the application of any linearization method on the other hand it turns out under suitable conditions to be equivalent to a nonlinear integral equation arising in extended kinetic theory the equivalence between the two nonlinear problems is exploited in particular conditions which need to be prescribed to establish such an equivalence are considered backlund transformations are applied to study the problem of interest specifically it is proved that the nonlinear differential equation enjoys an invariance property when the source term s x is represented by a solution of a suitable functional equation the latter is discussed and some solutions are explicitly written thus the corresponding backlund charts are depicted to show the obtained new invariances

color analysis theory of γ graded integrable evolution equations and super nijenhuis operators using the generalized grassmann variables or color variables a theory of graded integrable evolution equations is presented by elevating the treatments of magri gel fand dorfman and fuchssteiner of nonlinear integrable bosonic evolution equations to the graded case as an example it is shown that kupershmidt s super kdv is characterized by a z2 graded nijenhuis operator compatible with the underlying z2 graded hamiltonian structure

on reciprocal transformation of a 3 component camassa holm type system abstract we construct a reciprocal transformation to connect a 3 component camassa holm type system proposed by geng and xue with the first negative flow in a generalized mkdv hierarchy we discuss the hamiltonian pair and infinitely many conserved quantities for the generalized mkdv hierarchy

compacton and generalized kink wave solutions of the ch dp equation the ch dp equation is investigated by using the bifurcation method of planar systems and simulation method of differential equations the bifurcation phase portraits are drawn in different regions of parameter plane the planar graphs of compactons and generalized kink waves are simulated by using mathematical software maple exact explicit parameter expressions of compactons and implicit expressions of generalized kink wave solutions are given and the dynamic characters of these solutions are investigated

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

blow up phenomena and local well posedness for a generalized camassa holm equation with cubic nonlinearity abstract in this paper we consider the cauchy problem for a generalized camassa holm equation with cubic nonlinearity in besov spaces we first establish the local well posedness of the equation in the besov space b p r s by using the littlewood paley theory then under a sign condition we reach the sign preserved property and a precise blow up criterion applying this precise criterion we present a blow up result and the precise blow up rate for strong solutions to the equation

averaging principle for perturbed random evolution equations and corresponding dirichlet problems we continue the study of exit problems for perturbed random evolution equations corresponding to the weakly coupled elliptic pde systems in the present paper we consider the cases where the corresponding random evolutions stay in a given domain for ever with probability one but do not hinder the exit of the perturbed process we treat such problems by methods based on the averaging principle in such a way we also study the asymptotic behavior of the solutions of the corresponding perturbed dirichlet problems

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

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

initial boundary value problems for nonlinear dispersive wave equations in this paper we study initial value boundary problems of two types of nonlinear dispersive wave equations on the half line and on a finite interval subject to homogeneous dirichlet boundary conditions we first prove local well posedness of the rod equation and of the b equation for general initial data furthermore we are able to specify conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite life span in the case of finite time singularities we are able to describe the precise blow up scenario of breaking waves our approach is based on sharp extension results for functions on the half line or on a finite interval and several symmetry preserving properties of the equations under discussion

two types of bounded traveling wave solutions of a two component camassa holm equation the camassa holm equation ch is a well known integrable equation with is used to describe the velocity dynamics of shallow water waves this equation exhibits spontaneous emergence of peakons from smooth initial conditions the ch equation has been recently extended to a two component integrable system ch2 which includes both velocity and density variables in dynamics in this paper the bifurcation method of dynamical systems and numerical approach of differential equations are employed to study ch2 equation two types of bounded traveling waves are found one of them is called compacton another is called generalized kink wave the planar graphs of compactons and generalized kink awaves are simulated by using software maple exact explicit parameter expressions of the compactons and implicit expressions of the generalized kink waves solutions are given

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

solitons and discrete eigenfunctions of the recursion operator of non linear evolution equations i the caudrey dodd gibbon sawada kotera equation the solution of the third order isospectral equation of the caudrey dodd gibbon sawada kotera equation cdgske for soliton potential is obtained recursively from the riccati equation derived by iterating once the auto backlund transformation it is then shown that the discrete eigenfunctions of the sixth order recursion operator for this equation can be written in terms of the solutions of the isospectral equation the behaviour of the 1 soliton solution which has certain novel features is studied a sine gordon like equation resembling the double sine gordon equation is derived from the cdgske

remarks on the well posedness of camassa holm type equations in besov spaces abstract in this paper we prove the solution map of the cauchy problem of camassa holm type equations depends continuously on the initial data in nonhomogeneous besov spaces in the sense of hadamard by using the littlewood paley theory and the method introduced by kato 37 and danchin 21

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

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

properties of the solutions to the two component b family systems abstract in this paper some properties of the solutions are considered to the two component b family systems which cover the two component camassa holm equation and the two component degasperis procesi equation this paper is concerned with some aspects of the existence of global solutions persistence properties and propagation speed one first discusses the local well posedness and blow up scenario then establishes the sufficient conditions on the global existence of the solutions finally the persistence properties and propagation speed of strong solutions will also be investigated

path integration over the n dimensional euclidean group the path integral for the n dimensional free particle is considered according to the underlying symmetry the short time propagator is expanded in zonal spherical functions of the euclidean group g t n so n with respect to the subgroup h so n the group theoretical approach to path integration including the radial part is explicitly demonstrated

the abundant symmetry structure of hierarchies of nonlinear equations obtained by reciprocal links explicit computation for a kawamoto type equation shows that there is a rich associated symmetry structure for four separate hierarchies of nonlinear integrodifferential equations contrary to the general belief that symmetry groups for nonlinear evolution equations in 1 1 dimensions have to be abelian it is shown that in this case the symmetry group is noncommutative its semisimple part is isomorphic to the affine lie algebra a 1 1 associated to sl 2 c in two of the additional hierarchies that were found an explicit dependence of the independent variable occurs surprisingly the generic invariance for the kawamoto type equation obtained in rogers and carillo phys scr 36 865 1987 via a reciprocal link to the mobius invariance of the singularity equation of the kaup kupershmidt kk equation only holds for one of the additional hierarchies of symmetry groups thus the generic invariance is not a universal property for the complete symmetry group of equations obtained by reciprocal links in

a formula for obtaining new hereditary symmetries and new integrable equations in this paper we give a simple formula to obtain hereditary symmetries related to the isospectral eigenvalue problem of integrable equations using this formula i we prove that eigenfunctions of hereditary strong symmetry are symmetries for the whole hierarchy which improves the result of fokas and anderson j math phys 23 1066 1982 ii we find new integrable equations and iii we give strong symmetries of these new integrable equations and various equations for eigenfunctions of the isospectral eigenvalue problems

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

on sharp global well posedness and ill posedness for a fifth order kdv bbm type equation abstract we consider the cauchy problem associated to the recently derived higher order hamiltonian model for unidirectional water waves and prove global existence for given data in the sobolev space h s s 1 we also prove an ill posedness result by showing that the flow map is not continuous if the given data has sobolev regularity s 1 the results obtained in this work are sharp

on the use of isospectral eigenvalue problems for obtaining hereditary symmetries for hamiltonian systems we present an algorithmic method for obtaining an hereditary symmetry the generalized squared eigenfunction operator from a given isospectral eigenvalue problem this method is applied to the n n eigenvalue problem considered by ablowitz and haberman and to the eigenvalue problem considered by alonso the relevant hamiltonian formulations are also determined finally an alternative method is presented in the case two evolution equations are related by a miura type transformation and their hamiltonian formulations are known

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

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

path integration on hyperbolic spaces quantum mechanics on the hyperbolic spaces of rank one is discussed using a path interaction technique hyperbolic spaces are multi dimensional generalizations of the hyperbolic plane i e the poincare upper half plane endowed with a hyperbolic geometry the author evaluates the path integral on s1 equivalent to so n 1 so n and s2 equivalent to su n 1 s u 1 u n in a particular coordinate system yielding explicitly the wavefunctions and the energy spectrum furthermore one can exploit a general property of all these spaces namely that they can be parameterized by a pseudopolar coordinate system

blow up phenomena and global existence for the nonuniform weakly dissipative b equation abstract we first establish the local well posedness for the nonuniform weakly dissipative b equation which includes both the weakly dissipative camassa holm equation and the weakly dissipative degasperis procesi equation as its special cases we then study the blow up phenomena and the long time behavior of the solutions two blow up results are established for certain initial profiles moreover two sufficient conditions for the decay of the solutions are presented

on the periodic intermediate long wave equation the authors discuss some properties of a physically interesting nonlinear integro differential equation with periodic boundary conditions it is the natural periodic analogue of the intermediate long wave equation and it provides a periodic analogue of the benjamin ono equation in the appropriate limit due to the speciality of the integral operator the equation admits a backlund transformation an infinity of motion constants etc two simple periodic solutions are exhibited the equation may be transformed into more than one kind of bilinear equation

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

local symplectic operators and structures related to them matrices with entries being differential operators that endow the phase space of an evolution system with a pre symplectic structure are considered special types of such structures are explicitly described links with integrability geometry of loop spaces and backlund transformations are traces

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

multisolitons or the discrete eigenfunctions of the recursion operator of nonlinear evolution equations ii background for pt i see ibid vol 19 p 3755 1986 different definitions of the notion of the multisoliton are investigated it is demonstrated that even in more complicated cases like the cdgsk equation multisolitons can be completely characterised in terms of the discrete spectrum of the strong symmetry and that degeneracy of this spectrum leads to resonance solitons yielding singular solitons in case of the kdv equation and non singular ones in other cases furthermore a method is described which allows us to compute strong symmetries out of auto backlund transformations for nonlinear systems

theory of small aspect ratio waves in deep water in the limit of small values of the aspect ratio parameter or wave steepness which measures the amplitude of a surface wave in units of its wave length a model equation is derived from the euler system in infinite depth deep water without potential flow assumption the resulting equation is shown to sustain periodic waves which on the one side tend to the proper linear limit at small amplitudes on the other side possess a threshold amplitude where wave crest peaking is achieved an explicit expression of the crest angle at wave breaking is found in terms of the wave velocity by numerical simulations stable soliton like solutions experiencing elastic interactions propagate in a given velocities range on the edge of which they tend to the peakon solution published physica d 211 2005 377 390

bifurcations and some new traveling wave solutions for the ch γ equation in this paper the ch c equation is investigated by employing the bifurcation theory and the method of phase portraits analysis the dynamical behavior of equilibrium points and the bifurcations of phase portraits of the traveling wave system corresponding to this equation are discussed under some parameter conditions some bounded traveling wave solutions such as solitary waves peakons and periodic cusp waves are presented furthermore based on the auxiliary equation various new traveling wave solutions of parametric form are given the previous results for this equation are extended

on the inverse scattering approach and action angle variables for the dullin gottwald holm equation abstract the poisson brackets for the scattering data of the dullin gottwald holm equation are computed then the action angle variables are expressed in terms of the scattering data we also discuss on the inverse scattering problem for this equation

separation of variables in path integrals and path integral solution of two potentials on the poincare upper half plane the author discusses how to separate variables in path integrals it is assumed that a one dimensional problem with potential v x has an exact solution with energy levels elambda and wavefunctions psi lambda in order to perform the separation of variables a time transformation is performed back and forth in the path integral which allows one to insert the path integral solution corresponding to the potential v x the author illustrates the method by discussing some specific potential problems on the poincare upper half plane

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

new hierarchies of nonlinear completely integrable systems related to a change of variables for evolution parameters abstract several new hierarchies of completely integrable systems together with their respective hereditary recursion operators are given systematic methods for the construction of coupled completely integrable systems are developed it is shown that for most systems a change of variable for the evolution parameter is compatible with complete integrability bi hamiltonian structure and action angle variable representation a lorentz invariant recursion operator for the sine gordon equation is constructed

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

lie algebras for systems with mixed spectra i the scattering poschl teller potential starting from an n body quantum space we consider the lie algebraic framework where the poschl teller hamiltonian 1 2 2 c sech2 s csch2 is the single sp 2 r casimir operator the spectrum of this system is mixed it contains a finite number of negative energy bound states and a positive energy continuum of free states it is identified with the clebsch gordan series of the d d representation coupling the wave functions are the sp 2 r clebsch gordan coefficients of that coupling in the parabolic basis using only lie algebraic techniques we find the asymptotic behavior of these wave functions for the special pure trough potential s 0 we derive thus the transmission and reflection amplitudes of the scattering matrix

path integration over compact and noncompact rotation groups applications of group theoretical methods in the path integral formalism of nonrelativistic quantum theory are considered analysis of the symmetry of the lagrangian leads to the expansion of the short time propagator in matrix elements of unitary irreducible representations of the symmetry group identification of the coordinates with the group parameters transforms the path integral to integrals over the group manifold the integration is performed using the orthogonality of the representations compact and noncompact rotation groups are considered where the corresponding path integral is embedded in euclidean and pseudo euclidean spaces respectively the unit sphere and unit hyperboloid may either be viewed as the group manifold itself or at least as a group quotient in the first case fourier analysis leads to an expansion in group characters in the second case an expansion in zonal spherical functions is obtained as examples the groups so n su 2 so n 1 1 and su 1 1 are explicitly discussed the path integral on so n m and so n m in bispherical coordinates is also treated

a novel noncommutative kdv type equation its recursion operator and solitons a noncommutative kdv type equation is introduced extending the backlund chart in carillo et al symmetry integrability geom methods appl 12 087 2016 this equation called meta mkdv here i

direct dynamical energy cascade in the modified kdv equation in this study we examine the energy transfer mechanism during the nonlinear stage of the modulational instability mi in the modified korteweg de vries equation the particularity of this study consists in considering the problem essentially in the fourier space a dynamical energy cascade model of this process originally proposed for the focusing nls type equations is transposed to the mkdv setting using the existing connections between the kdv type and nls type equations the main predictions of the d cascade model are outlined and thoroughly discussed finally the obtained theoretical results are validated by direct numerical simulations of the mkdv equation using the pseudo spectral methods a general good agreement is reported in this study the nonlinear stages of the mi evolution are also investigated for the mkdv equation

optimal control of a class of nonlocal dispersive equations abstract in this paper the optimal control problem for a class of nonlocal dispersive equations is considered we introduce a kind of the definition of a weak solution to this equation and prove the existence of a unique weak solution in a special hilbert space s 0 t we also discuss the existence of optimal control solution to a class of nonlocal dispersive equations with the cost functional adopting the dubovitskii milyutin functional analytical approach we prove the necessary optimality condition of optimal control local maximum principle for the investigative system in fixed final horizon case

stochastic perturbations to conservative dynamical systems on the plane i convergence of invariant distributions we consider a nonlinear system on the plane given by an oscil lator with homoclinic orbits the above system is subjected to a perturbation composed of a deterministic part and a random white noise part assum ing the existence of a finite invariant measure to the perturbed system we deal with the convergence of the measures to a limit measure as the pertur bation parameter tends to zero the limit measure is constructed in terms of the action function of the unperturbed oscillator and the strong local l2 convergence of the associated densities is proved

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