rational recursion operators for integrable differential difference equations in this paper we introduce the concept of prehamiltonian pairs of difference operators demonstrate their connections with nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential difference equations we begin with a rigorous setup of the problem in terms of the skew field of rational pseudo difference operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational pseudo difference operators in particular we give a criteria for a rational operator to be weakly nonlocal a difference operator is called prehamiltonian if its image is a lie subalgebra with respect to the lie bracket on the difference field two prehamiltonian operators form a prehamiltonian pair if any linear combination of them is prehamiltonian then we show that a prehamiltonian pair naturally leads to a nijenhuis operator and a nijenhuis operator can be represented in terms of a prehamiltonian pair this provides a systematic method to check whether a rational operator is nijenhuis as an application we construct a prehamiltonian pair and thus a nijenhuis recursion operator for the differential difference equation recently discovered by adler and postnikov the nijenhuis operator obtained is not weakly nonlocal we prove that it generates an infinite hierarchy of local commuting symmetries we also illustrate our theory on the well known examples including the toda the ablowitz ladik and the kaup newell differential difference equations

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

smooth multisoliton solutions and their peakon limit of novikov s camassa holm type equation with cubic nonlinearity we consider novikov s camassa holm type equation with cubic nonlinearity in particular we present a compact parametric representation of the smooth bright multisolution solutions on a constant background and investigate their structure we find that the tau functions associated with the solutions are closely related to those of a model equation for shallow water waves sww introduced by hirota and satsuma this novel feature is established by applying the reciprocal transformation to the novikov equation we also show by specifying a complex phase parameter that the smooth soliton is converted to a novel singular soliton with single cusp and double peaks we demonstrate that both the smooth and singular solitons converge to a peakon as the background field tends to zero whereby we employ a method that has been developed for performing a similar limiting procedure for the multisoliton solutions of the camassa holm equation in the subsequent asymptotic analysis of the two and n soliton solutions we confirm their solitonic behavior remarkably the formulas for the phase shifts of the solitons as well as their peakon limits coincide formally with those of the degasperis procesi equation last we derive an infinite number of conservation laws of the novikov equation by using a relation between solutions of the novikov equation and those of the sww equation in appendix we prove various bilinear identities associated with the tau functions of the multisoliton solutions of the sww equation

nonlocal symmetries and a darboux transformation for the camassa holm equation we announce two new structures associated with the camassa holm ch equation a lie algebra of nonlocal symmetries and a darboux transformation for this important equation which we construct using only our symmetries we also extend our results to the associated camassa holm equation introduced by j schiff 1998 physica d 121 24 43

interpretation of af c algebras in łukasiewicz sentential calculus on donne un critere pour la non simplicite d une af c algebre 7b u en termes de proprietes de la theorie de la recursivite du groupe dimension k 0 7b u

traveling wave solutions of the camassa holm equation all weak traveling wave solutions of the camassa holm equation are classified we show that in addition to smooth solutions there are a multitude of traveling waves with singularities peakons cuspons stumpons and composite waves

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

canards in r3 abstract we give a geometric analysis of canard solutions in three dimensional singularly perturbed systems with a folded two dimensional critical manifold by analysing the reduced flow we obtain singular canard solutions passing through a singularity on the fold curve we classify these singularities called canard points as folded saddles folded nodes and folded saddle nodes we prove the existence of canard solutions in the case of the folded saddle we show the existence of canards in the folded node case provided a generic non resonance condition is satisfied and in a subcase of the folded saddle node the proof is based on the blow up method

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 good eq algebras a special algebra called eq algebra has been recently introduced by vilem novak its original motivation comes from fuzzy type theory in which the main connective is fuzzy equality eq algebras have three binary operations meet multiplication and fuzzy equality and a unit element they open the door to an alternative development of fuzzy many valued logic with the basic connective being a fuzzy equality instead of an implication this direction is justified by the idea presented by g w leibniz that a fully satisfactory logical calculus must be an equational one in this paper we continue the study of eq algebras and their special cases we introduce and study the prefilters and filters of separated eq algebras we give great importance to the study of good eq algebras as we shall see in this paper the goodness property and thus separateness is necessary for reasonably behaving algebras we enrich good eq algebras with a unary operation d the so called baaz delta fulfilling some additional assumptions that are heavily used in fuzzy logic literature we show that the characterization theorem obtained until now for representable good eq algebras also hold for the enriched algebra

the hidden dynamics of the rossler attractor abstract the fast slow dynamics of the rossler model in the chaotic regime are compared to the dynamics of the reduced slow model governing the flow along the slow invariant manifold on which the trajectory is restrained to evolve during the slow part of the attractor it is shown that the dynamics of the reduced model incorporate the slow time scales of the full model however instead of the fast dissipative time scale of the full model that restrains the trajectory on the slow invariant manifold the reduced model generates a new time scale that i relates to the curvature of the manifold and ii becomes faster as higher order corrections are incorporated in the approximation of the manifold and the reduced model it is shown that this new time scale does not characterize the motion of the solution on the manifold but it can be employed as a signal for the strengthening or weakening of the manifold depending on whether it relates to components of the reduced model that tend to lead its solution towards equilibrium or away from it finally it is demonstrated that the new time scale introduced by the reduced model has a significant role in determining the maximum accuracy that can be delivered by the singular perturbation methodology

hulls of ordered algebras projectability strong projectability and lateral completeness abstract there has been compelling evidence during the past decade that lattice ordered groups l groups play a far more significant role in the study of algebras of logic than it had been previously anticipated their key role has emerged on two fronts first a number of research articles have established that some of the most prominent classes of algebras of logic may be viewed as l groups with a modal operator second and perhaps more importantly recent research has demonstrated that the foundations of the conrad program for l groups can be profitably extended to a much wider class of algebras namely the variety of e cyclic residuated lattices that is residuated lattices that satisfy the identity x e e x here the term conrad program refers to paul conrad s approach to the study of l groups that analyzes the structure of individual or classes of l groups by primarily focusing on their lattices of convex l subgroups the present article building on the aforementioned works studies existence and uniqueness of the laterally complete projectable and strongly projectable hulls of e cyclic residuated lattices while these hulls first made their appearance in the context of functional analysis and in particular the theory of riesz spaces their introduction into the study of algebras of logic adds new tools and techniques in the area and opens up possibilities for a deep exploration of their logical counterparts

zero curvature formulations of dual hierarchies zero curvature formulations are given for the dual hierarchies of standard soliton equation hierarchies recently introduced by olver and rosenau including the physically interesting fuchssteiner fokas camassa holm hierarchy

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

dynamics of director fields an asymptotic equation for weakly nonlinear hyperbolic waves governed by variational principles is derived and analyzed the equation is used to study a nonlinear instability in the director field of a nematic liquid crystal it is shown that smooth solutions of the asymptotic equation break down in finite time also constructed are weak solutions of the equation that are continuous despite the fact that their spatial derivative blows up

pseudo bck algebras as partial algebras it is well known that the representation of several classes of residuated lattices involves lattice ordered groups an often applicable method to determine the representing group or groups from a residuated lattice is based on partial algebras the monoidal operation is restricted to those pairs that fulfil a certain extremality condition and else left undefined the subsequent construction applied to the partial algebra is easy transparent and leads directly to the structure needed for representation in this paper we consider subreducts of residuated lattices the monoidal and the meet operation being dropped the resulting algebras are pseudo bck semilattices assuming divisibility we can pass on to partial algebras also in this case to reconstruct the underlying group structure from this partial algebra if applicable is again straightforward we demonstrate the elegance of this method for two classes of pseudo bck semilattices semilinear divisible pseudo bck algebras and cone algebras

measures states and de finetti maps on pseudo bck algebras in this paper we extend the notions of states and measures presented in dvurecenskij and pulmannova 2000 12 to the case of pseudo bck algebras and study similar properties we prove that under some conditions the notion of a state in the sense of dvurecenskij and pulmannova 2000 12 coincides with the bosbach state and we extend to the case of pseudo bck algebras some results proved by kuhr only for pseudo bck semilattices we characterize extremal states and show that the quotient pseudo bck algebra over the kernel of a measure can be embedded into the negative cone of an archimedean group additionally we introduce a borel state and using results by kuhr and mundici 2007 28 we prove a relationship between de finetti maps bosbach states and borel states

splittings in gbl algebras ii the representable case abstract in this paper we continue our investigations on splitting algebras in varieties of residuated semi lattices carried out in 2 and 3 the focus is on representable varieties of gbl algebras i e varieties generated by their totally ordered members in this context several characterizations of splitting algebras are obtained

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

periodic orbits and chaos in fast slow systems with bogdanov takens type fold points abstract the existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast slow type having bogdanov takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation method and the blow up method in particular the blow up method is effectively used for analyzing the flow near the bogdanov takens type fold point in order to show that a slow manifold near the fold point is extended along the boutroux s tritronquee solution of the first painleve equation in the blow up space

functional completeness of bounded structures of fuzzy logic with wvt operators a simple criterion for a functional completeness of partially ordered commutative integral monoids with a weak very true operator is presented

some new integrable systems constructed from the bi hamiltonian systems with pure differential hamiltonian operators abstract when both hamiltonian operators of a bi hamiltonian system are pure differential operators we show that the generalized kupershmidt deformation gkd developed from the kupershmidt deformation in 10 offers an useful way to construct new integrable system starting from the bi hamiltonian system we construct some new integrable systems by means of the generalized kupershmidt deformation in the cases of harry dym hierarchy classical boussinesq hierarchy and coupled kdv hierarchy we show that the gkd of harry dym equation gkd of classical boussinesq equation and gkd of coupled kdv equation are equivalent to the new integrable rosochatius deformations of these soliton equations with self consistent sources we present the lax pair for these new systems therefore the generalized kupershmidt deformation provides a new way to construct new integrable systems from bi hamiltonian systems and also offers a new approach to obtain the rosochatius deformation of soliton equation with self consistent sources

backlund transformation and smooth multisoliton solutions for a modified camassa holm equation with cubic nonlinearity we present a compact parametric representation of the smooth bright multisoliton solutions for the modified camassa holm mch equation with cubic nonlinearity we first transform the mch equation to an associated mch equation through a reciprocal transformation and then find a novel backlund transformation between solutions of the associated mch equation and a model equation for shallow water waves sww introduced by ablowitz et al we combine this result with the expressions of the multisoliton solutions for the sww and modified korteweg de vries equations to obtain the multisoliton solutions of the mch equation subsequently we investigate the properties of the one and two soliton solutions as well as the general multisoliton solutions we show that the smoothness of the solutions is assured only if the amplitude parameters of solitons satisfy certain conditions we also find that at a critical value of the parameter beyond which the solution becomes singular the soliton solution exhibits a different feature from that of the peakon solution of the ch equation then by performing an asymptotic analysis for large time we obtain the formula for the phase shift and confirm the solitonic nature of the multisoliton solutions finally we use the backlund transformation to derive an infinite number of conservation laws of the mch equation

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

local analysis near a folded saddle node singularity abstract folded saddle nodes occur generically in one parameter families of singularly perturbed systems with two slow variables we show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems canards and delayed hopf bifurcations we combine techniques from geometric singular perturbation theory the blow up technique and from delayed hopf bifurcation theory complex time path analysis to analyze the flow near such folded saddle nodes in particular we show the existence of canards as intersections of stable and unstable slow manifolds to derive these canard results we extend the singularly perturbed vector field into the complex domain and study it along elliptic paths this enables us to extend the invariant slow manifolds beyond points where normal hyperbolicity is lost furthermore we define a way in way out function describing the maximal delay expected for generic solutions passing through a folded saddle node singularity branch points associated with the change from a complex to a real eigenvalue structure in the variational equation along the critical slow manifold make our analysis significantly different from the classical delayed hopf bifurcation analysis where these eigenvalues are complex only

the first second and fourth painleve equations on weighted projective spaces abstract the first second and fourth painleve equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces c p 3 p q r s with suitable weights p q r s determined by the newton diagrams of the equations or the versal deformations of vector fields singular normal forms of the equations a simple proof of the painleve property and symplectic atlases of the spaces of initial conditions are given with the aid of the orbifold structure of c p 3 p q r s in particular for the first painleve equation a well known painleve s transformation is geometrically derived which proves to be the darboux coordinates of a certain algebraic surface with a holomorphic symplectic form the affine weyl group dynkin diagram and the boutroux coordinates are also studied from a view point of the weighted projective space

representable pseudo bck algebras and integral residuated lattices abstract pseudo bck algebras arise as the 1 subreducts of integral residuated lattices in this note we characterize pseudo bck algebras that are subdirect products of linearly ordered pseudo bck algebras

well posedness and analyticity for the cauchy problem for the generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem for the generalized camassa holm equation which was proposed by hakkaev and kirchev s hakkaev k kirchev local well posedness and orbital stability of solitary wave solutions for the generalized camassa holm equation comm partial differential equations 30 2005 761 781 we establish the local well posedness for a range of besov spaces moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time which extends some results of danchin r danchin a few remarks on the camassa holm equation differential integral equations 14 2001 953 988 and himonas and misiolek a himonas g misiolek analyticity of the cauchy problem for an integrable evolution equation math ann 327 2003 575 584 to more general equations

relaxation oscillations in r3 abstract the existence of periodic relaxation oscillations in singularly perturbed systems with two slow and one fast variable is analyzed geometrically it is shown that near a singular periodic orbit a return map can be defined which has a one dimensional slow manifold with a stable invariant foliation under a natural hyperbolicity assumption on the singular periodic orbit this allows to prove the existence of a periodic relaxation orbit for small values of the perturbation parameter additionally the existence of an invariant torus is proved for the periodically forced van der pol oscillator the analysis is based on methods from geometric singular perturbation theory the blow up method is used to analyze the dynamics near the fold curves

an integrable semi discretization of the camassa holm equation and its determinant solution an integrable semi discretization of the camassa holm ch equation is presented the keys of its construction are bilinear forms and determinant structure of solutions of the ch equation determinant formulas of n soliton solutions of the continuous and semi discrete camassa holm equations are presented based on determinant formulas we can generate multi soliton multi cuspon and multi soliton cuspon solutions numerical computations using the integrable semi discrete camassa holm equation are performed it is shown that the integrable semi discrete camassa holm equation gives very accurate numerical results even in the cases of cuspon cuspon and soliton cuspon interactions the numerical computation for an initial value condition which is not an exact solution is also presented

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

nearest matrix with prescribed eigenvalues and its applications consider an n n matrix a and a set consisting of k n prescribed complex numbers lippert 2010 in a challenging article studied geometrically the spectral norm distance from a to the set of matrices whose spectra included specified set and constructed a perturbation matrix with minimum spectral norm such that a had in its spectrum this paper presents an easy practical computational method for constructing the optimal perturbation by improving and extending the methodology necessary definitions and lemmas of previous related works also some conceivable applications of this issue are provided

nonlocal symmetry darboux transformation and soliton cnoidal wave interaction solution for the shallow water wave equation abstract in classical shallow water wave sww theory there exist two integrable one dimensional sww equation hirota satsuma hs type and ablowitz kaup newell segur akns type in the boussinesq approximation in this paper we mainly focus on the integrable sww equation of akns type the nonlocal symmetry in form of square spectral function is derived starting from its lax pair infinitely many nonlocal symmetries are presented by introducing the arbitrary spectrum parameter these nonlocal symmetries can be localized and the sww equation is extended to enlarged system with auxiliary dependent variables then darboux transformation for the prolonged system is found by solving the initial value problem similarity reductions related to the nonlocal symmetry and explicit group invariant solutions are obtained it is shown that the soliton cnoidal wave interaction solution which represents soliton lying on a cnoidal periodic wave background can be obtained analytically interesting characteristics of the interaction solution between soliton and cnoidal periodic wave are displayed graphically

some properties of pseudo bck and pseudo bci algebras abstract pseudo bci algebras generalize both bci algebras and pseudo bck algebras which are a non commutative generalization of bck algebras in this paper following 37 we show that pseudo bci algebras are the residuation subreducts of semi integral residuated po monoids and characterize those pseudo bci algebras which are direct products of pseudo bck algebras and groups regarded as pseudo bci algebras we also show that the quasivariety of pseudo bci algebras is relatively congruence modular in fact we prove that this holds true for all relatively point regular quasivarieties which are relatively ideal determined in the sense that the kernels of relative congruences can be described by means of ideal terms

canard cycles in the presence of slow dynamics with singularities we study the cyclicity of limit periodic sets that occur in families of vector flelds of slow fast type the limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point at this turning point a stability change takes place on one side of the turning point the dynamics point strongly towards the curve of singularities on the other side the dynamics point away from the curve of singularities the presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point i e orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve all existing results deal with a non zero slow movement permitting to get a good estimate of the cyclicity by considering the slow divergence integral along the curve of singularities in this paper we study what happens when the slow dynamics exhibit singularities in particular our study includes the cyclicity of the slow fast 2 saddle cycle formed by a regular saddle connection the fast part and a part of the curve of singularities the slow part we see that the relevant information is no longer merely contained in the slow divergence integral this paper concerns the study of the cyclicity of limit periodic sets in a quite general class of slow fast vector flelds on a 2 manifold m we are interested in families of vector flelds x possibly depending on other parameters as well where the unperturbed fast vector fleld x0 has a curve of singular points called a critical curve we call a point p on normally attracting resp normally repelling when dx0 p has a strictly negative resp strictly positive eigenvalue corresponding to an eigendirection not tangent to when has both normally attracting points and normally repelling points it may occur that x0 has orbits connecting two such points let f be such a fast orbit so that the limit and fi limit lie on we study the limit periodic set lf formed by f and the piece of going from the limit of f to the fi limit of f of course the nonzero eigenvalue of dx0 p must bifurcate along this piece of assume that this happens in a unique point p called a turning point one also says normal hyperbolicity of x0 is lost in p in such a situation it is possible that the limit periodic set perturbs into one or more isolated periodic orbits such orbits are called canard cycles

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

the conrad program from l groups to algebras of logic abstract a number of research articles have established the significant role of lattice ordered groups l groups in logic the fact that underpins these studies is the realization that important algebras of logic may be viewed as l groups with a modal operator these connections are just the tip of the iceberg the purpose of the present article is to lay the groundwork for and provide significant initial contributions to the development of a conrad type approach to the study of algebras of logic the term conrad program refers to paul conrad s approach to the study of l groups which analyzes the structure of individual or classes of l groups by primarily using strictly lattice theoretic properties of their lattices of convex l subgroups the present article demonstrates that large parts of the conrad program can be profitably extended in the setting of e cyclic residuated lattices that is residuated lattices that satisfy the identity x e e x an indirect benefit of this work is the introduction of new tools and techniques in the study of algebras of logic and the enhanced role of the lattice of convex subalgebras of a residuated lattice

generalized mv algebras we generalize the notion of an mv algebra in the context of residuated lattices to include noncommutative and unbounded structures we investigate a number of their properties and prove that they can be obtained from lattice ordered groups via a truncation construction that generalizes the chang mundici functor this correspondence extends to a categorical equivalence that generalizes the ones established by d mundici and a dvurey the decidability of the equational theory of the variety of generalized mv algebras follows from our analysis

finite propagation speed for the camassa holm equation we prove that any classical solution of the camassa holm equation will have compact support if its initial data has this property

splittings in gbl algebras i the general case abstract we study and characterize splitting algebras in varieties of integral residuated semi lattices the main result is a complete characterization of the splitting algebras in the variety of gbl e w algebras i e integral bounded commutative and divisible residuated lattices

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

lax integrability and the peakon problem for the modified camassa holm equation peakons are special weak solutions of a class of nonlinear partial differential equations modelling non linear phenomena such as the breakdown of regularity and the onset of shocks we show that the natural concept of weak solutions in the case of the modified camassa holm equation studied in this paper is dictated by the distributional compatibility of its lax pair and as a result it differs from the one proposed and used in the literature based on the concept of weak solutions used for equations of the burgers type subsequently we give a complete construction of peakon solutions satisfying the modified camassa holm equation in the sense of distributions our approach is based on solving certain inverse boundary value problem the solution of which hinges on a combination of classical techniques of analysis involving stieltjes continued fractions and multi point pade approximations we propose sufficient conditions needed to ensure the global existence of peakon solutions and analyze the large time asymptotic behaviour whose special features include a formation of pairs of peakons that share asymptotic speeds as well as toda like sorting property

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