third order differential equations describing pseudospherical surfaces third order differential equations which describe pseudospherical surfaces are considered the complete classification of a class of such equations is given a linear problem with one or more parameters also known as zero curvature representation for which the equation is the integrability condition is explicitly given the classification provides five large families of differential equations third order nonlinear dispersive wave equations such as the camassa holm equation and degasperis procesi equation are examples contained in the classification many other explicit examples are included

hamiltonian formulation of surfaces with constant gaussian curvature dirac s method for constrained hamiltonian systems is used to describe surfaces of constant gaussian curvature a geometrical free energy for which these surfaces are equilibrium states is introduced and interpreted as an action an equilibrium surface can then be generated by the evolution of a closed space curve since the underlying action depends on second derivatives the velocity of the curve and its conjugate momentum must be included in the set of phase space variables furthermore the action is linear in the acceleration of the curve and possesses a local symmetry reparametrization invariance which implies primary constraints in the canonical formalism these constraints are incorporated into the hamiltonian through lagrange multiplier functions that are identified as the components of the acceleration of the curve the formulation leads to four first order partial differential equations one for each canonical variable with the appropriate choice of parametrization only one of these equations has to be solved to obtain the surface which is swept out by the evolving space curve to illustrate the formalism several evolutions of pseudospherical surfaces are discussed

high frequency nonlinear optics from the nonlinear schrodinger approximation to ultrashort pulses equations after a brief introduction and physical motivation we show how the nonlinear schrodinger nls equation can be derived from a general class of nonlinear hyperbolic systems its purpose is to describe the behaviour of high frequency oscillating wave packets over a large time scale that requires us to take into account diffractive effects we then show that the nls approximation fails for short pulses and propose some alternative models including a modified schrodinger equation with improved frequency dispersion it turns out that these models have better properties and are quite accurate for short pulses for ultrashort pulses however they must also be abandoned for more complex approaches we give the main steps for such an analysis and explain one striking fact about ultrashort pulses their dynamics in dispersive media is linear

a non local elliptic hyperbolic system related to the short pulse equation abstract in this paper we prove the well posedness of a non local elliptic hyperbolic system related to the short pulse equation it is a model which describes the evolution of the electrical field of linearly polarized continuum spectrum pulses in optical waveguides including fused silica telecommunication type or photonic crystal fibers as well as hollow capillaries filled with transparent gases

on finite groups in which coprime commutators are covered by few cyclic subgroups the coprime commutators j and j were recently introduced as a tool to study properties of finite groups that can be expressed in terms of commutators of elements of coprime orders every element of a finite group g is both a 1 commutator and a 0 commutator now let j 2 and let x be the set of all elements of g that are powers of j 1 commutators an element g is a j commutator if there exist a x and b g such that g a b and a b 1 for j 1 let y be the set of all elements of g that are powers of j 1 commutators an element g is a j commutator if there exist a b y such that g a b and a b 1 the subgroups of g generated by all j commutators and all j commutators are denoted by j g and j g respectively for every j 2 the subgroup j g is precisely the last term g of the lower central series of g while for every j 1 the subgroup j g is precisely the last term of the lower central series of j 1 g that is j g j 1 g in the present paper we prove that if g possesses m cyclic subgroups whose union contains all j commutators of g then j g contains a subgroup of m bounded order which is normal in g and has the property that j g is cyclic if j 2 and g possesses m cyclic subgroups whose union contains all j commutators of g then the order of j g is m bounded

defocusing complex short pulse equation and its multi dark soliton solution in this paper we propose a complex short pulse equation of both focusing and defocusing types which governs the propagation of ultrashort pulses in nonlinear optical fibers it can be viewed as an analog of the nonlinear schrodinger nls equation in the ultrashort pulse regime furthermore we construct the multi dark soliton solution for the defocusing complex short pulse equation through the darboux transformation and reciprocal hodograph transformation one and two dark soliton solutions are given explicitly whose properties and dynamics are analyzed and illustrated

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

complex short pulse and coupled complex short pulse equations abstract in the present paper we propose a complex short pulse equation and a coupled complex short equation to describe ultra short pulse propagation in optical fibers they are integrable due to the existence of lax pairs and infinite number of conservation laws furthermore we find their multi soliton solutions in terms of pfaffians by virtue of hirota s bilinear method one and two soliton solutions are investigated in details showing favorable properties in modeling ultra short pulses with a few optical cycles especially same as the coupled nonlinear schrodinger equation there is an interesting phenomenon of energy redistribution in soliton interactions it is expected that for the ultra short pulses the complex and coupled complex short pulses equation will play the same roles as the nonlinear schrodinger equation and coupled nonlinear schrodinger equation

fourth order evolution equations which describe pseudospherical surfaces abstract differential equations that describe pseudospherical surfaces are considered these equations are equivalent to the structure equations of a metric with gaussian curvature k 1 they can also be described as the compatibility condition of an associated linear problem also referred to as a zero curvature representation a complete and explicit classification of a class of fourth order evolution equations is given the classification provides four huge classes referred to as types i iv of fourth order evolution equations that describe pseudospherical surfaces together with the associated one or more parameter linear problems the differential equations of each type are determined by choosing certain arbitrary differentiable functions fourth order member of the burgers hierarchy and a modified kuramoto sivashinsky equation are examples of equations described by types i and iv respectively many other explicit examples are presented

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

conservation laws and calapso guichard deformations of equations describing pseudo spherical surfaces the relation between the chern and tenenblat approach to conservation laws of equations describing pseudo spherical surfaces conservation laws from pseudo spherical structure and the more familiar riccati equation approach conservation laws from associated linear problem is investigated two examples cylindrical korteweg de vries kdv and lund regge equations are presented chern and tenenblat s point of view is then connected with the theory of soliton surfaces a generalization of the original chern tenenblat construction of conservation laws results and a reasonable family of large deformations for scalar equations describing pseudo spherical surfaces the equations describing calapso guichard surfaces can be introduced it is shown that these equations are also the integrability condition of linear problems

a class of quasilinear second order partial differential equations which describe spherical or pseudospherical surfaces abstract second order partial differential equations which describe spherical surfaces ss or pseudospherical surfaces pss are considered these equations are equivalent to the structure equations of a metric with gaussian curvature k 1 or k 1 respectively and they can be seen as the compatibility condition of an associated su 2 valued or sl 2 r valued linear problem also referred to as a zero curvature representation under certain assumptions we give a complete and explicit classification of equations of the form z t t a z z x z t z x x b z z x z t z x t c z z x z t describing pss or ss in terms of some arbitrary differentiable functions several examples of such equations are provided by choosing the arbitrary functions in particular well known equations which describe pseudospherical surfaces such as the short pulse and the constant astigmatism equations as well as their generalizations and their spherical analogues are included in the paper

large time asymptotics for the reduced ostrovsky equation we consider the cauchy problem for the reduced ostrovsky equation u tx u left u 3 right xx with real valued initial data u left 0 right u 0 we introduce the factorization for the free evolution group to prove the global existence of solutions also we show that the large time asymptotics of solutions has a logarithmic correction in the phase comparing with the corresponding linear case

asymptotic analysis of weakly nonlinear bessel gaus beams abstract in this paper we investigate the propagation of conical waves in nonlinear media in particular we are interested in the effects resulting from applying a gaussian apodization to an ideal nondiffracting wave first we present a multiple scales approach to derive amplitude equations for weakly nonlinear conical waves from a governing equation of cubic nonlinear schrodinger type from these equations we obtain asymptotic solutions for the linear and the weakly nonlinear problem for which we state several uniform estimates that describe the deviation from the ideal nondiffracting solution moreover we show numerical simulations based on an implementation of our amplitude equations to support and illustrate our analytical results

complex mathcal pt symmetric extensions of the nonlinear ultra short light pulse model the short pulse equation is symmetric which arises in nonlinear optics for the ultra short pulse case we present a family of new complex symmetric extensions of the short pulse equation based on the complex symmetric extension principle some properties of these equations with some chosen parameters are studied including the hamiltonian structures and exact solutions such as solitary wave solutions doubly periodic wave solutions and compacton solutions our results may be useful to understand complex symmetric nonlinear physical models this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

local geometric invariants of integrable evolution equations the integrable hierarchy of commuting vector fields for the localized induction equation of 3d hydrodynamics and its associated recursion operator are used to generate families of integrable evolution equations which preserve local geometric invariants of the evolving curve or swept out surface

a robust numerical integrator for the short pulse equation near criticality abstract the short pulse equation was introduced by schafer wayne 2004 for modeling the propagation of ultrashort optical pulses while it can describe a wide range of solutions its ultrashort pulse solutions with a few cycles which the conventional nonlinear schrodinger equation does not possess have drawn much attention in such a region the solutions can become quite close to a singularity and thus existing numerical methods cease to work stably or even if they do they require high computational cost in this paper we propose a robust numerical integration method which is obtained by combining a hodograph transformation and some structure preserving methods the resulting scheme successfully works even when the singularity occurs and thus gives a highly robust method for resolving near singular solutions of interest it is confirmed by numerical experiments and some new insights about derivative blow up are obtained

nonexistence of the usual scattering states for the generalized ostrovsky hunter equation we consider the cauchy problem for the generalized ostrovsky hunter equation utx u u 1u xx we prove the nonexistence of the usual scattering states

few optical cycle dissipative solitons by using a powerful reductive perturbation technique or multiscale analysis a generalized modified korteweg de vries partial differential equation is derived which describes the physics of few optical cycle dissipative solitons beyond the slowly varying envelope approximation numerical simulations of the formation of stable dissipative solitons from arbitrary breather like few cycle pulses are also given

on groups in which commutators are covered by finitely many cyclic subgroups abstract we show that if g is a group in which all commutators are contained in finitely many cyclic subgroups then g is either finite or cyclic

long time asymptotics for the short pulse equation abstract in this paper we analyze the long time behavior of the solution of the initial value problem ivp for the short pulse sp equation as the sp equation is a completely integrable system which posses a wadati konno ichikawa wki type lax pair we formulate a 2 2 matrix riemann hilbert problem to this ivp by using the inverse scattering method since the spectral variable k is the same order in the wki type lax pair we construct the solution of this ivp parametrically in the new scale y t whereas the original scale x t is given in terms of functions in the new scale in terms of the solution of this riemann hilbert problem however by employing the nonlinear steepest descent method of deift and zhou for oscillatory riemann hilbert problems we can get the explicit leading order asymptotic of the solution of the short pulse equation in the original scale x t as time t goes to infinity

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

pseudopotentials lax equations and backlund transformations for nonlinear evolution equations it is shown how to generate both lax equations and backlund transformations for nonlinear evolution equations using the concept of pseudopotentials associated with the properties of the riccati equation several examples are given modified korteweg de vries harry dym kaup kupershmidt and sawada kotera equations

a self adaptive moving mesh method for the camassa holm equation a self adaptive moving mesh method is proposed for the numerical simulations of the camassa holm equation it is an integrable scheme in the sense that it possesses the exact n soliton solution it is named a self adaptive moving mesh method because the non uniform mesh is driven and adapted automatically by the solution once the non uniform mesh is evolved the solution is determined by solving a tridiagonal linear system due to these two superior features of the method several test problems give very satisfactory results even if by using a small number of grid points

local isometric immersions of pseudospherical surfaces described by evolution equations in conservation law form abstract for the pseudospherical surfaces described by a class of second order evolution equations of the form z t a x t z z 2 b x t z z 1 we consider the problem of local isometric immersion into the 3 dimensional euclidean space e 3 with a second fundamental form depending on finite order jets of solutions z of the considered equations we also provide an extension of our analysis to the case of k th order evolution equations in conservation law form examples of equations admitting such local isometric immersions are provided by equations like burgers murray svinolupov sokolov kuramoto sivashinsky sawada kotera kaup kupershmidt as well as hierarchies of evolution equations in conservation law form like burgers mkdv and kdv

preliminary classification of qt f q qx qxx qxxx the preliminary classification of qt f q qx qxx qxxx is given the results are compared with fokas s symmetry and mikhailov shabat sokolov formal symmetry approaches

global existence and local well posedness of the single cycle pulse equation in this paper we mainly study the cauchy problem of the integrable single cycle pulse equation in the periodic domain we first establish the local well posedness of the equation in hs s s 2 by using an orthogonal projection of the equation onto mean zero functions and the kato method then we derive a relationship between the single cycle pulse equation and the sine gordon equation rigorously and reveal some useful conservation laws of the sine gordon equation next we deduce a new conservative quantity of the single cycle pulse equation and get the upper bound of the h2 norm control of the solution by a precise blow up criterion we finally get a global existence result for the equation

local well posedness and blow up phenomena of the generalized short pulse equation in this paper we investigate the cauchy problem of the generalized short pulse equation in periodic domain we first establish the local well posedness of the equation in hs s s 2 by taking advantage of the kato method and present some useful conservation laws then we obtain the boundedness of the solution by virtue of the conservation laws finally we give a precise blow up criterion for 0 or 0 and get several blow up results provided the initial data satisfies some certain conditions

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

nonlocal symmetries and the kaup kupershmidt equation computational and geometric aspects of nonlocal infinitesimal symmetries of nonlinear partial differential equations are considered in particular the relation of nonlocal symmetries with classical generalized and internal symmetries is briefly discussed a nonlocal symmetry for the kaup kupershmidt equation is introduced and studied in some detail some explicit particular solutions are found with its help and a darboux like transformation is also obtained

constant sign periodic and almost periodic solutions of a system of difference equations we consider the following system of difference equations u i k l ig i k l f l u 1 l u 2 l u n l k i 1 i n where i is a subset of our aim is to establish criteria such that the above system has a constantsign periodic and almost periodic solution u 1 u 2 u n the above problem is also extended to that on u i k l g i k l f i l u 1 l u 2 l u n l k 1 i n

on differential systems describing surfaces of constant curvature abstract the geometric notion of a differential system describing surfaces of constant nonzero gaussian curvature is introduced the nonlinear schrodinger equation nls with 1 and 1 is shown to describe a family of spherical surfaces s s and pseudospherical surfaces p s s respectively the schrodinger flow of maps into s 2 the hf model and its generalized version the landau lifschitz equation are shown to describe spherical surfaces the schrodinger flow of maps into h 2 the m hf model provides another example of a system describing pseudo spherical surfaces new differential systems describing surfaces of nonzero constant gaussian curvature are obtained furthermore we give a characterization of evolution systems which describe surfaces of nonzero constant gaussian curvature in particular we determine all differential systems of type u t v xx h 11 u v u x h 12 u v v x h 13 u v v t u xx h 21 u v u x h 22 u v v x h 23 u v which describe pseudospherical or spherical surfaces as an application we obtain four parameter family of such systems for a complex valued function q u iv given by iq t q xx i q 2 q x i q x q 2 q q 0 where 0 if 0 particular cases of this family obtained by the vanishing of the parameters are the linear equations the nls equation the derivative nonlinear schrodinger equation dnls and the mixed nls dnls equation

multiple positive solutions of some boundary value problems abstract we study the existence of multiple positive solutions of the equations u t u subject to linear boundary conditions we show that there are at least two positive solutions if t u is superlinear at one end point zero or infinity and sublinear at the other applications of these results are provided to yield multiple positive solutions of some elliptic boundary value problems on an annulus

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

on nonlocal symmetries of some shallow water equations a recent construction of nonlocal symmetries for the korteweg de vries camassa holm and hunter saxton equations is reviewed and it is pointed out that in the camassa holm and hunter saxton case these symmetries can be considered as nonlocal symmetries of integro differential equations

conservation laws for a class of third order evolutionary differential systems conservation laws of third order quasi linear scalar evolution equations are studied via exterior differential system and characteristic cohomology we find a subspace of 2 forms in the infinite prolongation space in which every conservation law has a unique representative analysis of the structure of this subspace based upon the symbol of the differential equation leads to a universal integrability condition for an evolution equation to admit any higher order weight conservation laws as an example we give a complete classification of a class of evolution equations which admit conservation laws of the first three consecutive weights 1 1 3 the differential system describing the flow of a curve in the plane by the derivative of its curvature with respect to the arc length is also shown to exhibit the kdv property i e an infinite sequence of conservation laws of distinct weights

an integrable coupled short pulse equation an integrable coupled short pulse csp equation is proposed for the propagation of ultra short pulses in optical fibers based on two sets of bilinear equations to a two dimensional toda lattice linked by a backlund transformation and an appropriate hodograph transformation the proposed csp equation is derived meanwhile its n soliton solutions are given by the casorati determinant in a parametric form the properties of one and two soliton solutions are investigated in detail same as the short pulse equation the two soliton solution turns out to be a breather type if the wave numbers are complex conjugate we also illustrate an example of soliton breather interaction

fifth order evolution equations describing pseudospherical surfaces we consider differential equations which describe pseudospherical surfaces with associated 1 forms i fi1dx fi2dt 1 i 3 we characterize all such equations of type ut uxxxxx g u ux uxxxx whose associated 1 forms satisfy fp1 pf11 p p p r 2 p 3 in addition to a generic technical assumption we also classify all of these equations which are independent of any of the real parameters p p obtaining as particular cases the fifth order korteweg de vries the sawada kotera and the kaup kupershmidt equations we determine huge classes of equations describing pseudospherical surfaces and their respective linear problems in which particular cases are obtained by merely specifying certain functions which depend on u and its derivatives with respect to x

the cavalcante tenenblat equation does the equation admits physical significance the main purpose of the given paper is to analyze a less studied third order non linear partial differential equation the so called cavalcante tenenblat equation cte in the following form u t u x 1 2 x x u x 3 2 since general class of solutions are of basic interest a complete characterization of the group properties is given the traveling wave ansatz restricts the solution manifold to special class of solutions and hence a generalize algorithm is necessary we determine the lie point symmetry vector fields and calculate similarity ansatze further we also derive a few non linear transformations and some similarity solutions are obtained explicitly due to the complexity of some similarity solutions a numerical procedure is of advantage moreover the non classical case potential symmetries is studied to the first time and further we show how the cte leads to approximate symmetries and we apply the method to the first time we call the disturbed equation the cte e equation and we show how to derive new class of solutions finally the equation does not pass the painleve test and is therefore not soluble by the inverse scattering transform method ist hence suitable alternative algebraic approaches are necessary to derive class of solutions explicitly

exact solutions for some nonlinear evolution equations which describe pseudo spherical surfaces for some nonlinear evolution equations which describe pseudo spherical surfaces two new exact solution classes are generated from known solutions either the seed solution is constant or a traveling wave exact traveling wave and solitary wave solutions for burgers equation and also for a fifth order evolution equation containing the fifth order korteweg de vries equation and also the sawada kotera equation are obtained by using an improved sine cosine method and the wu s elimination method we also apply backlund transformations to these solutions and generate new soliton solution classes

almost automorphic solutions of non autonomous difference equations abstract in the present paper we study the non autonomous difference equations given by u k 1 a k u k f k and u k 1 a k u k g k u k for k z where a k is a given non singular n n matrix with elements a i j k 1 i j n f z e n is a given n 1 vector function g z e n e n and u k is an unknown n 1 vector with components u i k 1 i n we obtain the existence of a discrete almost automorphic solution for both the equations assuming that a k and f k are discrete almost automorphic functions and the associated homogeneous system admits an exponential dichotomy also assuming the function g satisfies a global lipschitz type condition we prove the existence and uniqueness of an almost automorphic solution of the nonlinear difference equation

on exact solutions of the schafer wayne short pulse equation wki eigenvalue problem we consider a new equation recently found by sch afer and wayne hereafter named schshort pulse equation swspe describing the propagation of an ultrashort pulse in nonlinear media using some vanishing boundary conditions we construct and discuss the n soliton solutions to the previous equation by means of the wadati konno ichikawa wki method which is arguably more direct than the map through the sine gordon equation investigated much earlier by sakovich and sakovich 2005 j phys soc japan 74 239 2006 j phys a math gen 39 l361 we particularly focus our attention on the two soliton solution as an application as a result the collision process for two soliton solutions with similar amplitudes exhibits very different behaviour from the case when the amplitudes are dissimilar

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

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

soliton curvatures of surfaces and spaces an intrinsic geometry of surfaces and three dimensional riemann spaces is discussed in the geodesic coordinates the gauss equation for two dimensional riemann spaces surfaces is reduced to the one dimensional schrodinger equation where the gaussian curvature plays a role of potential the use of this fact provides an infinite set of explicit expressions for curvature and metric of surface a special case is governed by the kdv equation for the gaussian curvature integrable dynamics of curvature via the kdv equation higher kdv equations and 2 1 dimensional integrable equations with breaking solitons is considered for a special class of three dimensional riemann spaces the relation between metric and scalar curvature is given by the two dimensional stationary schrodinger or perturbed string equations this provides us an infinite family of riemann spaces with explicit scalar curvature and metric particular class of spaces and their integrable evolutions are described by the nizhnik veselov novikov equation and its higher analogs surfaces and three dimensional riemann spaces with large curvature and slow dependence on the variable are considered they are associated with the burgers and kadomtsev petviashvili equations respectively

scalar second order evolution equations possessing an irreducible sl2 valued zero curvature representation we find all scalar second order evolution equations that possess an sl2 valued zero curvature representation irreducible to a proper subalgebra of sl2 none of these zero curvature representations depends on a parameter that could serve as the spectral parameter

correspondence theorems for hierarchies of equations of pseudo spherical type hierarchies of evolution equations of pseudo spherical type are introduced thereby generalizing the notion of a single equation describing pseudo spherical surfaces due to s s chern and k tenenblat and providing a connection between differential geometry and the study of hierarchies of equations which are the integrability condition of sl 2 r valued linear problems as an application it is shown that there exist local correspondences between any two suitably generic solutions of arbitrary hierarchies of equations of pseudo spherical type

derivative nonlinear schrodinger equations and hermitian symmetric spaces it is shown that to each hermitian symmetric space there corresponds an integrable system of generalised derivative nonlinear schrodinger equations nls the nonlinear terms are related to the curvature tensor of the associated symmetric space the hamiltonian form of the equations is presented the results are an extension of those presented in an earlier paper 1983 on generalised nls equations associated with symmetric and reductive homogeneous spaces

on differential equations describing pseudo spherical surfaces abstract we give a complete classification of the evolution equations u t f u u x k u x k which describe pseudo spherical surfaces without any a priori assumptions on the presence of a spectral parameter we also prove a local existence theorem to the effect that given two differential equations describing pseudo spherical surfaces not necessarily evolutionary there exists under a technical assumption a smooth mapping transforming any suitably generic solution of one equation into a solution of the other

lie symmetry analysis and exact solutions for the short pulse equation abstract in this paper the lie symmetry analysis and the generalized symmetry method are performed for a short pulse equation spe the symmetries for this equation are given for the traveling wave solutions the exact parametric representations are investigated to guarantee the existence of the above solutions all parameter conditions are determined furthermore the exact analytic solutions are obtained by using the power series method

on zero curvature representations of evolution equations for zero curvature representations zcrs at bx ab ba 0 of evolution equations ut f x u ux ux x we develop a description which is invariant under gauge transformations a sas 1 sxs 1 and b sbs 1 sts 1 where a b and s are matrix functions of x u ux uxx we prove that every fixed matrix a of any dimension and order in ux x determines a continual class of evolution equations which admit zcrs with this a then we quote examples illustrating how a dependence of a on an essential parameter restricts classes of represented equations one of our examples shows that some non integrable systems can admit parametric lax pairs and infinitely many non trivial conservation laws

positive almost periodic solutions of nonlinear discrete systems with finite delay the existence of positive almost periodic solutions of nonlinear discrete systems with finite delay is obtained by using a variant of hilbert s projective metric on a subset of the space of almost periodic sequences

blow up solutions and peakons to a generalized μ camassa holm integrable equation considered here is a generalized type integrable equation which can be regarded as a generalization to both the camassa holm and modified camassa holm equations it is shown that the proposed equation is formally integrable with the lax pair and the bi hamiltonian structure and its scale limit is an integrable model of hydrodynamical systems describing short capillary gravity waves local well posedness of the cauchy problem in the suitable sobolev space is established by the viscosity method existence of peaked traveling wave solutions and formation of singularities of solutions for the equation are investigated it is found that the equation admits single and multi peaked traveling wave solutions the effects of varying camassa holm and modified camassa holm nonlocal nonlinearities on blow up criteria and wave breaking are illustrated in detail our analysis relies on the method of characteristics and conserved quantities and is proceeded with a priori differential estimates

novel exact solutions to the short pulse equation in this paper the bridge connecting the short pulse equation spe for short with the sine gordon equation is applied to construct the novel solutions to the short pulse equation it is shown that the solutions of the sine gordon equation can be used to obtain many different kinds of solutions to the short pulse equation with the aid of symbolic computation and plot representation of maple

the soliton content of classical jackiw teitelboim gravity it is pointed out that every generic in a sense to be made precise in section 2 solution to an arbitrary equation describing pseudo spherical surfaces or equivalently an arbitrary equation which is the integrability condition of a sl 2 r valued linear problem determines pseudo riemannian surfaces of constant scalar curvature and therefore classical solutions to the jackiw teitelboim field equations for two dimensional gravity in particular this observation explains why some standard soliton equations appear in this theory

lax representations for matrix short pulse equations the lax representation for different matrix generalizations of short pulse equations spes is considered the four dimensional lax representations of four component matsuno feng and dimakis muller hoissen matsuno equations are obtained the four component feng system is defined by generalization of the two dimensional lax representation to the four component case this system reduces to the original feng equation to the two component matsuno equation or to the yao zang equation the three component version of the feng equation is presented the four component version of the matsuno equation with its lax representation is given this equation reduces the new two component feng system the two component dimakis muller hoissen matsuno equations are generalized to the four parameter family of the four component spe the bi hamiltonian structure of this generalization for special values of parameters is defined this four component spe in special cases reduces to the new two component spe

nonlinear geometric optics for short pulses abstract this paper studies the propagation of pulse like solutions of semilinear hyperbolic equations in the limit of short wavelength the pulses are located at a wavefront 0 where satisfies the eikonal equation and d lies on a regular sheet of the characteristic variety the approximate solutions are u e approx u t x t x e where u t x r is a smooth function with compact support in r when u satisfies a familiar nonlinear transport equation from geometric optics it is proved that there is a family of exact solutions u e exact such that u e approx has relative error o e as e 0 while the transport equation is familiar the construction of correctors and justification of the approximation are different from the analogous problems concerning the propagation of wave trains with slowly varying envelope

pseudo spherical surfaces and integrability of evolution equations abstract a complete classification of evolution equations u t f x t u u x u x k which describe pseudo spherical surfaces is given thus providing a systematic procedure to determine a one parameter family of linear problems for which the given equation is the integrability condition it is shown that for every second order equation which admits a formal symmetry of infinite rank formal integrability such a family exists kinematic integrability it is also shown that this result cannot be extended as proven to third order formally integrable equations this fact notwithstanding a special case is proven and moreover several equations of interest including the harry dym cylindrical kdv and a family of equations solved by inverse scattering by calogero and degasperis are shown to be kinematically integrable conservation laws of equations describing pseudo spherical surfaces are studied and several examples are given

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

transformations of solutions for equations and hierarchies of pseudo spherical type it is known that if an equation describes non trivial one parameter families of pseudo spherical surfaces its conservation laws generalized nonlocal symmetries and backlund transformations can be studied by geometrical means 4 10 in this letter it is pointed out that there exist correspondences or generalized backlund transformations between arbitrary solutions satisfying some genericity conditions of any two single equations describing pseudo spherical surfaces then the notion of a hierarchy of equations of pseudo spherical type is introduced and a theorem stating that there also exist correspondences between arbitrary solutions of any two such hierarchies is presented a full account of these results appears elsewhere 12 13

non abelian pseudopotentials and conservation laws of reaction diffusion equations the authors apply the wahlquist estabrook technique to reaction diffusion equations a representation for non abelian pseudopotentials is obtained an infinite hierarchy of conservation laws is derived from these pseudopotentials the class of equations associated with such pseudopotentials is determined and examples are presented

integrable discretizations of the short pulse equation in this paper we propose integrable semi discrete and full discrete analogues of the short pulse sp equation the key construction is the bilinear form and determinant structure of solutions of the sp equation we also give the determinant formulas of n soliton solutions of the semi discrete and full discrete analogues of the sp equations from which the multi loop and multi breather solutions can be generated in the continuous limit the full discrete sp equation converges to the semi discrete sp equation and then to the continuous sp equation based on the semi discrete sp equation an integrable numerical scheme i e a self adaptive moving mesh scheme is proposed and used for the numerical computation of the short pulse equation

the initial value problem for the sequence of generalized korteweg de vries equations on considere le probleme aux valeurs initiales pour l equation u t g m u x k m u m 1

conservation laws for nonlinear evolution equations a method to derive conservation laws for evolution equations that describe pseudospherical surfaces is introduced based on a geometrical property of these surfaces a new third order evolution equation is obtained as a first example for a nongeneric case in the classification given by chern and tenenblat stud appl math 74 1 1986

orbital stability of periodic peakons to a generalized μ camassa holm equation in this paper we study the orbital stability of the periodic peaked solitons of the generalized camassa holm equation with nonlocal cubic and quadratic nonlinearities the equation is a version of a linear combination of the camassa holm equation and the modified camassa holm equation it is also integrable with the lax pair and bi hamiltonian structure and admits the single peakons and multi peakons by constructing an inequality related to the maximum and minimum of solutions with the conservation laws we prove that even in the case that the camassa holm energy counteracts in part the modified camassa holm energy the shapes of periodic peakons are still orbitally stable under small perturbations in the energy space

asymptotic behavior of solutions to the short pulse equation with critical nonlinearity abstract we consider the cauchy problem for the short pulse equation u t x u u 3 x x x r t 0 u 0 x u 0 x x r where u 0 is a real valued function we prove the global existence of small solutions to the short pulse equation moreover we give the l time decay estimate u t l c 1 t 1 2 and the asymptotic behavior of solutions

a classification of pseudospherical surface equations of type ut uxxx g u ux uxx the equations of type ut uxxx g u ux uxx which describe pseudospherical surfaces are characterized a new class of such equations is obtained this class together with the korteweg de vries kdv equation the modified korteweg de vries mkdv equation and the linear equation completes the classification

solitary wave solutions of the short pulse equation an exact nonsingular solitary wave solution of the schafer wayne short pulse equation is derived from the breather solution of the sine gordon equation by means of a transformation between these two integrable equations

the long wave limit for the water wave problem i the case of zero surface tension the korteweg de vries equation boussinesq equation and many other equations can be formally derived as approximate equations for the two dimensional water wave problem in the limit of long waves here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension we prove that the solutions of the water wave problem in the long wave limit split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries equation our result allows us to describe the nonlinear interaction of solitary waves 2000 john wiley sons inc

the korteweg de vries hierarchy and long water waves by using the multiple scale method with the simultaneous introduction of multiple times we study the propagation of long surface waves in a shallow inviscid fluid as a consequence of the requirements of scale invariance and absence of secular terms in each order of the perturbative expansion we show that the korteweg de vries hierarchy equations do play a role in the description of such waves finally we show that this procedure of eliminating secularities is closely related to the renormalization technique introduced by kodama and taniuti

continuity properties of the solution map for the generalized reduced ostrovsky equation abstract it is shown that the data to solution map for the generalized reduced ostrovsky gro equation is not uniformly continuous on bounded sets in sobolev spaces on the circle with exponent s 3 2 considering that for this range of exponents the gro equation is well posed with continuous dependence on initial data this result makes the continuity of the solution map an optimal property however if a weaker h r topology is used then it is shown that the solution map becomes holder continuous in h s

on the exactly solvable equation s_t beta s gamma 2 s_x _x alpha beta s gamma 2 s_x occurring in two phase flow in porous media the symmetry approach of fokas j math phys 21 1980 pp 1318 1325 is used to show that the most general equation of the form s t g s s xx f s s x dg ds ne 0 which is exactly solvable is equivalent to the equation s t beta s gamma 2 s x x alpha beta s gamma 2 s x this equation is then mapped to the linear heat equation through a series of backlund transformations the above result is used to solve an initial boundary value problem modeling the process of waterflooding i e the displacement of oil initially in a porous reservoir by continuously injected water

the method of darboux transformation matrix for solving the landau lifschitz equation for a spin chain with an easy plane the landau lifschitz equation for a spin chain with an easy plane is solved by the method of the darboux transformation matrix in terms of a particular parameter k jost solutions and darboux matrices are generated in a recursive manner the jost solutions are shown to satisfy the corresponding lax equations by a suitable choice of the constants involved in the darboux matrices a system of linear equations is derived and can yield the expressions for multi soliton solutions asymptotic behaviour in the limits as t to or infinity is derived an expression for the one soliton solution is given in terms of elementary functions of x and t as an example

verbal sets and cyclic coverings abstract we consider groups g such that the set of all values of a fixed word w in g is covered by a finite set of cyclic subgroups fernandez alcober and shumyatsky studied such groups in the case when w is the word x 1 x 2 and proved that in this case the corresponding verbal subgroup g is either cyclic or finite answering a question asked by them we show that this is far from being the general rule however we prove a weaker form of their result in the case when w is either a lower commutator word or a non commutator word showing that in the given hypothesis the verbal subgroup w g must be finite by cyclic even this weaker conclusion is not universally valid it fails for verbose words

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

on equations of type uxt f u ux which describe pseudospherical surfaces the equations uxt f u ux which describe pseudospherical surfaces are characterized in particular when f does not depend on ux the sine gordon sinh gordon and liouville equations are essentially obtained moreover it is shown that an equation uxt f u has a self baclund transformation if and only if it describes an pseudospherical surface