nonlocal symmetries related to backlund transformation and their applications starting from nonlocal symmetries related to backlund transformation bt many interesting results can be obtained taking the well known potential kdv pkdv equation as an example a new type of nonlocal symmetry in an elegant and compact form which comes from bt is presented and used to perform research works in two main subjects the nonlocal symmetry is localized by introducing suitable and simple auxiliary dependent variables to generate new solutions from old ones and to consider some novel group invariant solutions some other models both in finite and infinite dimensions are generated under new nonlocal symmetry the finite dimensional models are completely integrable in liouville sense which are shown equivalent to the results given through the nonlinearization method for lax pair

conformal invariance and integrable models usually an integrable nonlinear partial differential equation can be transformed to its conformal invariant form schwartz form using the conformal invariance of the integrable models we can obtain many interesting results in this paper we will focus mainly in obtaining new symmetries and new integrable models starting from the conformal invariance of an integrable model one can obtain infinitely many non local symmetries many types of 1 1 and 2 1 dimensional new sine gordon or sinh gordon extensions are obtained from the conformal flow equations of the koerteweg de vries type equations many other kinds of integrable models can be obtained from the conformal constraints of the known integrable models

explicit solutions from eigenfunction symmetry of the korteweg de vries equation in nonlinear science it is very difficult to find exact interaction solutions among solitons and other kinds of complicated waves such as cnoidal waves and painlevwaves actually even if for the most well known prototypical models such as the kortewet de vries kdv equation and the kadomtsev petviashvili kp equation this kind of problem has not yet been solved in this paper the explicit analytic interaction solutions betweensolitarywavesandcnoidalwavesareobtainedthroughthelocalizationprocedureofnonlocalsymmetries which are related to darboux transformation for the well known kdv equation the same approach also yields some other types of interaction solutions among different types of solutions such as solitary waves rational solutions bessel function solutions and or general painlev e ii solutions

infinitely many lax pairs and symmetry constraints of the kp equation starting from a known lax pair one can get some infinitely many coupled lax pairs infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways in this paper taking the well known kadomtsev petviashvili kp equation as a special example we show that infinitely many nonhomogeneous linear lax pairs can be obtained by using infinitely many symmetries differentiating the spectral functions with respect to the inner parameters using a known lax pair and the darboux transformations dt infinitely many nonhomogeneous nonlinear lax pairs can also be obtained by means of the infinitely many lax pairs dt and the conformal invariance of the schwartz form of the kp equation infinitely many new nonlocal symmetries can be obtained naturally infinitely many integrable models in 1 1 dimensions 2 1 dimensions 3 1 dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the kp equation related to the infinitely many lax pairs

non local symmetries via darboux transformations darboux transformations are used to find non local symmetries of integrable nonlinear models which include the kdv kp cdgks and 2 1 dimensional cdgks equations

backlund transformations for the isospectral and nonisospectral akns hierarchies by converting the usual lax pairs for the isospectral and nonisospectral akns hierarchies into lax pairs in riccati forms a unified explicit form of backlund transformations for these hierarchies of nonlinear evolution equations can be obtained in the isospectral case it is an auto backlund transformation however in the nonisospectral case it is not an auto backlund transformation

the lie algebraic structure of symmetries generated by hereditary symmetries based on the works of li zhu and co workers 1985 a general lie algebraic structure is given for symmetries of integrable systems which are generated by hereditary symmetries

deformations of the riccati equation by using miura type transformations using some different miura type transformations a c integrable ordinary differential equation the riccati equation is deformed to some different s integrable models such as the 1 1 dimensional and 2 1 dimensional sinh gordon equations and mikhailov dodd bullough equations

syntactic and rees indices of subsemigroups abstract we define two different notions of index for subsemigroups of semigroups the right syntactic index and the rees index we investigate the relationships between them and with the group index in particular we show that the syntactic index is a generalisation of both the group index and the rees index we use this fact to prove further similarities between the group index and the rees index it turns out however that very few of the nice properties of the group index are inherited by the syntactic index

the short pulse hierarchy we study a new hierarchy of equations containing the short pulse equation which describes the evolution of very short pulses in nonlinear media and the elastic beam equation which describes nonlinear transverse oscillations of elastic beams under tension we show that the hierarchy of equations is integrable we obtain the two compatible hamiltonian structures we construct an infinite series of both local and nonlocal conserved charges a lax description is presented for both systems for the elastic beam equations we also obtain a nonstandard lax representation

technical note probability of success in the search for a moving target the probability of detection in the search for a randomly moving target is calculated for the case of a target whose motion is a diffusion process and known searcher path the probability of detection can be calculated by solving a backward diffusion equation corwin corwin t 1980 on multiplicative functionals on diffusion processes siam j appl math 38 112 119 gives a solution of the backward equation for a special case in general exact solutions do not exist and other methods are needed in this paper the backward equation is solved approximately by using a formal asymptotic method valid when the intensity of the random motion is small the general solution is illustrated for the case of spatially homogeneous drift and diffusion coefficients in this case the asymptotic solution can be evaluated analytically

on the nonlocal equations and nonlocal charges associated with the harry dym hierarchy a large class of nonlocal equations and nonlocal charges for the harry dym hierarchy is exhibited they are obtained from nonlocal casimirs associated with its bi hamiltonian structure the lax representation for some of these equations is also given

symmetries of the kdv equation and four hierarchies of the integrodifferential kdv equations using the inverse strong symmetry of the korteweg de vries kdv equation on the trivial symmetry and 0 symmetry one gets four new sets of symmetries of the kdv equation these symmetries are expressed explicitly by the multi integrations of the jost function of the kdv equation and constitute an infinite dimensional lie algebra together with two hierarchies of the known symmetries contrary to the general belief the time independent symmetry groups of the kdv and mkdv equations are non abelian and the infinite dimensional lie algebras of the kdv and mkdv equations are nonisomorphic though two equations are related by the miura transformation starting from these sets of symmetries four hierarchies of the integrodifferential kdv equations which can be solved by the schrodinger inverse scattering transformation method are obtained some of these hierarchies enjoy a common strong symmetry and or same local conserved densities

abundant symmetries for the 1 1 dimensional classical liouville field theory explicit calculation shows us that there exists a rich associated symmetry structure and abundant strong symmetry configuration for the 1 1 dimensional classical liouville field theory starting from these symmetries and strong symmetries various integrable hierarchies can be obtained the liouville i iii hierarchies the modified korteweg de vries kdv hierarchy the fordy gibbons i and fordy gibbons ii hierarchies are just special cases while the kdv hierarchy caudry dodd gibbon sawada kotera hierarchy and kaup kupershmidt hierarchy can be obtained by using a same miura transformation from the modified kdv the fordy gibbons i and fordy gibbons ii hierarchies respectively two types of the generalized riccati hierarchies are also presented

hierarchy of negative order equation and its lax pair a simple method is presented to construct the hierarchy of a negative order equations from a given eigenvalue problem the hierarchy of the negative order equation we obtained is the natural continuation of a corresponding hierarchy of a higher order equation although most of the negative order equations are complicated we give their lax representations three examples are discussed to illustrate the effectiveness of our method

formal variable separation approach for nonintegrable models using a formal variable separation approach a nonlinear partial differential equation can be solved by solving ordinary different equations or even algebraic equations taking the kdv burgers and modified kdv burgers equations with background interaction as simple examples some explicit solitary wave solutions which are induced by background source and nonlinearity or dispersion are obtained

soliton solutions for a negative order non isospectral akns equation abstract in this paper the bilinear form of a negative order non isospectral akns equation is given the soliton solutions are obtained through hirota s direct method and the wronskian technique respectively

negative semi discrete kp and bkp hierarchies via nonlocal symmetries in this paper nonlocal symmetries for the bilinear semi discrete kadomtsev petviashvili kp and bilinear semi discrete b type kp bkp equations are derived by expanding these nonlocal symmetries in power series of each of two parameters we have derived bilinear negative semi discrete kp and bkp hierarchies an interesting observation is that bilinear positive and negative semi discrete kp hierarchies may be derived from the same nonlocal symmetry for the semi discrete kp equation

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

negative powers of olver recursion operators olver s concept of a recursion operator for symmetries of an evolution equation is extended to include negative powers of the operator some negative order kdv equations are derived in this way in particular the inverse image of the trivial zero symmetry generator has an elegant formulation in terms of an eigenfunction of the associated schrodinger equation used in the inverse scattering solution of the kdv equation and this formulation is used to show that this new equation is the miura transform of a sinh gordon equation

nonlocal symmetries of the hirota satsuma coupled korteweg de vries system and their applications exact interaction solutions and integrable hierarchy the nonlocal symmetry is derived from the known darboux transformation dt of the hirota satsuma coupled korteweg de vries hs ckdv system and infinitely many nonlocal symmetries are given by introducing the internal parameters by extending the hs ckdv system to an auxiliary system with five dependent variables the prolongation is found to localize the so called seed nonlocal symmetry related to the dt by applying the general lie point symmetry method to this enlarged system we obtain two main results a new type of finite symmetry transformation is derived which is different from the initial dt and can generate new solutions from old ones some novel exact interaction solutions among solitons and other complicated waves including periodic cnoidal waves and painleve waves are computed through similarity reductions in addition two kinds of new integrable models are proposed from the obtained nonlocal symmetry the negative hs ckdv hierarchy by introducing the internal parameters the integrable models both in lower and higher dimensions by restricting the symmetry constraints

numerical complexiton solutions for the complex kdv equation by the homotopy perturbation method in this paper the homotopy perturbation method is extended to investigate the numerical complexiton solutions of the complex kdv equation by constructing special forms of initial conditions three new types of realistic numerical solutions are obtained numerical positon solution expressed by the trigonometric functions numerical negaton solution expressed by the hyperbolic functions and particularly the numerical analytical complexiton solutions expressed by combinations of the two kinds of functions all these numerical solutions obtained can rapidly converge to the exact solutions obtained by lou et al illustrative numerical figures are exhibited the efficiency of the proposed method

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

nonlocal topological solitons of the sine gordon equation a special nonlocal symmetry related to the b cklund transformation is obtained for the 1 1 dimensional sine gordon sg equation in its polynomial form which can further generate infinitely many nonlocal symmetries of the sg equation due to the inclusion of an arbitrary spectrum parameter this nonlocal symmetry can be localized and hence turn into a point symmetry for the augmented system through the introduction of an auxiliary function consequently finite transformation similarity reductions and explicit invariant solutions related to the nonlocal symmetry are obtained it is known that the sg equation has topological solitons or kinks or antikinks based on the group invariant solutions a different kind of topological soliton which is a topological soliton situated on a cnoidal wave background is obtained analytically and graphically displayed

the bilinear negative kadomtsev petviashvili system and its kac moody virasoro symmetry group the symmetry transformation group of the bilinear negative kadomtsev petviashvili system is studied by means of a direct method the kac moody virasoro type lie point symmetry algebra is found to be a special infinitesimal form of the symmetry group

on subsemigroups of finitely presented semigroups the purpose of this paper is to investigate properties of subsemigroups of finitely presented semigroups particularly with respect to the property of being finitely presented more precisely we will consider various instances of the following main problem let s be a finitely presented semigroup i is every subsemigroup ideal one sided ideal of s finitely generated ii is every subsemigroup ideal one sided ideal of s which is finitely generated as a semigroup finitely presented iii is every subsemigroup ideal one sided ideal of finite index in s finitely presented for the definition of index see below for example if s is finite all these questions have an affirmative answer less trivially if s is a commutative semigroup then ii has an affirmative w x w x answer see 12 also in 2 it has been proved that a two sided ideal of finite index in a finitely presented semigroup is always finitely presented

infinitely many nonlocal symmetries and conservation laws for the 1 1 dimensional sine gordon equation abstract infinitely many nonlocal symmetries and infinitely many local and nonlocal conservation laws of the 1 1 dimensional sine gordon sg equation are derived in terms of its backlund transformation bt some special nonlocal symmetries and nonlocal conservation laws are obtained from the linearized equations of the sg equation and its bt furthermore one can derive infinitely many nonlocal symmetries from a known nonlocal symmetry but also infinitely many nonlocal conservation laws from a known nonlocal conservation law in addition infinitely many local and nonlocal conservation laws can be directly generated from the bt through the parameter expansion procedure

k symmetries and tau symmetries of evolution equations and their lie algebras on the basis of tu s work 1951 the author establishes a more general skeleton on k symmetries and tau symmetries of evolution equations and their lie algebraic structures and discuss carefully the corresponding symmetries and lie algebras for kdv and jaulent miodek hierarchies

sine gordon and modified korteweg de vries charges the sine gordon and the integrated modified korteweg de vries equations are shown to conserve the same infinite set of charges the charges are determined by a recursion relation as a consequence the solutions of all the equations generated by the charges have in common all time independent properties