classical solutions of the chiral model unitons and holomorphic vector bundles this paper deals with classical solutions of thesu 2 chiral model on 2 and of a generalized chiral model on 2 1 such solutions are shown to correspond to certain holomorphic vector bundles over minitwistor space with an appropriate boundary condition the solutions called 1 unitons in 9 correspond to bundles over a compact 2 dimensional complex manifold and the problem becomes one of algebraic geometry

the quasiclassical limit of the modified kp hierarchy in the quasiclassical limit the mkp hierarchy and its lax representation turn into a modified benney hierarchy and its poisson representation a miura map is constructed and shown to be canonical from the modified benney hierarchy into the unmodified one the modified hierarchy is given both hydrodynamical and kinetic representations and the miura map is given a kinetic form explicit combinatorial formulae are proved for the infinite number of conserved densities of the modified benney hierarchy

on twistor solutions of the dkp equation the factorization problem for the group of canonical transformations close to the identity and the corresponding twistor equations for an ample family of canonical variables are considered a method to deal with these reductions is developed for the construction of classes of nontrivial solutions of the dkp equation

deformation of algebras and solution of self duality equation the solutions of the self duality equation depending on a set of functions of three independent variables are constructed in an explicit way the obtained solutions are shown to be connected with two dimensional systems determined by the operators of the lax pair

reductions and hodograph solutions of the dispersionless kp hierarchy a general scheme for analysing reductions of dispersionless integrable hierarchies is presented it is based on a method for determining the s function by means of a system of first order differential equations compatibility systems of nonlinear partial differential equations of bourlet type characterizing both reductions and hodograph solutions of the dkp hierarchy are obtained wide classes of illustrative explicit examples are exhibited

lump dynamics in the ℂ p 1 model on the torus the topology and geometry of the moduli space m 2 of degree 2 static solutions of the p 1 model on a torus spacetime t 2 are studied it is proved that m 2 is homeomorphic to the left coset space g g 0 where g is a certain eight dimensional noncompact lie group and g 0 is a discrete subgroup of order 4 low energy two lump dynamics is approximated by geodesic motion on m 2 with respect to a metric g defined by the restriction to m 2 of the kinetic energy functional of the model this lump dynamics decouples into a trivial centre of mass motion and nontrivial relative motion on a reduced moduli space it is proved that m 2 g is geodesically incomplete and has only finite diameter a low dimensional geodesic submanifold is identified and a full description of its geodesics obtained

backlund transformations ward solitons and unitons the ward equation also called the modified 2 1 chiral model is obtained by a dimension reduction and a gauge fixing from the self dual yang mills field equation on r it has a lax pair and is an integrable system ward constructed solitons whose extended solutions have distinct simple poles he also used a limiting method to construct 2 solitons whose extended solutions have a double pole ioannidou and zakrzewski and anand constructed more soliton solutions whose extended solutions have a double or triple pole some of the main results of this paper are i we construct algebraic backlund transformations bts that generate new solutions of the ward equation from a given one by an algebraic method ii we use an order k limiting method and algebraic bts to construct explicit ward solitons whose extended solutions have arbitrary poles and multiplicities iii we prove that our construction gives all solitons of the ward equation explicitly and the entries of ward solitons must be rational functions in x y and t iv since stationary ward solitons are unitons our method also gives an explicit construction of all k unitons from finitely many rational maps from c to c

on continuous triangular norms that are migrative in this paper we completely describe all continuous migrative triangular norms since the migrative property excludes both idempotent and nilpotent classes the characterization and construction is carried out by solving a functional equation for additive generators of strict t norms we also study cases when the construction results in a smooth generator

a note on the ordinal sum theorem and its consequence for the construction of triangular norms in this paper the well known ordinal sum theorem of semigroups is generalized and applied to construct new families of triangular subnorms and triangular norms t norms among them one can find several new families of left continuous t norms too

the moyal bracket and the dispersionless limit of the kp hierarchy a new lax equation is introduced for the kp hierarchy which avoids the use of pseudo differential operators as used in the sate approach this lax equation is closer to that used in the study of the dispersionless kp hierarchy and is obtained by replacing the poisson bracket with the moyal bracket the dispersionless limit under which the moyal bracket collapses to the poisson bracket is particularly simple

periodic and homoclinic solutions of the modified 2 1 chiral model we use algebraic backlund transformations bts to construct explicit solutions of the modified 2 1 chiral model from t2 r to su n where t2 is a 2 torus algebraic bts are parametrized by z c poles and holomorphic maps from t2 to gr k cn we apply backlund transformations with carefully chosen poles and s to construct infinitely many solutions of the 2 1 chiral model that are i doubly periodic in space variables and periodic in time i e triply periodic ii homoclinic in the sense that the solution u has the same stationary limit u0 as t and is tangent to a stable linear mode of u0 as t and is tangent to an unstable mode of u0 as t

bi differential calculi bi hamiltonian systems and conformal killing tensors the theory of dimakis and m ller hoissen dimakis a and m ller hoissen f 2000 j phys a math gen 33 957 74 concerning bi differential calculi and completely integrable systems is related to bi hamiltonian systems of the poisson nijenhuis type in the special case where the ambient manifold is a cotangent bundle one is able to recover and elucidate the theory of ibort et al ibort a magri f and marmo g 2000 j geom phys 33 210 23 which is in turn a reworking in the bi hamitonian context of benenti s theory of hamilton jacobi separable systems in particular it is shown that benenti s conformal killing tensor which is central to his theory has an even more special form than has hitherto been realized and that when it is converted into a field of endomorphisms by raising an index with the ambient metric it necessarily has vanishing nijenhuis torsion

rational solutions of the kadomtsev petviashvili hierarchy and the dynamics of their poles i new form of a general rational solution a new approach to the construction of the rational solutions to the hierarchy of the kadomtsev petviashvili equation is presented the generalization of the superposition formula is found to permit the construction of a new general solution from some partial ones the features of the general polynomial factorization and the types of scattering in the many body calogero moser problem are investigated

dispersionless scalar integrable hierarchies whitham hierarchy and the quasiclassical dressing method the quasiclassical limit of the scalar nonlocal problem is derived and a quasiclassical version of the dressing method is presented dispersionless kadomtsev petviashvili kp modified kp and dispersionless two dimensional toda lattice 2dtl hierarchies are discussed as illustrative examples it is shown that the universal whitham hierarchy is nothing but the ring of symmetries for the quasiclassical problem the reduction problem is discussed and in particular the d2dtl equation of b type is derived

conformal maps and integrable hierarchies we show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2d toda integrable hierarchy the maps are determined by a particular solution to the hierarchy singled out by the conditions known as string equations the same hierarchy locally solves the 2d inverse potential problem i e reconstruction of the domain out of a set of its harmonic moments this is the same solution which is known to describe 2d gravity coupled to c matter we also introduce a concept of the function for analytic curves

dispersionless limit of the noncommutative potential kp hierarchy and solutions of the pseudodual chiral model in 2 1 dimensions the usual dispersionless limit of the kp hierarchy does not work in the case where the dependent variable has values in a noncommutative e g matrix algebra passing over to the potential kp hierarchy there is a corresponding scaling limit in the noncommutative case which turns out to be the hierarchy of a pseudodual chiral model in 2 1 dimensions pseudodual to a hierarchy extending ward s modified integrable chiral model applying the scaling procedure to a method generating exact solutions of a matrix potential kp hierarchy from solutions of a matrix linear heat hierarchy leads to a corresponding method that generates exact solutions of the matrix dispersionless potential kp hierarchy i e the pseudodual chiral model hierarchy we use this result to construct classes of exact solutions of the su m pseudodual chiral model in 2 1 dimensions including various multiple lump configurations

the approach to the dispersionless kp hierarchy the dispersionless limit of the scalar nonlocal problem is derived it is given by a special class of nonlinear first order equations a quasiclassical version of the dressing method is presented it is shown that the algebraic formulation of dispersionless hierarchies can be expressed in terms of properties of beltrami type equations the universal whitham hierarchy and in particular the dispersionless kp hierarchy turn out to be rings of symmetries for the quasiclassical problem pacs number 02 30 uk

some observations about the convex combination of continuous triangular norms abstract the convex combination of continuous triangular norms is discussed let c 1 be the class of all nilpotent t norms c 2 be the class of all strict t norms and c 3 be the class of all continuous non archimedean t norms it is shown that for any t norms t 1 t 2 which are taken from two different classes of c i i 1 2 3 the convex combination t 1 t 1 t 2 is not associative for any 0 1 the case of two continuous non archimedean t norms is also investigated by our results the problem of whether or not two continuous t norms can be combined with each other is reduced to that of whether two strict or two nilpotent t norms can be combined with each other or not

the dispersionless lax equations and topological minimal models it is shown that perturbed rings of the primary chiral fields of the topological minimal models coincide with some particular solutions of the dispersionless lax equations the exact formulae for the tree level partition functions of an topological minimal models are found the virasoro constraints for the analogue of the function of the dispersionless lax equation correspond ing to these models are proved

symmetries and tau function of higher dimensional dispersionless integrable hierarchies a higher dimensional analogue of the dispersionless kp hierarchy is introduced in addition to the two dimensional phase space variables k x of the dispersionless kp hierarchy this hierarchy has extra spatial dimensions compactified to a two or any even dimensional torus integrability of this hierarchy and the existence of an infinite dimensional of additional symmetries are ensured by an underlying twistor theoretical structure or a nonlinear riemann hilbert problem an analogue of the tau function whose logarithm gives the f function free energy or prepotential in the context of matrix models and topological conformal field theories is constructed the infinite dimensional symmetries can be extended to this tau or f function the extended symmetries just like those of the dispersionless kp hierarchy obey an anomalous commutation relations

convex combinations of continuous t norms with the same diagonal function abstract the problem of whether a non trivial convex combination of two continuous t norms with the same diagonal function can be a t norm is studied it is shown that in both cases of two nilpotent and of two strict t norms a non trivial convex combination of t norms with common diagonal function is associative only if the two t norms involved coincide for general continuous t norms a similar result follows an example of a convex class of non continuous t norms is also included

extended wave solutions in an integrable chiral model in 2 1 dimensions there is a modification of the su 2 chiral model which is integrable in 2 1 dimensions j math phys 29 386 1988 in addition to localized lumps it admits extended wave solutions which move at constant velocity the interaction of two waves causes each to experience a phase shift in the interaction between a wave and a lump the wave suffers no phase shift but the lump changes shape

on the discrete spectrum of the nonstationary schrodinger equation and multipole lumps of the kadomtsev petviashvili i equation the discrete spectrum of the nonstationary schrodinger equation and localized solutions of the kadomtsev petviashvili i kpi equation are studied via the inverse scattering transform it is shown that there exist infinitely many real and rationally decaying potentials which correspond to a discrete spectrum whose related eigenfunctions have multiple poles in the spectral parameter an index or winding number is asssociated with each of these solutions the resulting localized solutions of kpi behave as collection of individual humps with nonuniform dynamics

hierarchy structure in integrable systems of gauge fields and underlying lie algebras an improved version of nakamura s self dual yang mills hierarchy is presentd and its symmetry contents are studied the new hierarchy as well as the previous one represents a set of commuting dynamical flows in an infinite dimensional manifolds of loop type but includes a large set of dependent variables because of new degrees of freedom the theory acquires a more symmetric form with richer structures for example it allows a large symmetry algebra of riemann hilbert type which is actually a direct sum of two subalgebras left and right this phenomenon is basically the same as observed recently by avan and bellon on the case of principal chiral models in addition to these rather familiar symmeties a new type of symmetries referred to as coordinate transformation type are also introduced generators of the above dynamical flows are all included therein these two types of symmetries altogether form a big lie algebra which lead to more satisfactory understanding of symmetry properties of integrable systems of guage fields

nontrivial soliton scattering in an integrable chiral model in 2 1 dimensions soliton solutions of an integrable chiral model in 2 1 dimensions are studied through the solution of a riemann problem with zeros solitons of this model have already been found r s ward j math phys 29 386 1988 which scatter in a trivial manner by exploiting a connection with the o 3 model it is shown that the model also has solitons that are not contained in the above construction and whose scattering properties are highly nontrivial soliton scattering is also discussed for a model which interpolates between this integrable model and the su 2 chiral model

bi differential calculi and integrable models the existence of an infinite set of conserved currents in completely integrable classical models including chiral and toda models as well as the kp and self dual yang mills equations is traced back to a simple construction of an infinite chain of closed respectively covariantly constant 1 forms in a gauged bi differential calculus the latter consists of a differential algebra on which two differential maps act in a gauged bi differential calculus these maps are extended to flat covariant derivatives

some results of weighted quasi arithmetic mean of continuous triangular norms the weighted quasi arithmetic mean of continuous triangular norms is investigated some results on convex combination of continuous triangular norms in ouyang and fang y ouyang j x fang some observations about the convex combination of continuous triangular norms nonlinear anal tma 68 2008 3382 3387 are generalized several open problems are posed

twistors in 2 1 dimensions the geometry of twistors for 2 1 dimensional flat space time is described functions on twistor space generate solutions of various field equations in space time as an illustrative example it is shown what a sine gordon soliton looks like in this twistor description

bicomplexes and backlund transformations a bicomplex is a simple mathematical structure in particular associated with completely integrable models the conditions defining a bicomplex are a special form of a parameter dependent zero curvature condition we generalize the concept of a darboux matrix to bicomplexes and use it to derive backlund transformations for several models the method also works for moyal deformed equations with a corresponding deformed bicomplex

hydrodynamic reductions of multidimensional dispersionless pdes the test for integrability a d 1 dimensional dispersionless pde is said to be integrable if its n component hydrodynamic reductions are locally parametrized by d 1 n arbitrary functions of one variable the most important examples include the four dimensional heavenly equation descriptive of self dual ricci flat metrics and its six dimensional generalization arising in the context of sdiff 2 self dual yang mills equations given a multidimensional pde which does not pass the integrability test the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which when added to the equation make it an integrable system in fewer dimensions as an example of this phenomenon we discuss the second commuting flow of the dispersionless kp hierarchy considered separately this is a four dimensional pde which does not pass the integrability test however the method of hydrodynamic reductions generates additional differential constraints which reconstruct the full 2 1 dimensional dis

functional representations of integrable hierarchies we consider a general framework for integrable hierarchies in lax form and derive certain universal equations from which functional representations of particular hierarchies such as kp discrete kp mkp akns i e formulations in terms of functional equations are systematically and quite easily obtained the formalism genuinely applies to hierarchies where the dependent variables live in a noncommutative typically matrix algebra the obtained functional representations can be understood as noncommutative analogues of fay identities for the kp hierarchy

pp is closed under truth table reductions beigel reingold and spielman j comput system sci 50 191 202 1995 showed that pp is closed under intersection and a variety of special cases of polynomial time truth table closure we extend their techniques to show that pp is closed under general polynomial time truth table reductions we also show that pp is closed under constant round truth table reductions

solutions of the modified chiral model in 2 1 dimensions in this paper we deal with classical solutions of the modified chiral model on r2 1 such solutions are shown to correspond to products of various factors which we call time dependent unitons then the problem of solving the system of second order partial differential equations for the chiral field is reduced to solving a sequence of systems of first order partial differential equations for the unitons

multi scale expansions in the theory of systems integrable by the inverse scattering transform abstract it is shown that using multi scale expansions conventionally employed in the theory of nonlinear waves one can transform systems integrable by the ist method into other systems of this type

on the miura map between the dispersionless kp and dispersionless modified kp hierarchies the miura map between the dispersionless kp and dispersionless modified kp hierarchies is investigated it is shown that the miura map is canonical with respect to their bi hamiltonian structures moreover inspired by the works of takasaki and takebe the twistor construction of solution structure for the dispersionless modified kp hierarchy is given

on the convex combination of td and continuous triangular norms the problem of when t l 1 l t d lt l 0 1 is a triangular norm where t d is the drastic product and t is a continuous triangular norm is studied it is shown that t l cannot be a triangular norm when t is nilpotent it is also shown that t l is a triangular norm if t is strict and its additive generator f satisfies f lx f x f l for all x 0 1 the cases that t t m and t is the ordinal sum of continuous archimedean summands are also discussed some left continuous t norms which can be combined with each other are given