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We investigate the quasi-integrability properties of various deformations of the Korteweg–de Vries (KdV) equation, depending on two parameters ε1 and ε2, which include among them the regularized long-wave (RLW) and modified regularized long-wave (mRLW) equations. We show, using analytical and numerical methods, that the charges, constructed from a deformation of the zero curvature equation for the KdV equation, are asymptotically conserved for various values of the deformation parameters. By this we mean that, despite the fact that the charges do vary in time during the scattering of solitons, they return after the scattering to the same values they had before it. This property was tested numerically for the scattering of two and three solitons, and analytically for the scattering of two solitons in the mRLW theory (ε2=ε1=1). In addition we show that for any values of ε1 and ε2 the Hirota method leads to analytical one-soliton solutions of our deformed equation but for ε1≠1 such solutions have the dispersion relation which depends on the parameter ε1. We also discuss some properties of soliton-radiation interactions seen in some of our simulations.

We study a recently proposed modification of the Skyrme model that possesses an exact self-dual sector leading to an infinity of exact Skyrmion solutions with arbitrary topological (baryon) charge. The self-dual sector is made possible by the introduction, in addition to the usual three SU(2) Skyrme fields, of six scalar fields assembled in a symmetric and invertible three dimensional matrix h. The action presents quadratic and quartic terms in derivatives of the Skyrme fields, but instead of the group indices being contracted by the SU(2) Killing form, they are contracted with the h-matrix in the quadratic term, and by its inverse on the quartic term. Due to these extra fields the static version of the model, as well as its self-duality equations, are conformally invariant on the three dimensional space R^3. We show that the static and self-dual sectors of such a theory are equivalent, and so the only non-self-dual solution must be time dependent. We also show that for any configuration of the Skyrme SU(2) fields, the h-fields adjust themselves to satisfy the self-duality equations, and so the theory has plenty of non-trivial topological solutions. We present explicit exact solutions using a holomorphic rational ansatz, as well as a toroidal ansatz based on the conformal symmetry. We point to possible extensions of the model that break the conformal symmetry as well as the self-dual sector, and that can perhaps lead to interesting physical applications., Comment: 25 pages, no figures, version published in JHEP 09 (2020) 031

We introduce a Skyrme type model with the target space being the 3-sphere S^3 and with an action possessing, as usual, quadratic and quartic terms in field derivatives. The novel character of the model is that the strength of the couplings of those two terms are allowed to depend upon the space-time coordinates. The model should therefore be interpreted as an effective theory, such that those couplings correspond in fact to low energy expectation values of fields belonging to a more fundamental theory at high energies. The theory possesses a self-dual sector that saturates the Bogomolny bound leading to an energy depending linearly on the topological charge. The self-duality equations are conformally invariant in three space dimensions leading to a toroidal ansatz and exact self-dual Skyrmion solutions. Those solutions are labelled by two integers and, despite their toroidal character, the energy density is spherically symmetric when those integers are equal and oblate or prolate otherwise., Comment: 14 pages, 3 figures, a reference added

We establish that the Wu-Yang monopole needs the introduction of a magnetic point source at the origin in order for it to be a solution of the differential and integral equations for the Yang-Mills theory. That result is corroborated by the analysis through distribution theory, of the two types of magnetic fields relevant for the local and global properties of the Wu-Yang solution. The subtlety lies on the fact that with the non-vanishing magnetic point source required by the Yang-Mills integral equations, the Wu-Yang monopole configuration does not violate, in the sense of distribution theory, the differential Bianchi identity., 24 pages, one figure, matches version published in Journal of Physics A 52 (2019) 155202

We use ideas of generalized self-duality conditions to construct real scalar field theories in (1 + 1)-dimensions with exact self dual sectors. The approach is based on a pre-potential U that defines the topological charge and the potential energy of these theories. In our algebraic method to construct the required pre-potentials we use the representation theory of Lie groups. This approach leads naturally to an infinite set of degenerate vacua and so to topologically non-trivial self-dual solutions of these models. We present explicit examples for the groups SU(2), SU(3) and SO(5) and discuss some properties of these solutions., Comment: 39 pages, 17 figures

We look at simple BPS systems involving more than one field. We discuss the conditions that have to be imposed on various terms in Lagrangians involving many fields to produce BPS systems and then look in more detail at the simplest of such cases. We analyse in detail BPS systems involving 2 interacting Sine-Gordon like fields, both when one of them has a kink solution and the second one either a kink or an antikink solution. We take their solitonic static solutions and use them as initial conditions for their evolution in Lorentz covariant versions of such models. We send these structures towards themselves and find that when they interact weakly they can pass through each other with a phase shift which is related to the strength of their interaction. When they interact strongly they repel and reflect on each other. We use the method of a modified gradient flow in order to visualize the solutions in the space of fields., Comment: 27 pages, 17 figures

We use the SU(2) 't Hooft-Polyakov monopole configuration, and its BPS version, to test the integral equations of the Yang-Mills theory. Those integral equations involve two (complex) parameters which do not appear in the differential Yang-Mills equations, and if they are considered to be arbitrary it then implies that non-abelian gauge theories (but not abelian ones) possess an infinity of integral equations. For static monopole configurations only one of those parameters is relevant. We expand the integral Yang-Mills equation in a power series of that parameter and show that the 't Hooft-Polyakov monopole and its BPS version satisfy the integral equations obtained in first and second order of that expansion. Our results points to the importance of exploring the physical consequences of such an infinity of integral equations on the global properties of the Yang-Mills theory.

We propose a modification of the Skyrme model that supports a self-dual sector possessing exact non-trivial finite energy solutions. The action of such a theory possesses the usual quadratic and quartic terms in field derivatives, but the couplings of the components of the Maurer-Cartan form of the Skyrme model is made by a non-constant symmetric matrix, instead of the usual Killing form of the SU(2) Lie algebra. The introduction of such a matrix make the self-duality equations conformally invariant in three space dimensions, even though it may break the global internal symmetries of the original Skyrme model. For the case where that matrix is proportional to the identity we show that the theory possesses exact self-dual Skyrmions of unity topological charges., 12 pages, no figures

We consider deformations of the $SU(3)$ Affine Toda theory (AT) and investigate the integrability properties of the deformed theories. We find that for some special deformations all conserved quantities change to being conserved only asymptotically, {\it i.e.} in the process of the scattering of two solitons these charges do vary in time, but they return, after the scattering, to the values they had prior to the scattering. This phenomenon, which we have called quasi-integrability, is related to special properties of the two-soliton solutions under space-time parity transformations. Some properties of the AT solitons are discussed, especially those involving interesting static multi-soliton solutions. We support our analytical studies with detailed numerical ones in which the time evolution has been simulated by the 4th order Runge-Kutta method. We find that for some perturbations the solitons repel and for the others they form a quasi-bound state. When we send solitons towards each other they can repel when they come close together with or without `flipping' the fields of the model. The solitons radiate very little and appear to be stable. These results support the ideas of quasi-integrability, {\it i.e.} that many effects of integrability also approximately hold for the deformed models., a small mistyping corrected

We report analytical and numerical results on breather-like field configurations in a theory which is a deformation of the integrable sine-Gordon model in (1 + 1) dimensions. The main motivation of our study is to test the ideas behind the recently proposed concept of quasi-integrability, which emerged from the observation that some field theories possess an infinite number of quantities which are asymptotically conserved in the scattering of solitons, and periodic in time in the case of breather-like configurations. Even though the mechanism responsible for such phenomena is not well understood yet, it is clear that special properties of the solutions under a space-time parity transformation play a crucial role. The numerical results of the present paper give support for the ideas on quasi-integrability, as it is found that extremely long-lived breather configurations satisfy these parity properties. We also report on a mechanism, particular to the theory studied here, that favours the existence of long lived breathers even in cases of significant deformations of the sine-Gordon potential. We also find numerically that our breather-like configurations decay through the gradual increase of their frequency of oscillations.

In this paper we present our preliminary results which suggest that some field theory models are ‘almost’ integrable; i.e. they possess a large number of ‘almost’ conserved quantities. First we demonstrate this, in some detail, on a class of models which generalise sine-Gordon model in (1+1) dimensions. Then, we point out that many field configurations of these models look like those of the integrable systems and others are very close to being integrable. Finally we attempt to quantify these claims looking in particular, both analytically and numerically, at some long lived field configurations which resemble breathers.

The Bullough-Dodd model is an important two dimensional integrable field theory which finds applications in physics and geometry. We consider a conformally invariant extension of it, and study its integrability properties using a zero curvature condition based on the twisted Kac-Moody algebra A_2^{(2)}. The one and two-soliton solutions as well as the breathers are constructed explicitly . We also consider integrable extensions of the Bullough-Dodd model by the introduction of spinor (matter) fields. The resulting theories are conformally invariant and present local internal symmetries. All the one-soliton solutions, for two examples of those models, are constructed using an hybrid of the dressing and Hirota methods. One model is of particular interest because it presents a confinement mechanism for a given conserved charge inside the solitons., Comment: 48 pages, 3 eps figures, latex

We evaluate the gauge invariant, dynamically conserved charges, recently obtained from the integral form of the Yang-Mills equations, for the BPS multi-dyon solutions of a Yang-Mills-Higgs theory associated to any compact semi-simple gauge group G. Those charges are shown to correspond to the eigenvalues of the next-to-leading term of the asymptotic form of the Higgs field at spatial infinity, and so coinciding with the usual topological charges of those solutions. Such results show that many of the topological charges considered in the literature are in fact dynamical charges, which conservation follows from the global properties of classical Yang-Mills theories encoded into their integral dynamical equations. The conservation of those charges can not be obtained from the differential form of Yang-Mills equations., Comment: Version to be published in JHEP, Journal of High Energy Physics (19 pages, no figures, some examples added)

It has been shown recently that deformations of some integrable field theories in (1+1)-dimensions possess an infinite number of charges that are asymptotically conserved in the scattering of soliton like solutions. Such charges are not conserved in time and they do vary considerably during the scattering process, however they all return in the remote future (after the scattering) to the values they had in the remote past (before the scattering). Such non-linear phenomenon was named quasi-integrability, and it seems to be related to special properties of the solutions under a space-time parity transformation. In this paper we investigate, analytically and numerically, such phenomenon in the context of deformations of the integrable Bullough-Dodd model. We find that a special class of two-soliton like solutions of such deformed theories do present an infinite number of asymptotically conserved charges., Comment: 33 pages, 5 pdf figures

We show that the Skyrme theory possesses a submodel with an infinite number of local conserved currents. The constraints leading to the submodel explore a decomposition of SU(2) with a complex field parametrizing the symmetric space SU(2)/U(1) and a real field in the direction of U(1). We demonstrate that the Skyrmions of topological charges $\pm 1$ belong to such integrable sector of the theory. Our results open ways to the development of exact methods, compensating for the non-existence of a BPS type sector in the Skyrme theory., 9 pages, LaTeX, version to appear in Phys. Lett. B

We consider a (3+1)-dimensional local field theory defined on the sphere. The model possesses exact soliton solutions with non trivial Hopf topological charges, and infinite number of local conserved currents. We show that the Poisson bracket algebra of the corresponding charges is isomorphic to that of the area preserving diffeomorphisms of the sphere. We also show that the conserved currents under consideration are the Noether currents associated to the invariance of the Lagrangian under that infinite group of diffeomorphisms. We indicate possible generalizations of the model.

We consider a two-dimensional integrable and conformally invariant field theory possessing two Dirac spinors and three scalar fields. The interaction couples bilinear terms in the spinors to exponentials of the scalars. Its integrability properties are based on the sl(2) affine Kac-Moody algebra, and it is a simple example of the so-called conformal affine Toda theories coupled to matter fields. We show, using bosonization techniques, that the classical equivalence between a U(1) Noether current and the topological current holds true at the quantum level, and then leads to a bag model like mechanism for the confinement of the spinor fields inside the solitons. By bosonizing the spinors we show that the theory decouples into a sine-Gordon model and free scalars. We construct the two-soliton solutions and show that their interactions lead to the same time delays as those for the sine-Gordon solitons. The model provides a good laboratory to test duality ideas in the context of the equivalence between the sine-Gordon and Thirring theories., LaTeX, 26 pages

In this paper we construct explicitly an infinite number of Hopfions (static, soliton solutions with non-zero Hopf topological charges) within the recently proposed 3+1-dimensional, integrable and relativistically invariant field theory. Two integers label the family of Hopfions we have found. Their product is equal to the Hopf charge which provides a lower bound to the soliton's finite energy. The Hopfions are constructed explicitly in terms of the toroidal coordinates and shown to have a form of linked closed vortices.

We analyze the integrability properties of models defined on the symmetric space SU(2)/U(1) in 3+1 dimensions, using a recently proposed approach for integrable theories in any dimension. We point out the key ingredients for a theory to possess an infinite number of local conservation laws, and discuss classes of models with such property. We propose a 3+1-dimensional, relativistic invariant field theory possessing a toroidal soliton solution carrying a unit of topological charge given by the Hopf map. Construction of the action is guided by the requirement that the energy of static configuration should be scale invariant. The solution is constructed exactly. The model possesses an infinite number of local conserved currents. The method is also applied to the Skyrme-Faddeev model, and integrable submodels are proposed., LaTeX, 14 pgs

The algebraic matrix hierarchy approach based on affine Lie $sl (n)$ algebras leads to a variety of 1+1 soliton equations. By varying the rank of the underlying $sl (n)$ algebra as well as its gradation in the affine setting, one encompasses the set of the soliton equations of the constrained KP hierarchy. The soliton solutions are then obtained as elements of the orbits of the dressing transformations constructed in terms of representations of the vertex operators of the affine $sl (n)$ algebras realized in the unconventional gradations. Such soliton solutions exhibit non-trivial dependence on the KdV (odd) time flows and KP (odd and even) time flows which distinguishes them from the conventional structure of the Darboux-B\"{a}cklund Wronskian solutions of the constrained KP hierarchy., Comment: LaTeX, 13pgs

The zero curvature representation for two dimensional integrable models is generalized to spacetimes of dimension d+1 by the introduction of a d-form connection. The new generalized zero curvature conditions can be used to represent the equations of motion of some relativistic invariant field theories of physical interest in 2+1 dimensions (BF theories, Chern-Simons, 2+1 gravity and the CP^1 model) and 3+1 dimensions (self-dual Yang-Mills theory and the Bogomolny equations). Our approach leads to new methods of constructing conserved currents and solutions. In a submodel of the 2+1 dimensional CP^1 model, we explicitly construct an infinite number of previously unknown nontrivial conserved currents. For each positive integer spin representation of sl(2) we construct 2j+1 conserved currents leading to 2j+1 Lorentz scalar charges., Comment: 52 pages, 4 figures, shortened version to appear in NPB

The solutions of a large class of hierarchies of zero-curvature equations that includes Toda and KdV type hierarchies are investigated. All these hierarchies are constructed from affine (twisted or untwisted) Kac-Moody algebras~$\ggg$. Their common feature is that they have some special ``vacuum solutions'' corresponding to Lax operators lying in some abelian (up to the central term) subalgebra of~$\ggg$; in some interesting cases such subalgebras are of the Heisenberg type. Using the dressing transformation method, the solutions in the orbit of those vacuum solutions are constructed in a uniform way. Then, the generalized tau-functions for those hierarchies are defined as an alternative set of variables corresponding to certain matrix elements evaluated in the integrable highest-weight representations of~$\ggg$. Such definition of tau-functions applies for any level of the representation, and it is independent of its realization (vertex operator or not). The particular important cases of generalized mKdV and KdV hierarchies as well as the abelian and non abelian affine Toda theories are discussed in detail., 27 pages, plain TeX

Following our attempts to define quasi-integrability in which we related this concept to a particular symmetry of the two-soliton function we check this condition in three classes of modified Sine-Gordon models in (1+1) dimensions. We find that the numerical results seen in various scatterings of two solitons and in the time evolution of breather-like structures support our ideas about the symmetry of the field configurations and its effects on the anomalies of the conservation laws of the charges., 33 pages, 19 figures, Reference added

We propose a new Skyrme-like model with fields taking values on the sphere S^3 or, equivalently, on the group SU(2). The action of the model contains a quadratic kinetic term plus a quartic term which is the same as that of the Skyrme-Faddeev model. The novelty of the model is that it possess a first order Bogomolny type equation whose solutions automatically satisfy the second order Euler-Lagrange equations. It also possesses a lower bound on the static energy which is saturated by the Bogomolny solutions. Such Bogomolny equation is equivalent to the so-called force free equation used in plasma and solar Physics, and which possesses large classes of solutions. An old result due to Chandrasekhar prevents the existence of finite energy solutions for the force free equation on the entire tridimensional space R^3. We construct new exact finite energy solutions to the Bogomolny equations for the case where the space is the three-sphere S^3, using toroidal like coordinates., Comment: 12 pages, no figures

We show that certain field theory models, although non-integrable according to the usual definition of integrability, share some of the features of integrable theories for certain configurations. Here we discuss our attempt to define a "quasi-integrable theory", through a concrete example: a deformation of the (integrable) sine-Gordon potential. The techniques used to describe and define this concept are both analytical and numerical. The zero-curvature representation and the abelianisation procedure commonly used in integrable field theories are adapted to this new case and we show that they produce asymptotically conserved charges that can then be observed in the simulations of scattering of solitons., Submitted to the Proceedings of the 2nd International Workshop on Nonlinear and Modern Mathematical Physics, March 9-11, 2013, Tampa, Florida, USA

We investigate higher grading integrable generalizations of the affine Toda systems. The extra fields, associated to non zero grade generators, obey field equations of the Dirac type and are regarded as matter fields. The models possess soliton configurations, which can be interpreted as particles of the theory, on the same footing as those associated to fundamental fields. A special subclass of these models is remarkable. They possess a $U(1)$ Noether current which, after a special gauge fixing of the conformal symmetry, is proportional to a topological current. This leads to the confinement of the matter field inside the solitons, which can be regarded as a one dimensional bag model for QCD. These models are also relevent to the study of electron self--localization in (quasi)-one-dimensional electron--phonon systems., Comment: 51 pages, LaTeX

Despite the fact that the integral form of the equations of classical electrodynamics is well known, the same is not true for non-Abelian gauge theories. The aim of the present paper is threefold. First, we present the integral form of the classical Yang-Mills equations in the presence of sources and then use it to solve the long-standing problem of constructing conserved charges, for any field configuration, which are invariant under general gauge transformations and not only under transformations that go to a constant at spatial infinity. The construction is based on concepts in loop spaces and on a generalization of the non-Abelian Stokes theorem for two-form connections. The third goal of the paper is to present the integral form of the self-dual Yang-Mills equations and calculate the conserved charges associated with them. The charges are explicitly evaluated for the cases of monopoles, dyons, instantons and merons, and we show that in many cases those charges must be quantized. Our results are important in the understanding of global properties of non-Abelian gauge theories.

We consider modifications of the nonlinear Schrodinger model (NLS) to look at the recently introduced concept of quasi-integrability. We show that such models possess an infinite number of quasi-conserved charges which present intriguing properties in relation to very specific space-time parity transformations. For the case of two-soliton solutions where the fields are eigenstates of this parity, those charges are asymptotically conserved in the scattering process of the solitons. Even though the charges vary in time their values in the far past and the far future are the same. Such results are obtained through analytical and numerical methods, and employ adaptations of algebraic techniques used in integrable field theories. Our findings may have important consequences on the applications of these models in several areas of non-linear science. We make a detailed numerical study of the modified NLS potential of the form V = |psi|^(2(2+epsilon)), with epsilon being a perturbation parameter. We perform numerical simulations of the scattering of solitons for this model and find a good agreement with the results predicted by the analytical considerations. Our paper shows that the quasi-integrability concepts recently proposed in the context of modifications of the sine-Gordon model remain valid for perturbations of the NLS model., 39 pages, 12 figures

We construct new classes of vortexlike solutions of the CP{sup N} model in (3+1) dimensions and discuss some of their properties. These solutions are obtained by generalizing to (3+1) dimensions the techniques well established for the two-dimensional CP{sup N} models. We show that as the total energy of these solutions is infinite, they describe evolving vortices and antivortices with the energy density of some configurations varying in time. We also make some further observations about the dynamics of these vortices.

We present a class of solutions of the $C{P}^{N}$ model in ($3+1$) dimensions. We suggest that they represent vortexlike configurations. We also discuss some of their properties. We show that some configurations of vortices have a divergent energy per unit length while for the others such an energy has a minimum for a very special orientation of vortices. We also discuss the Noether charge densities of these vortices.

The construction of axially symmetric soliton solutions with non-zero Hopf topological charges according to a theory known as the extended Skyrme-Faddeev model, was performed in [1]. In this paper we show how masses of glueballs are predicted within this model.

We look at properties of vortex solutions of the extended CP^N Skyrme-Faddeev model. We show that only holomorphic solutions of the CP^N model are also solutions of the Skyrme-Faddeev model. As the total energy of these solutions is infinite these solutions should be interpreted as describing time dependent vortices. We describe their dynamics and, in partcular, point out that one of the terms in the energy density is related to the Noether charge of the model., Comment: 20 pages, 6 figures

Invariance under non-linear ${\sf {\hat W}}_{\infty}$ algebra is shown for the two-boson Liouville type of model and its algebraic generalizations, the extended conformal Toda models. The realization of the corresponding generators in terms of two boson currents within KP hierarchy is presented., 10 pgs, LaTeX, IFT-P.038/93

We use Hirota's method formulated as a recursive scheme to construct complete set of soliton solutions for the affine Toda field theory based on an arbitrary Lie algebra. Our solutions include a new class of solitons connected with two different type of degeneracies encountered in the Hirota's perturbation approach. We also derive an universal mass formula for all Hirota's solutions to the Affine Toda model valid for all underlying Lie groups. Embedding of the Affine Toda model in the Conformal Affine Toda model plays a crucial role in this analysis., Comment: 36 pages, LaTeX

We consider a four dimensional field theory with target space being CP^N which constitutes a generalization of the usual Skyrme-Faddeev model defined on CP^1. We show that it possesses an integrable sector presenting an infinite number of local conservation laws, which are associated to the hidden symmetries of the zero curvature representation of the theory in loop space. We construct an infinite class of exact solutions for that integrable submodel where the fields are meromorphic functions of the combinations (x^1+i x^2) and (x^3+x^0) of the Cartesian coordinates of four dimensional Minkowski space-time. Among those solutions we have static vortices and also vortices with waves traveling along them with the speed of light. The energy per unity of length of the vortices show an interesting and intricate interaction among the vortices and waves., Comment: 21 pages, plain latex, no figures

We use the deformed sine-Gordon models recently presented by Bazeia et al to discuss possible definitions of quasi-integrability. We present one such definition and use it to calculate an infinite number of quasi-conserved quantities through a modification of the usual techniques of integrable field theories. Performing an expansion around the sine-Gordon theory we are able to evaluate the charges and the anomalies of their conservation laws in a perturbative power series in a small parameter which describes the "closeness" to the integrable sine-Gordon model. Our results indicate that in the case of the two-soliton scattering the charges are conserved asymptotically, i.e. their values are the same in the distant past and future, when the solitons are well separated. We back up our results with numerical simulations which also demonstrate the existence of long lived breather-like and wobble-like states in these models., Comment: 40 pages, 47 ps figures

Made available in DSpace on 2022-04-29T08:43:23Z (GMT). No. of bitstreams: 0 Previous issue date: 1992-12-01 The authors establish a precise relation between the structures of Lie and Jordan algebras by presenting a method of constructing one type of algebra from the other. The examples of the Lie algebras associated to simple Jordan algebras Mm(n) and Clifford algebras are discussed in detail. The generalization of such arguments to infinite-dimensional Lie algebras in terms of Fermi fields is also discussed. Inst. de Fisica Teorica, Sao Paulo

As recently shown the conformal affine Toda models can be obtained via hamiltonian reduction from a two-loop Kac-Moody algebra. In this paper we propose a systematic procedure to analyze the higher spin symmetries of the conformal affine Toda models. The method is based on an explicit construction of infinite towers of extended conformal symmetry generators. Two fundamental building blocks of this construction are special spin-one and -two primary fields characterizing the conformal structure of these models. The connection to the algebra of area preserving diffeomorphisms on a two-manifold (w∞ algebra) is established.

We construct static soliton solutions with non-zero Hopf topological charges to a theory which is an extension of the Skyrme-Faddeev model by the addition of a further quartic term in derivatives. We use an axially symmetric ansatz based on toroidal coordinates, and solve the resulting two coupled non-linear partial differential equations in two variables by a successive over-relaxation (SOR) method. We construct numerical solutions with Hopf charge up to four, and calculate their analytical behavior in some limiting cases. The solutions present an interesting behavior under the changes of a special combination of the coupling constants of the quartic terms. Their energies and sizes tend to zero as that combination approaches a particular special value. We calculate the equivalent of the Vakulenko and Kapitanskii energy bound for the theory and find that it vanishes at that same special value of the coupling constants. In addition, the model presents an integrable sector with an infinite number of local conserved currents which apparently are not related to symmetries of the action. In the intersection of those two special sectors the theory possesses exact vortex solutions (static and time dependent) which were constructed in a previous paper by one of the authors. It is believed that such model describes some aspects of the low energy limit of the pure SU(2) Yang-Mills theory, and our results may be important in identifying important structures in that strong coupling regime., 22 pages, 42 figures, minor corrections

We review our proposal to generalize the standard two-dimensional flatness construction of Lax-Zakharov-Shabat to relativistic field theories in d+1 dimensions. The fundamentals from the theory of connections on loop spaces are presented and clarified. These ideas are exposed using mathematical tools familiar to physicists. We exhibit recent and new results that relate the locality of the loop space curvature to the diffeomorphism invariance of the loop space holonomy. These result are used to show that the holonomy is abelian if the holonomy is diffeomorphism invariant. These results justify in part and set the limitations of the local implementations of the approach which has been worked out in the last decade. We highlight very interesting applications like the construction and the solution of an integrable four dimensional field theory with Hopf solitons, and new integrability conditions which generalize BPS equations to systems such as Skyrme theories. Applications of these ideas leading to new constructions are implemented in theories that admit volume preserving diffeomorphisms of the target space as symmetries. Applications to physically relevant systems like Yang Mills theories are summarized. We also discuss other possibilities that have not yet been explored., 64 pages, 8 figures

We construct static and time-dependent exact soliton solutions with non-trivial Hopf topological charge for a field theory in 3+1 dimensions with the target space being the two dimensional sphere S**2. The model considered is a reduction of the so-called extended Skyrme-Faddeev theory by the removal of the quadratic term in derivatives of the fields. The solutions are constructed using an ansatz based on the conformal and target space symmetries. The solutions are said self-dual because they solve first order differential equations which together with some conditions on the coupling constants, imply the second order equations of motion. The solutions belong to a sub-sector of the theory with an infinite number of local conserved currents. The equation for the profile function of the ansatz corresponds to the Bogomolny equation for the sine-Gordon model., Comment: plain latex, no figures, 23 pages

We construct exact vortex solutions in 3+1 dimensions to a theory which is an extension, due to Gies, of the Skyrme-Faddeev model, and that is believed to describe some aspects of the low energy limit of the pure SU(2) Yang-Mills theory. Despite the efforts in the last decades those are the first exact analytical solutions to be constructed for such type of theory. The exact vortices appear in a very particular sector of the theory characterized by special values of the coupling constants, and by a constraint that leads to an infinite number of conserved charges. The theory is scale invariant in that sector, and the solutions satisfy Bogomolny type equations. The energy of the static vortex is proportional to its topological charge, and waves can travel with the speed of light along them, adding to the energy a term proportional to a U(1) Noether charge they create. We believe such vortices may play a role in the strong coupling regime of the pure SU(2) Yang-Mills theory., 12 pages, enlarged version published in JHEP05(2009)001

We study various solutions of the Sine Gordon model in (1+1) dimensions. We use the Hirota method to construct some of them and then show that the wobble, discussed in detail by K\"alberman, is one of such solutions. We concentrate our attention on a kink and its bound states with one or two breathers. We study their stability and some aspects of their scattering properties on potential wells and on fixed boundary conditions., Comment: 17 pages, 7 ps figures, latex

A method is presented for constructing the general solution to higher Hamiltonians (nonquadratic in the momenta) of the Toda hierarchies of integrable models associated with a simple Lie group G. The method is representation independent and is based on a modified version of the Lax operator. It constitutes a generalization of the method used to construct the solutions of the Toda molecule models. The SL(3) and SL(4) cases are discussed in detail.

A construction relating the structures of super Lie and super Jordan algebras is proposed. This may clarify the role played by field theoretical realizations of super Jordan algebras in constructing representations of super Kac-Moody algebras. The case of OSP (m, n) and super Clifford algebras involving independent Fermi fields and symplectic bosons is discussed in detail.

We present a general review of the dynamics of topological solitons in 1 and 2 dimensions and then discuss some recent work on the scattering of various solitonic objects (such as kinks and breathers etc) on potential obstructions., based on the talk given by W.J. Zakrzewski at QTS5. To appear in the Proceedings in a special issue of Journal of Physics A

We consider a very large class of hierarchies of zero-curvature equations constructed from affine Kac-Moody algebras Ĵ. We argue that one of the basic ingredients for the appearance of soliton solutions in such theories is the existence of “vacuum solutions” corresponding to Lax operators lying in some abelian (up to central term) subalgebra of Ĵ. Using the dressing transformation procedure we construct the solutions in the orbit of those vacuum solutions, and conjecture that the soliton solutions correspond to some special points in those orbits. The generalized tau-function for those hierarchies are defined for integrable highest weight representations of Ĵ, and it applies for any level of the representation and it is independent of its realization. We illustrate our methods with the recently proposed non abelian Toda models coupled to matter fields. A very special class of such theories possess a U (1) Noether charge that, under a suitable gauge fixing of the conformal symmetry, is proportional to a topological charge. That leads to a mechanism that confines the matter fields inside the solitons.

We construct the vertex operator representation for the Affine Kac-Moody ŝl (M+K+1) algebra, which is relevant for the construction of the soliton solutions of the constrained KP hierarchies. The oscillators involved in the vertex operator construction are provided by the Heisenberg subalgebras of ŝl(M+K+1) realized in the unconventional gradations. The well-known limiting cases are the homogeneous Heisenberg subalgebra of ŝl(M+1) and the principal Heisenberg subalgebra of ŝl(K+1). The explicit example of M=K=1 is discussed in detail and the corresponding soliton solutions and tau-functions are given.