Self-dual Equations Research Articles

Generalized self-duality for the Yang-Mills-Higgs system

Self-duality is a very important concept in the study and applications of topological solitons in many areas of Physics. The rich mathematical structures underlying it lead, in many cases, to the development of exact and non-perturbative methods. We present a generalization of the Yang-Mills-Higgs system by the introduction of scalar fields assembled in a symmetric and invertible matrix h of the same dimension as the gauge group. The coupling of such new fields to the gauge and Higgs fields is made by replacing the Killing form, in the contraction of the group indices, by the matrix h in the kinetic term for the gauge fields, and by its inverse in the Higgs field kinetic term. The theory is conformally invariant in the three dimensional space R^3. An important aspect of the model is that for practically all configurations of the gauge and Higgs fields the new scalar fields adjust themselves to solve the modified self-duality equations. We construct solutions using a spherically symmetric ans\"atz and show that the 't Hooft-Polyakov monopole becomes a self-dual solution of such modified Yang-Mills-Higgs system. We use an ans\"atz based on the conformal symmetry to construct vacuum solutions presenting non-trivial toroidal magnetic fields.

Solitons and exotic instantons

In this paper, we study the 't Hooft type instantons in eight dimensions, which satisfy the (anti)self-dual equations $F\wedge F=\pm\ast_8F\wedge F$. Using various designs of such instantons, we find new soliton solutions to the low-energy effective theory of the heterotic fivebrane. We investigate conditions under which these instanton configurations can be identified with the $D$-instantons embedded in the $D7$-brane world volume. Finally, we discuss the relationship between eight-dimensional periodic instantons and monopoles.

Exotic Instantons in Eight Dimensions

In this paper, we study the (anti-)self-duality equations $\ast F\wedge F=\pm F\wedge F$ in the eight-dimensional Euclidean space. Using properties of the Clifford algebra $Cl_{0,8}(\mathbb{R})$, we find a new solution to these equations.

Reduction of the Self-dual Yang-Mills Equations to Sinh-Poisson Equation and Exact Solutions

The geometric properties of differential systems are used to demonstrate how the sinh-poisson equation describes a surface with a constant negative curvature in this paper. The canonical reduction of 4-dimensional self dual Yang Mills theorem is the sinh-poisson equation, which explains pseudo spherical surfaces. We derive the B¨acklund transformations and the travelling wave solution for the sinh-poisson equation in specific. As a result, we discover exact solutions to the self-dual Yang-Mills equations.

Uniformization of branched surfaces and Higgs bundles

Given a compact connected Riemann surface [Formula: see text] of genus [Formula: see text], and an effective divisor [Formula: see text] on [Formula: see text] with [Formula: see text], there is a unique cone metric on [Formula: see text] of constant negative curvature [Formula: see text] such that the cone angle at each point [Formula: see text] is [Formula: see text] [R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988) 222–224; M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821]. We describe the Higgs bundle on [Formula: see text] corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on [Formula: see text] parametrized by a nonempty open subset of [Formula: see text] that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin’s results in [N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59–126] for the case [Formula: see text].

Self-dual solitons in a Born-Infeld baby Skyrme model

We show the existence of self-dual (topological) solitons in a gauged version of the baby Skyrme model in which the Born-Infeld term governs the gauge field dynamics. The successful implementation of the Bogomol'nyi-Prasad-Sommerfield formalism provides a lower bound for the energy and the respective self-dual equations whose solutions are the solitons saturating such a limit. The energy lower bound (Bogomol'nyi bound) is proportional to the topological charge of the Skyrme field and therefore quantized. In contrast, the total magnetic flux is a nonquantized quantity. Furthermore, the model supports three types of self-dual solitons profiles: the first describes compacton solitons, the second follows a Gaussian decay law, and the third portrays a power-law decay. Finally, we perform numerical solutions of the self-dual equations and depicted the soliton profiles for different values of the parameters controlling the nonlinearity of the model.

Discrete nonlocal N-fold Darboux transformation and soliton solutions in a reverse space-time nonlocal nonlinear self-dual network equation

In this paper, we construct a discrete nonlocal integrable lattice hierarchy related to a reverse space-time nonlocal nonlinear self-dual network equation which may have the potential applications in designing nonlocal electrical circuits and understanding the propagation of electrical signals. By means of nonlocal version of [Formula: see text]-fold Darboux transformation (DT) technique, discrete multi-soliton solutions in determinant form are constructed for the reverse space-time nonlocal nonlinear self-dual network equation. Through the asymptotic and graphic analysis, unstable soliton structures of one-, two- and three-soliton solutions are discussed graphically. We observe that the single components in this nonlocal equation display instability while the combined potential terms with nonlocal [Formula: see text]-symmetry show stable soliton structures. It is shown that these nonlocal solutions are clearly different from those of its corresponding local equation. The results given in this paper may explain the soliton propagation in electrical signals.

Topological solutions of the self-dual equations for the generalized Maxwell-Chern-Simons-Higgs model

In this paper, we consider the self-dual equations arising from the generalized Maxwell-Chern-Simons-Higgs model. This model is one of the k-field models that have nonstandard kinetic terms, and is known to be useful in the study of cosmology. By using a variational argument, we show that there exists a topological solution for any coupling constants.

Discrete multi-soliton solutions and dynamics for a reverse-time nonlocal nonlinear self-dual network equation

Inspired by the idea of Ablowitz and Musslimani, a reverse-time nonlocal nonlinear self-dual network equation is proposed under certain nonlocal symmetry reduction. N-fold Darboux transformation technique is used to construct discrete unstable and stable multi-soliton solutions in zero seed background for this system. Based on the asymptotic and graphic analysis, structures of one-, two- and three-soliton solutions are shown graphically. It is found that the individual components in this nonlocal equation show unstable temporal soliton structures whereas the combined potential terms exhibit stable soliton structures. Dynamical behaviors for one-soliton solutions are investigated via numerical simulations. It is worth noting that the solutions of this nonlocal equation are clearly different from those of its corresponding local part. Results obtained in this paper may be helpful for the explanation of the propagation of electrical signals.

E 9 exceptional field theory. Part II. The complete dynamics

We construct the first complete exceptional field theory that is based on an infinite-dimensional symmetry group. E9 exceptional field theory provides a unified description of eleven-dimensional and type IIB supergravity covariant under the affine Kac-Moody symmetry of two-dimensional maximal supergravity. We present two equivalent formulations of the dynamics, which both rely on a pseudo-Lagrangian supplemented by a twisted self-duality equation. One formulation involves a minimal set of fields and gauge symmetries, which uniquely determine the entire dynamics. The other formulation extends {mathfrak{e}}_9 by half of the Virasoro algebra and makes direct contact with the integrable structure of two-dimensional supergravity. Our results apply directly to other affine Kac-Moody groups, such as the Geroch group of general relativity.

On the symbolic computation of exact solutions and conservation laws of a generalized (2+1)-dimensional Calogaro-Bogoyavlenskii-Schiff equation

New exact solutions of the (2+1)-dimensional Calogaro-Bogoyavlenskii-Schiff equation are presented. This equation was first erected by Bogoyavlenskii and Schiff in unique ways, namely Bogoyavlenskii used the modified Lax formalism, whereas Schiff employed the same equation by reducing the self-dual Yang Mills equation. We also derive conserved vectors of the above mentioned equation by invoking the multiplier method.

Modulational instability and dynamics of discrete rational soliton and mixed interaction solutions for a higher-order nonlinear self-dual network equation

In this paper, the higher-order nonlinear self-dual network equation is investigated. Firstly, an integrable lattice hierarchy associated with this equation is constructed from a discrete matrix spectral problem without the denominator. Secondly, the condition of modulational instability for this equation is given. Thirdly, the infinitely many conservation laws are constructed on the basis of its new Lax representation. Finally, the discrete generalised $$(m, N-m)$$ -fold Darboux transformation (DT) with non-zero constant as seed solution is used to derive new rational soliton (RS) and mixed interaction solutions. As an application, RS solutions can be obtained by the discrete generalised $$(1, N-1)$$ -fold DT (i.e., generalised $$(m, N-m)$$ -fold DT with $$m=1$$ ), and mixed interaction solutions of the usual sech-type soliton (US) and RS can be derived by the discrete generalised $$(2, N-2)$$ -fold DT (i.e., generalised $$(m, N-m)$$ -fold DT with $$m=2$$ ). We also perform the numerical simulations for such soliton solutions to explore their dynamical behaviours. Results given in this paper may have some prospective applications for understanding the propagation of electrical signals.

Bubbling solutions for the gravitational Maxwell gauged [formula omitted] model in [formula omitted

In this paper, we construct type II nontopological bubbling solutions of the self-dual equation for the Maxwell gauged O(3) sigma model coupled with gravity in R2. Our solutions blow up at some of the string points or the anti-string points as the coupling parameter tends to zero and they are asymptotically radial near each blow-up point.

Solitons of Some Nonlinear Sigma-Like Models

We present a set of differential identities for some class of matrices. These identities are used to derive the N-soliton solutions for the Pohlmeyer nonlinear sigma-model, two-dimensional self-dual Yang-Mills equations and some modification of the vector Calapso equation.

Infinite-Parameter ADHM Transform

Abstract The Atiyah–Drinfeld–Hitchin–Manin transform and its various generalizations are examples of nonlinear integral transforms between finite-dimensional moduli spaces. This note describes a natural infinite-dimensional generalization, where the transform becomes a map from boundary data to a family of solutions of the self-duality equations in a domain.

Soliton interactions and asymptotic state analysis in a discrete nonlocal nonlinear self-dual network equation of reverse-space type* *Project supported by the National Natural Science Foundation of China (Grant Nos. 12071042 and 61471406), the Beijing Natural Science Foundation, China (Grant No. 1202006), and Qin Xin Talents Cultivation Program of Beijing Information Science and Technology University (QXTCP-B201704).

We propose a reverse-space nonlocal nonlinear self-dual network equation under special symmetry reduction, which may have potential applications in electric circuits. Nonlocal infinitely many conservation laws are constructed based on its Lax pair. Nonlocal discrete generalized (m, N – m)-fold Darboux transformation is extended and applied to solve this system. As an application of the method, we obtain multi-soliton solutions in zero seed background via the nonlocal discrete N-fold Darboux transformation and rational solutions from nonzero-seed background via the nonlocal discrete generalized (1, N – 1)-fold Darboux transformation, respectively. By using the asymptotic and graphic analysis, structures of one-, two-, three- and four-soliton solutions are shown and discussed graphically. We find that single component field in this nonlocal system displays unstable soliton structure whereas the combined potential terms exhibit stable soliton structures. It is shown that the soliton structures are quite different between discrete local and nonlocal systems. Results given in this paper may be helpful for understanding the electrical signals propagation.

BPS solitons with internal structure in the gauged O(3) sigma model

We investigate the existence of self-dual solitons with internal structure in a gauged $O(3)$ nonlinear sigma model immersed in a dielectric medium generated by a real scalar field (dubbed the source field). We consider rotationally symmetric configurations and applying the {Bogomol'nyi-Prasad-Sommerfield} formalism to obtain the energy lower bound and the respective {first-order differential equations (or self-dual equations).} By solving such a system of equations for three different dielectric media, we find the internal structure generates relevant changes in the soliton profiles when compared with the ones obtained without the presence of the dielectric medium.

Nonlinear self-dual network equations: modulation instability, interactions of higher-order discrete vector rational solitons and dynamical behaviours.

The nonlinear self-dual network equations that describe the propagations of electrical signals in nonlinear LC self-dual circuits are explored. We firstly analyse the modulation instability of the constant amplitude waves. Secondly, a novel generalized perturbation (M, N - M)-fold Darboux transform (DT) is proposed for the lattice system by means of the Taylor expansion and a parameter limit procedure. Thirdly, the obtained perturbation (1, N - 1)-fold DT is used to find its new higher-order rational solitons (RSs) in terms of determinants. These higher-order RSs differ from those known results in terms of hyperbolic functions. The abundant wave structures of the first-, second-, third- and fourth-order RSs are exhibited in detail. Their dynamical behaviours and stabilities are numerically simulated. These results may be useful for understanding the wave propagations of electrical signals.

Self-duality in the context of the Skyrme model

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 ℝ3. 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.

Self-dual solitons in a Maxwell-Chern-Simons baby Skyrme model

We have studied the existence de self-dual solitons in a gauged version of the baby Skyrme model in which the gauge field dynamics is governed by the Maxwell-Chern-Simons action. For such a purpose, we have developed a detailed implementation of the Bogomol'nyi-Prasad-Sommerfield formalism providing the self-dual equations whose solutions saturate the energy lower bound. Such a bound related to the topological charge of the Skyrme field becomes quantized whereas both the total magnetic flux and the total electrical charge are not. We have found two types of self-dual Skyrme field profiles: the first is described by a solution which decays following an exponential-law ($e^{-\alpha r^2}$, $\alpha>0$); the second is portrayed by a solution having a power-law decay ($r^{-\beta}$, $\beta>0$). On other hand, in both cases the asymptotic behavior of the gauge field is similar to the one presented in the context of the Abelian Higgs models describing Abrikosov-Nielsen-Olesen charged vortices. Other interesting feature we highlight is the localized magnetic flux inversion, a property does not observed in others gauged baby Skyrme models already studied in literature. Numerical results are presented for rotationally symmetrical field configurations by remarking some of its essential features.