Self-dual Equations Research Articles

Remarks on Solutions of the Generalized Jackiw-Pi Model with Self-Dual Potential

We derive the self-dual equations of the generalized Jackiw-Pi model with the self-dual potential (1.7). We also show that the finite energy solution of the generalized Jackiw-Pi model with special self-dual potential (1.9) satisfies the self-dual system which can be transformed into the Liouville type equation.

Soliton interactions and their dynamics in a higher-order nonlinear self-dual network equation

Under investigation in this paper is a higher-order nonlinear self-dual network equation, which may simulate the wave propagation in a ladder type electric circuit. By means of the N-fold Darboux transformation and symbolic computation, the N-soliton solutions in determinant form are obtained. Based on the asymptotic and graphic analysis, the elastic interaction phenomena between/among two-, three- and four-soliton solutions are discussed, and some important physical quantities are accurately analyzed. Numerical simulations are used to explore the dynamical stability of one- and two-soliton solutions. Results might be helpful for understanding the propagation and interaction properties of electrical signals in a ladder type nonlinear self-dual network.

Higher solutions of Hitchin’s self-duality equations

AbstractSolutions of Hitchin’s self-duality equations correspond to special real sections of the Deligne–Hitchin moduli space—twistor lines. A question posed by Simpson in 1997 asks whether all real sections give rise to global solutions of the self-duality equations. An affirmative answer would in principle allow for complex analytic procedures to obtain all solutions of the self-duality equations. The purpose of this article is to construct counter examples given by certain (branched) Willmore surfaces in three-space (with monodromy) via the generalized Whitham flow. Though these sections do not give rise to global solutions of the self-duality equations on the whole Riemann surface M, they induce solutions on an open and dense subset of it. This suggest a connection between Willmore surfaces, i.e., rank 4 harmonic maps theory, with the rank 2 self-duality theory.

Fractional derivative of inverse matrix and its applications to soliton theory

In this paper, a formula of the local fractional partial derivative of inverse matrix is presented and proved. With the help of the derived formula, two new non-linear PDE are derived including the local fractional non-isospectral self-dual Yang-Mills equation and the local fractional principal chiral field equation. It is shown that the formula of the local fractional partial derivative of inverse matrix can be used to derive some other local fractional non-linear PDE in soliton theory.

The non-abelian self-dual string

We argue that the relevant higher gauge group for the non-abelian generalization of the self-dual string equation is the string 2-group. We then derive the corresponding equations of motion and discuss their properties. The underlying geometric picture is a string structure, i.e. a categorified principal bundle with connection whose structure 2-group is the string 2-group. We readily write down the explicit elementary solution to our equations, which is the categorified analogue of the 't Hooft-Polyakov monopole. Our solution passes all the relevant consistency checks; in particular, it is globally defined on $\mathbb{R}^4$ and approaches the abelian self-dual string of charge one at infinity. We note that our equations also arise as the BPS equations in a recently proposed six-dimensional superconformal field theory and we show that with our choice of higher gauge structure, the action of this theory can be reduced to four-dimensional supersymmetric Yang-Mills theory.

Rational solutions for three semi-discrete modified Korteweg–de Vries-type equations

In this paper, we consider three semi-discrete modified Korteweg–de Vries type equations which are the nonlinear lumped self-dual network equation, the semi-discrete lattice potential modified Korteweg–de Vries equation and a semi-discrete modified Korteweg–de Vries equation. We derive several kinds of exact solutions, in particular rational solutions, in terms of the Casorati determinant for these three equations, respectively. For some rational solutions, we present the related asymptotic analysis to understand their dynamics better.

Rational solutions for the potential nonlinear lumped self-dual network equation

In this letter we investigate the potential nonlinear lumped self-dual network equation, which is always used to model one-dimensional anharmonic chain of atoms. By the bilinear method we construct the rational solutions for the aforesaid equation. These solutions are presented in terms of Casoratian. The dynamical behaviors for the first three non-trivial rational solutions are analyzed with graphical illustration.

A New SDIE Based on CFIE for Electromagnetic Scattering From IBC Objects

This article presents a new self-dual integral equation (SDIE) for electromagnetic scattering from arbitrarily impedance boundary condition (IBC) objects including partly coated objects. The proposed SDIE is constructed by using the combined field integral equation (CFIE) and IBC, shorted as C-SDIE. To overcome the difficulty of discontinuous surface impedance from nonuniform IBC/partly coated objects, the discontinuous Galerkin (DG) method is applied to discretize the C-SDIE. Numerical experiments confirm that the DG-C-SDIE has promising numerical performance in terms of accuracy and efficiency. Furthermore, the domain decomposition preconditioning based on DG is employed to further enhance the proposed DG-C-SDIE for large-scale, multi-scale objects. The numerical results demonstrate the capability of the proposed DG-C-SDIE.

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

We have shown the existence of self-dual solitons in a type of generalized Chern-Simons baby Skyrme model where the generalized function (depending only in the Skyrme field) is coupled to the sigma-model term. The consistent implementation of the Bogomol'nyi-Prasad-Sommerfield (BPS) formalism requires the generalizing function becomes the superpotential defining properly the self-dual potential. Thus, we have obtained a topological energy lower-bound (Bogomol'nyi bound) and the self-dual equations satisfied by the fields saturating such a bound. The Bogomol'nyi bound being proportional to the topological charge of the Skyrme field is quantized whereas the total magnetic flux is not. Such as expected in a Chern-Simons model the total magnetic flux and the total electrical charge are proportional to each other. Thus, by considering the superpotential a well-behaved function in the whole target space we have shown the existence of three types of self-dual solutions: compacton solitons, soliton solutions whose tail decays following an exponential-law $e^{-\alpha r^{2}}$ ($\alpha>0$), and solitons having a power-law decay $r^{-\beta}$ ($\beta>0$). The profiles of the two last solitons can exhibit a compactonlike behavior. The self-dual equations have been solved numerically and we have depicted the soliton profiles, commenting on the main characteristics exhibited by them.

A Brief Survey of Higgs Bundles

Considering a compact Riemann surface of genus greater or equal than two, a Higgs bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin’s work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as Higgs fields because in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name Higgs bundle for a holomorphic bundle together with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror symmetry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief but complete construction of the moduli space of Higgs bundles.

From the conformal self-duality equations to the Manakov–Santini system

Under two separate symmetry assumptions, we demonstrate explicitly how the equations governing a general anti-self-dual conformal structure in four dimensions can be reduced to the Manakov–Santini system, which determines the three-dimensional Einstein–Weyl structure on the space of orbits of symmetry. The two symmetries investigated are a non-null translation and a homothety, which are previously known to reduce the second heavenly equation to the Laplace’s equation and the hyper-CR system, respectively. Reductions on the anti-self-dual null-Kähler condition are also explored in both cases.

A Discontinuous Galerkin Self-Dual Integral Equation Method for Scattering From IBC Objects

A discontinuous Galerkin self-dual integral equation (DG-SDIE) method is presented for accurately calculating electromagnetic scattering from objects with impedance boundary condition (IBC) surfaces. The working mechanisms of each term in the DG-SDIE formulation are studied physically and numerically. Based on the mechanisms, we propose a new efficient nonsymmetric DG-SDIE formulation (NDG-SDIE) for IBC problems. A comprehensive study is made to compare the NDG-SDIE with the previous symmetric DG-SDIE (SDG-SDIE) and antisymmetric DG-SDIE (ADG-SDIE) extension formulations for IBC. Numerical results demonstrate that NDG-SDIE is more efficient than SDG-SDIE and ADG-SDIE while generally maintaining the similar accuracy. In addition, we show that the DG methods are more attractive than the standard SDIE due to its flexible performance on large complex multiscale objects.

Infinite number of conservation laws and Darboux transformations for a 6-field integrable lattice system

In this paper, we study a 6-field integrable lattice system, which, in some special cases, can be reduced to the self-dual network equation, the discrete second-order nonlinear Schrödinger equation and the relativistic Volterra lattice equation. With the help of the Lax pair, we construct infinitely many conservation laws and a new Darboux transformation for system. Exact solutions resulting from the obtained Darboux transformation are presented by using a given seed solution. Further, we generate the soliton solutions and plot the figures of one-soliton solutions with properly parameters.

Existence of Radial Mixed Type Solutions in Chern–Simons Theories of Rank 2 in $$\mathbb {R}^2$$

We are concerned with Chern–Simons gauge theories of rank 2 in $$\mathbb {R}^2$$ such as SU(3), SO(5), and $$G_2$$ Chern–Simons theory. One can reduce the self-dual equations of these models to a $$2\times 2$$ system of elliptic equations. There exist three types of solutions of these elliptic system classified according to their asymptotic behaviors, namely, topological, nontopological, and mixed type solutions. In this paper, we consider the existence of radial mixed type solutions of these models. By counting the degree of the corresponding operator, we show the existence under a reasonable condition. Our result on the SU(3) Chern–Simons theory is an improvement of the existence result in Choe et al. (2017). Also, our results for the SO(5) and $$G_2$$ Chern–Simons theories are entirely new, to which the variational approach developed in Choe et al. (2017) cannot be applied.

Degenerate Solutions of the Nonlinear Self-Dual Network Equation**Supported by the Natural Science Foundation of Zhejiang Province under Grant No. LY15A010005, the Natural Science Foundation of Ningbo under Grant No. 2018A610197, the NSF of China under Grant No. 11671219, K.C. Wong Magna Fund in Ningbo University

The N-fold Darboux transformation (DT) of the nonlinear self-dual network equation is given in terms of the determinant representation. The elements in determinants are composed of the eigenvalues and the corresponding eigenfunctions of the associated Lax equation. Using this representation, the N-soliton solutions of the nonlinear self-dual network equation are given from the zero “seed” solution by the N-fold DT. A general form of the N-degenerate soliton is constructed from the determinants of N-soliton by a special limit and by using the higher-order Taylor expansion. For 2-degenerate and 3-degenerate solitons, approximate orbits are given analytically, which provide excellent fit of exact trajectories. These orbits have a time-dependent “phase shift”, namely .

Chern-Simons gauged sigma model into <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{H}^2 $\end{document}</tex-math></inline-formula> and its self-dual equations

We propose the Chern-Simons gauged sigma model from $\mathbb{R}^{1+2}$ into the hyperbolic plane $\mathbb{H}^2$. We seek a static configuration of this model and derive self-dual equations. We also establish some existence results for solutions of the self-dual equations under appropriate boundary conditions near $\infty$

Minitwistors and 3d Yang-Mills-Higgs theory

We construct a minitwistor action for Yang–Mills–Higgs (YMH) theory in three dimensions. The Feynman diagrams of this action will construct perturbation theory around solutions of the Bogomolny equations in much the same way that MHV (maximally helicity violating) diagrams describe perturbation theory around the self-dual Yang Mills equations in four dimensions. We also provide a new formula for all tree amplitudes in YMH theory (and its maximally supersymmetric extension) in terms of degree d maps to minitwistor space. We demonstrate its relationship to the Roiban-Spradlin-Volovich-Witten (RSVW) formula in four dimensions and show that it generates the correct MHV amplitudes at d = 1 and factorizes correctly in all channels for all degrees.

An angular dependent supersymmetric quantum mechanics with a Z2-invariant potential

We generalize the conformally invariant topological quantum mechanics of a particle propagating on a punctured plane by introducing a potential that breaks both the rotational and the conformal invariance down to a Z2 angular-dependent discrete symmetry. We derive a topological quantum mechanics whose localization gauge functions give interesting self-dual equations. The model contains an order parameter and exhibits a critical phenomenon with the existence of two ground states above a critical scale. Unlike the ordinary O(2)-invariant Higgs potential, an angular-dependence is found and saddle points, instead of local maxima, appear, posing subtle questions about the existence of instantons. The supersymmetric quantum mechanical model is constructed in both the path integral and the operatorial frameworks.

PEC 또는 PMC 영역을 포함하는 불균일 임피던스 매질의 산란 해석을 위한 수정된 SDIE MLFMM 방법

A coated perfect electric conductor(PEC) interface to reduce scattered fields can be efficiently modeled by using the impedance boundary condition. The self-dual integral equation(SDIE) proposed by Yan et al. may be an efficient multi-level fast multipole method (MLFMM) formulation for the impedance object. This equation can be applied to an inhomogeneous impedance material, but its accuracy can be degenerated when the material contains a PEC or perfect magnetic conductor(PMC) region. In this paper, we modify the original SDIE formulation for an inhomogeneous object containing a PEC or PMC region and numerically verify its accuracy.

On Non-Relativistic 3D Spin-1 Theories

We describe non-relativistic limits of the 3D Proca and square-root Proca theories that yield spin-1 Schroedinger equations. Analogous results are found by generalized null reduction of the 4D Maxwell or complex self-dual Maxwell equations. We briefly discuss the extension to spin-2.