Self-dual Yang-Mills Equations Research Articles

Lie symmetry analysis, optimal system and exact solutions for a NLPDE from the reduced quasi-classical self-dual Yang–Mills equation

Lie symmetry analysis, optimal system and exact solutions for a NLPDE from the reduced quasi-classical self-dual Yang–Mills equation

On Recursion Operators for Full-Fledged Nonlocal Symmetries of the Reduced Quasi-classical Self-dual Yang–Mills Equation

AbstractWe introduce the idea of constructing recursion operators for full-fledged nonlocal symmetries and apply it to the reduced quasi-classical self-dual Yang–Mills equation. It turns out that the discovered recursion operators can be interpreted as infinite-dimensional matrices of differential functions which act on the generating vector functions of the nonlocal symmetries simply by matrix multiplication. To the best of our knowledge, there are no other examples of such recursion operators in the literature so far, so our approach is completely innovative. Further, we investigate the algebraic properties of the discovered operators and discuss the $${\mathbb {R}}$$ R -algebra structure on the set of all recursion operators for full-fledged nonlocal symmetries of the equation in question. Finally, we illustrate the action of the obtained recursion operators on particularly chosen full-fledged symmetries and emphasize their advantages compared to the action of traditionally used recursion operators for shadows.

Solutions to the SU([formula omitted]) self-dual Yang–Mills equation

In this paper we aim to derive solutions for the SU(N) self-dual Yang–Mills (SDYM) equation with arbitrary N. A set of noncommutative relations are introduced to construct a matrix equation that can be reduced to the SDYM equation. It is shown that these relations can be generated from two different Sylvester equations, which correspond to the two Cauchy matrix schemes for the (matrix) Kadomtsev–Petviashvili hierarchy and the (matrix) Ablowitz–Kaup–Newell–Segur hierarchy, respectively. In each Cauchy matrix scheme we investigate the possible reductions that can lead to the SU(N) SDYM equation and also analyze the physical significance of some solutions, i.e. being Hermitian, positive-definite and of determinant being one.

Cauchy matrix approach to the SU(2) self‐dual Yang–Mills equation

AbstractThe Cauchy matrix approach is developed to solve the self‐dual Yang–Mills (SDYM) equation. Starting from a Sylvester matrix equation coupled with certain dispersion relations for an infinite number of coordinates, we derive some new relations that give rise to the SDYM equation under Yang's formulation. By imposing further constraints on complex independent variables, a broad class of explicit solutions of the equation under Yang's formulation are obtained.

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.

Exact multi-instanton solutions to self-dual Yang–Mills equation on curved spaces

We find exact multi-instanton solutions to the self-dual Yang–Mills equation on a large class of curved spaces with [Formula: see text] isometry, generalizing the results previously found on [Formula: see text]. The solutions are featured with explicit multi-centered expressions and topological properties. As examples, we demonstrate the approach on several different curved spaces, including the Einstein static universe and [Formula: see text], and show that the exact multi-instanton solutions exist on these curved backgrounds.

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.

Isospectral deformation of the reduced quasi-classical self-dual Yang–Mills equation

We derive a new four-dimensional partial differential equation with the isospectral Lax representation by shrinking the symmetry algebra of the reduced quasi-classical self-dual Yang–Mills equation and applying the technique of twisted extensions to the obtained Lie algebra. Then we find a recursion operator for symmetries of the new equation and construct a Bäcklund transformation between this equation and the four-dimensional Martínez Alonso–Shabat equation. Finally, we construct extensions of the integrable hierarchies associated to the hyper-CR equation for Einstein–Weyl structures, the reduced quasi-classical self-dual Yang–Mills equation, the four-dimensional universal hierarchy equation, and the four-dimensional Martínez Alonso–Shabat equation.

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.

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.

Theory of Multidimensional Delsarte–Lions Transmutation Operators. II

The differential-geometric and topological structures related to the Delsarte transmutation operators and the Gelfand–Levitan–Marchenko equations that describe these operators are studied by using suitable differential de Rham–Hodge–Skrypnik complexes. The correspondence between the spectral theory and special Berezansky-type congruence properties of the Delsarte transmutation operators is established. Some applications to multidimensional differential operators are presented, including the three-dimensional Laplace operator, the two-dimensional classical Dirac operator, and its multidimensional affine extension associated with self-dual Yang–Mills equations. The soliton solutions are discussed for a certain class of dynamical systems.

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.

5a-A-10 2次元Derivative Nonlinear Schrodinger方程式と自己双対Yang-Mills方程式

5a-A-10 2次元Derivative Nonlinear Schrodinger方程式と自己双対Yang-Mills方程式

Chiral higher spin theories and self-duality

We study recently proposed chiral higher spin theories — cubic theories of interacting massless higher spin fields in four-dimensional flat space. We show that they are naturally associated with gauge algebras, which manifest themselves in several related ways. Firstly, the chiral higher spin equations of motion can be reformulated as the self-dual Yang-Mills equations with the associated gauge algebras instead of the usual colour gauge algebra. We also demonstrate that the chiral higher spin field equations, similarly to the self-dual Yang-Mills equations, feature an infinite algebra of hidden symmetries, which ensures their integrability. Secondly, we show that off-shell amplitudes in chiral higher spin theories satisfy the generalised BCJ relations with the usual colour structure constants replaced by the structure constants of higher spin gauge algebras. We also propose generalised double copy procedures featuring higher spin theory amplitudes. Finally, using the light-cone deformation procedure we prove that the structure of the Lagrangian that leads to all these properties is universal and follows from Lorentz invariance.

Linearly degenerate hierarchies of quasiclassical SDYM type

We demonstrate that SDYM (self-dual Yang-Mills) equations for the Lie algebra of one-dimensional vector fields represent a natural reduction in the framework of a general linearly degenerate dispersionless hierarchy. We define the reduction in terms of wave functions and introduce a generating relation, Lax-Sato equations, and the dressing scheme for the reduced hierarchy. A multidimensional case is also discussed.

Wilson lines in the MHV action

The MHV action is the Yang-Mills action quantized on the light-front, where the two explicit physical gluonic degrees of freedom have been canonically transformed to a new set of fields. This transformation leads to the action with vertices being off-shell continuations of the MHV amplitudes. We show that the solution to the field transformation expressing one of the new fields in terms of the Yang-Mills field is a certain type of the Wilson line. More precisely, it is a straight infinite gauge link with a slope extending to the light-cone minus and the transverse direction. One of the consequences of that fact is that certain MHV vertices reduced partially on-shell are gauge invariant — a fact discovered before using conventional light-front perturbation theory. We also analyze the diagrammatic content of the field transformations leading to the MHV action. We found that the diagrams for the solution to the transformation (given by the Wilson line) and its inverse differ only by light-front energy denominators. Further, we investigate the coordinate space version of the inverse solution to the one given by the Wilson line. We find an explicit expression given by a power series in fields. We also give a geometric interpretation to it by means of a specially defined vector field. Finally, we discuss the fact that the Wilson line solution to the transformation is directly related to the all-like helicity gluon wave function, while the inverse functional is a generating functional for solutions of self-dual Yang-Mills equations.

SDYM equations on the self-dual background

We introduce the technique combining the features of integration schemes for self-dual Yang–Mills (SDYM) equations and multidimensional dispersionless integrable equations to get SDYM equations on the conformally self-dual background. Generating differential form is defined, and the dressing scheme is developed. Some special cases and reductions are considered.

Three New (2+1)-dimensional Integrable Systems and Some Related Darboux Transformations**Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Shandong Provincial Natural Science Foundation of China under Grant Nos. ZR2012AQ011, ZR2013AL016, ZR2015EM042, National Social Science Foundation of China under Grant No. 13BJY026, the Development of Science and Technology Project under Grant No. 2015NS1048 and A Project

We introduce two operator commutators by using different-degree loop algebras of the Lie algebra A1, then under the framework of zero curvature equations we generate two (2+1)-dimensional integrable hierarchies, including the (2+1)-dimensional shallow water wave (SWW) hierarchy and the (2+1)-dimensional Kaup-Newell (KN) hierarchy. Through reduction of the (2+1)-dimensional hierarchies, we get a (2+1)-dimensional SWW equation and a (2+1)-dimensional KN equation. Furthermore, we obtain two Darboux transformations of the (2+1)-dimensional SWW equation. Similarly, the Darboux transformations of the (2+1)-dimensional KN equation could be deduced. Finally, with the help of the spatial spectral matrix of SWW hierarchy, we generate a (2+1) heat equation and a (2+1) nonlinear generalized SWW system containing inverse operators with respect to the variables x and y by using a reduction spectral problem from the self-dual Yang-Mills equations.

A New Reduction of the Self-Dual Yang–Mills Equations and its Applications

Abstract Through imposing on space–time symmetries, a new reduction of the self-dual Yang–Mills equations is obtained for which a Lax pair is established. By a proper exponent transformation, we transform the Lax pair to get a new Lax pair whose compatibility condition gives rise to a set of partial differential equations (PDEs). We solve such PDEs by taking different Lax matrices; we develop a new modified Burgers equation, a generalised type of Kadomtsev–Petviasgvili equation, and the Davey–Stewartson equation, which also generalise some results given by Ablowitz, Chakravarty, Kent, and Newman.

Bianchi-IX, Darboux–Halphen and Chazy–Ramanujan

Bianchi-IX four metrics are SU(2) invariant solutions of vacuum Einstein equation, for which the connection-wise self-dual case describes the Euler top, while the curvature-wise self-dual case yields the Ricci flat classical Darboux–Halphen system. It is possible to see such a solution exhibiting Ricci flow. The classical Darboux–Halphen system is a special case of the generalized one that arises from a reduction of the self-dual Yang–Mills equation and the solutions to the related homogeneous quadratic differential equations provide the desired metric. A few integrable and near-integrable dynamical systems related to the Darboux–Halphen system and occurring in the study of Bianchi-IX gravitational instanton have been listed as well. We explore in details whether self-duality implies integrability.