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  • 1
    Electronic Resource
    Electronic Resource
    High symmetry solutions of the anti-self-dual Yang–Mills equations (1990)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 31 (1990), S. 2253-2257 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Beginning with the anti-self-dual Yang–Mills (ASDYM) equations for an arbitrary Lie algebra on Minkowski space, this paper specializes to the case in which the vector potentials are independent of all the space-time coordinates, i.e., are space-time constants. The resulting equations are three algebraic equations on the algebra. These equations are then simplified by using a null basis. Two of the equations can be immediately solved while the third remains, in general, quite difficult to deal with. Two general cases are considered: finite-dimensional Lie groups and the infinite-dimensional diffeomorphism groups on finite-dimensional manifolds. In a few of the special cases, e.g., SL(2,C) and the Virasoro algebra, the solutions can easily be found. The study of the the diffeomorphism groups leads unexpectedly to the Monge–Ampère equation. In particular, the four-dimensional volume preserving diffeomorphism group is identical with the vacuum anti-self-dual Einstein equations. In conclusion, the question of the associated Lax pair equations and its relation to the Riemann–Hilbert splitting problem on S2 is examined.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528633
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  • 2
    Electronic Resource
    Electronic Resource
    Green's functions of the edh operators (1989)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 30 (1989), S. 45-52 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The spin-weight s Green's functions for the operator (edh) and powers of (edh) are obtained. The extension of these Green's functions to negative values of s and to the (edh)¯n operators, as well as a procedure for obtaining the Green's function for any combination of products of (edh)n and (edh)¯n, is also given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528587
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  • 3
    Electronic Resource
    Electronic Resource
    On the construction of Hamiltonians (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 2430-2433 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: It is shown that, for the system of first-order ordinary differential equations of the form dxa/dt =f a(xb), it is always possible to construct a locally defined nondegenerate symplectic structure Ωab(x), and write the equations as the canonical equations for some Hamiltonian function H(x) such that f aΩab=∇bH. Furthermore, a Lagrangian L(x,x(overdot)), which is linear in the velocities dxa/dt, can be found, whose variation yields these same canonical equations. Finally we discuss, via the Dirac brackets, how the standard transition from a Lagrangian system to a Hamiltonian system works in this pathological case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528129
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  • 4
    Electronic Resource
    Electronic Resource
    Some reductions of the self-dual Yang–Mills equations to integrable systems in 2+1 dimensions (1995)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 36 (1995), S. 763-772 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A reduction of the self-dual Yang–Mills (SDYM) equations is studied by imposing two space–time symmetries and by requiring that the connection one-form belongs to a Lie algebra of formal matrix-valued differential operators in an auxiliary variable. In this article, the scalar case and the canonical cases for 2×2 matrices are examined. In the scalar case, it is shown that the field equations can be reduced to the forced Burgers equation. In the matrix case, several well-known 2+1 integrable equations are obtained. Also examined are certain transformation properties between the solutions of some of these 2+1 equations. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531155
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  • 5
    Electronic Resource
    Electronic Resource
    Special solutions of the Sparling equation (1986)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 27 (1986), S. 2043-2046 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The Sparling equation, a first-order, matrix-valued linear differential equation that is equivalent to the self-dual Yang–Mills equations for any group, has recently been solved by quadratures for the case of SL(2,C) or its subgroup. It is the purpose of this paper to show how for a series of special cases, rather than integrating the quadratures, the Sparling equation can be reduced to an algebraic equation and then solved, yielding the single- and multi-instanton fields parallel to isospace.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527024
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  • 6
    Electronic Resource
    Electronic Resource
    Gauge theories, the holonomy operator, and the Riemann–Hilbert problem (1986)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 27 (1986), S. 2797-2802 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The so-called Riemann–Hilbert problem has arisen and plays a major role in the study of many nonlinear integrable systems, such as the sine–Gordon equation, stationary axial symmetric Einstein equations, etc. Here it is shown how the Riemann–Hilbert problem arises naturally in the study of self-dual Yang–Mills fields in Minkowski space via a simple geometric construction of the holonomy operator on anti-self-dual planes. This Riemann–Hilbert problem is then converted to a linear homogeneous differential equation that is considerably simpler to study than the original problem. Finally it is shown that the nonlinear equation of Yang for self-dual fields is easily understood from the holonomy point of view.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527254
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  • 7
    Electronic Resource
    Electronic Resource
    Maxwell's equations and the bundle of null directions on Minkowski space (1985)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 26 (1985), S. 300-305 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We reexpress the Maxwell field as a cross section of a line bundle over M×S2, the six-dimensional space of null directions on Minkowski space. Maxwell's equations then become a pair of linear equations for a Herz-like scalar on M×S2. We obtain a deeper understanding of the simple, yet nontrivial relationship between the self-dual and the anti-self-dual parts of a real Maxwell field. Our results are then applied to study solutions which are globally regular (on M×S2) namely, the pure radiation solutions, as well as solutions associated with discrete sources (the Lienard–Wiechert fields).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.526658
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  • 8
    Electronic Resource
    Electronic Resource
    Bäcklund transformations for the anti-self-dual Yang–Mills equations (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 1005-1013 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Beginning from any given (local) solution of the GL(n,C) anti-self-dual Yang–Mills (ASDYM) equations on Minkowski space, a simple technique for the generation of large classes of solutions (perhaps in some sense all) is given. The origin of this technique is described in terms of two versions of the Ward construction. The resulting description of Bäcklund transformations is sufficiently simple that it is then possible to identify the group generated by the collection of all such Bäcklund transformations and the space on which it acts in terms of concrete functions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528014
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  • 9
    Electronic Resource
    Electronic Resource
    Yang–Mills equations and solvable groups. II (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 1022-1025 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: It is shown that the necessary and sufficient condition for the Yang–Mills equations (associated with an arbitrary group G) to be derivable from a Lagrangian (which is polynonial in the derivatives of the connection) is that the Lie algebra g of G possesses an invariant nondegenerate quadratic form γ. It is well known that for semisimple groups such a γ exists, namely the Killing form. What is not so well known is that such a γ exists for many other groups and in particular for many solvable and nilpotent groups. Several examples in this class are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528016
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  • 10
    Electronic Resource
    Electronic Resource
    Quadratures for self-dual GL(2,C) Yang–Mills fields (1987)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 28 (1987), S. 334-338 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: It is the purpose of this paper to show that the GL(2,C) Yang–Mills equations can be solved in terms of integrals over the characteristic initial data. The method is based on showing that enough gauge freedom exists in the choice of characteristic initial data so that the data can always be put into either upper or lower triangular form. With triangular form data the Sparling equation (a linear first-order equation equivalent to the self-dual Yang–Mills equations) can be solved by explicit quadratures.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527662
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