Abstract
The self-dual Yang-Mills equations play a central role in the study of integrable systems. In this paper we develop a formalism for deriving a four dimensional integrable hierarchy of commuting nonlinear flows containing the self-dual Yang-Mills flow as the first member. We show that upon appropriate reduction and suitable choice of gauge group it produces virtually all well known hierarchies of soliton equations in 1+1 and 2+1 dimensions and can be considered as a “universal” integrable hierarchy. Prototypical examples of reductions to classical soliton equations are presented and related issues such as recursion operators, symmetries, and conservation laws are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chodos, A., Hadjimichael, E., Tze, C. (eds.), Solitions in Nuclear and Elementary Particle Physics. Proc. of the Lewes Workshop. Singapore: World Scientific 1984
Ashtekar, A.: Radiative degrees of freedom of the gravitational field in exact general relativity. J. Math. Phys.22, 2885–2895 (1981)
Freed, David, S. Uhlenbeck, Karen, K.: Instantons and 4-manifolds. New York: Springer 1984
Chau, Ling Lie: Integrability properties of supersymmetric Yang-Mills fields and relations with other nonlinear systems. Proc. of the 13th Int. Coll. on Group Theoretical Methods in Physics, Zachary, W. W. (ed.), Singapore: World Scientific 1984
Beals, R., Coifman, R. R.: The D-Bar approach to Inverse Scattering and Nonlinear Evolutions. Physica18D, 242–249 (1986)
Bruschi, M., Levi, D., Ragnisco, O.: Nonlinear Partial Differential Equation and Bäcklund Transformations Related to the 4-Dimensional Self-Dual Yang-Mills Equation. Lett. Nuovo Cimento33, 263–266 (1982)
Ward, R. S.: Integrable and solvable systems and relations among them. Phil. Trans. Roy. Soc. Lond.A315, 451–457 (1985)
Atiyah, M. F.: Classical Geometry of Yang-Mills Fields. Fermi Lectures, Scuola Normale Pisa (1980)
Belavin, A. A., Zakharov, V. E.: Yang-Mills equations as inverse scattering problem. Phys. Lett.73B, 53–57 (1978)
Atiyah, M. F., Ward, R. S.: Instantons and algebraic geometry. Commun. Math. Phys.55, 117–124 (1977)
Schur, I.: Über vertauschbare lineare Differentialausdrücke. Gesammelte AbhandlungenI, 170–176, Springer 1905
Mulase, M.: KP equations, strings and Schottky problem. In: Algebraic Analysis, Kashiwara, M., Kawai, T. (eds.),II, 473–492, New York: Academic Press 1988
Sato, M., Sato, Y.: Soliton equations as dynamical systems on infinite dimensional Grassmann manifold. Lect. Notes in Num. Appl. Anal.5 259–271 (1982)
Faddeev, L., Takhtajan, L.: Hamiltonian Methods in the Theory of Solitons. Berlin, Heidelberg, New York: Springer 1987
Mason, J., Sparling, G.: Twistor correspondence for the soliton hierarchies. J. Geom. and Phys.8, 243–271 (1992)
Prasad, M. K., Sinha, A., Chau-Wang, L. L.: Non-local continuity equation for self-dualSU(N) Yang-Mills fields. Phys. Lett.87B, 237–238 (1979)
Mason, J., Sparling, G.: Nonlinear Schrödinger and Korteweg-de Vries equations are reductions of self-dual Yang-Mills equations. Phys. Lett.137A 29–33 (1989)
Zakharov, V. E., Manakov, S. V.: The theory of resonant interaction of wave packets in nonlinear media. Sov. Phys. JETP Lett.18, 243–245 (1973)
Zakharov, V. E., NOvikov, S. P., Manakov, S. V. Pitaevsky, L. P.: Theory of Solitons: The Inverse Scattering Method. New York: Consultant Bureau 1984
Ablowitz, M. J., Segur, H.: Solitons and Inverse Scattering Transform. Philadelphia: SIAM 1981
Chakravarty, S., Ablowitz, M. J.: On reduction of self-dual Yang-Mills equations. In: Painleve Transcendents, Their Asymptotics and Physical Applications, Levi, D., Winternitz, P. (eds.) London, New York: Plenum 1992
McIntosh, I.: Soliton equations and connections with self-dual curvature. Preprint (1991)
Zakharov, V. E., Shabat, A. B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media. Sov. Phys. JETP34, 62–69 (1972)
Gelfand, I. M., Dikii, L. A.: Asymptotic behaviour of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equations. Russ. Math. Surv.30:5, 77–113 (1975)
Adler, M.: On a trace functional for formal pseudodifferential operators and symplectic structure of the Korteweg-de Vries type equations. Invent. Math.50, 219–248 (1979)
Hermite, Ch.: Sur l'équation de Lamé. — Cours d'analyse de l'École Polytechn. Paris, 1872–1873, 32-e leçon; Oeuvres,III, 118–122, Paris 1912
Gardner, C. S., Greene, J. M., Kruskal, M. D., Miura, R. H.: Korteweg-de Vries equation and generalizations. VI. Methods for exact solution. Commun. Pure Appl. Math.27, 97–133 (1974)
Ablowitz, M. J., Kaup, D. J., Newell, A. C., Segur, H.: The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math.53 249–315 (1974)
Ablowitz, M. J., Haberman, R.: Nonlinear evolution equations — two and three dimensions. Phys. Rev. Lett.35, 1185–1188 (1975)
Zakharov, V. E., Shabat, A. B.: Integration of nonlinear equations of mathematical physics by the method of inverse scattering, II. Func. Anal. Appl.13, 166–176 (1979)
Ablowitz, M. J., Clarkson, P. A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Notes Series,149, Cambridge: Cambridge University Press 1992
Fokas, A. S., Gel'fand, I. M.: Bi-Hamiltonian Structures and Integrability. Preprint, INS 188 (1992)
Konopelchenko, B. G.: Nonlinear Integrable Equations: Recursion Operators, Group-Theoretical and Hamiltonian Structures of Soliton Equations. Lect. Notes in Physics,270, Berlin, Heidelberg, New York: Springer 1987
Fokas, A. S., Santini, P. M.: Recursion operators and bi-Hamiltonian structures in multidimensions, II. Commun. Math. Phys.116, 449–474 (1988)
Author information
Authors and Affiliations
Additional information
Communicated by N. Yu. Reshetikhin
Rights and permissions
About this article
Cite this article
Ablowitz, M.J., Chakravarty, S. & Takhtajan, L.A. A self-dual Yang-Mills hierarchy and its reductions to integrable systems in 1+1 and 2+1 dimensions. Commun.Math. Phys. 158, 289–314 (1993). https://doi.org/10.1007/BF02108076
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02108076