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Darboux transformations for a Lax integrable system in \(2n\) dimensions. (English) Zbl 0869.58029

Summary: A \(2 n\)-dimensional Lax integrable system is proposed by a set of specific spectral problems. It contains Takasaki equations, the self-dual Yang-Mills equations and its integrable hierarchy as examples. An explicit formulation of Darboux transformations is established for this Lax integrable system. The Vandermonde and generalized Cauchy determinant formulas lead to a description for deriving explicit solutions and thus some rational and analytic solutions are obtained.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q58 Other completely integrable PDE (MSC2000)
58J70 Invariance and symmetry properties for PDEs on manifolds
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References:

[7] Takasaki, K.: Integrable systems in gauge theory, Kähler geometry and super KP hierarchy - symmetries and algebraic point of view, Proc. Internat. Congr. Math., Kyoto 1990, Vol II, pp. 1205-1214. · Zbl 0747.53025
[20] Gu C.H. Generalized self-dual Yang-Mills .ows, explicit solutions and reductions, Preprint 1995. · Zbl 0838.58016
[21] Ward, R. S.: Nontrivial scattering of localized solitons in a (2+ 1)-dimensional integrable system, solv-int/9510004.
[23] Zhou, Z. X.: Soliton solutions for some equations in 1 + 2-dimensional hyperbolic su(N) AKNS system, Preprint, 1995.
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