Deformation of surfaces, integrable systems and Self-Dual Yang-Mills equation

Details Deformation of surfaces, integrable systems and Self-Dual Yang-Mills equation Deformation of surfaces, integrable systems and Self-Dual Yang-Mills equation

2002-07-25

arxiv.org/abs/nlin/0207046v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

7 pages, no figures

Scientific paper

We conjecture that many (maybe all) integrable equations and spin systems in 2+1 dimensions can be obtained from the (2+1)-dimensional Gauss-Mainardi-Codazzi and Gauss-Weingarten equations, respectively. We also show that the (2+1)-dimensional Gauss-Mainardi-Codazzi equation which describes the deformation (motion) of surfaces is the exact reduction of the Yang-Mills-Higgs-Bogomolny and Self-Dual Yang-Mills equations. On the basis of this observation, we suggest that the (2+1)-dimensional Gauss-Mainardi-Codazzi equation is a candidate to be integrable and the associated linear problem (Lax representation) with the spectral parameter is presented.

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