Some Aspects of Moyal Deformed Integrable Systems

Physics – High Energy Physics – High Energy Physics - Theory

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16 pages, Latex, Corrected Typos in p.9,10,12

Scientific paper

Besides its various applications in string and D-brane physics, the $\theta$-deformation of space (-time) coordinates (naively called the noncommutativity of coordinates), based on the $\star$-product, behaves as a more general framework providing more mathematical and physical informations about the associated system. Similarly to the Gelfand-Dickey framework of pseudo differential operators, the Moyal $\theta$-deformation applied to physical problems makes the study more systematic. Using these facts as well as the backgrounds of Moyal momentum algebra introduced in previous works [21, 25, 26], we look for the important task of studying integrability in the $\theta$-deformation framework. The main focus is on the $\theta$-deformation version of the Lax representation of two principal examples: the $sl_2$ KdV$_{\theta}$ equation and the Moyal $\theta$-version of the Burgers systems. Important properties are presented.

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