On the structure of $(2+1)$--dimensional commutative and noncommutative integrable equations

Details On the structure of $(2+1)$--dimensional commutative and noncommutative integrable equations On the structure of $(2+1)$--dimensional commutative and noncommutative integrable equations

2006-06-13

arxiv.org/abs/nlin/0606036v1

Journal of Mathematical Physics 47(11), 113508, 2006

Nonlinear Sciences

Exactly Solvable and Integrable Systems

17 pages

Scientific paper

We develop the symbolic representation method to derive the hierarchies of $(2+1)$-dimensional integrable equations from the scalar Lax operators and to study their properties globally. The method applies to both commutative and noncommutative cases in the sense that the dependent variable takes its values in ${\mathbb C}$ or a noncommutative associative algebra. We prove that these hierarchies are indeed quasi-local in the commutative case as conjectured by Mikhailov and Yamilov in 1998. We propose a ring extension based on the symbolic representation. As examples, we give noncommutative versions of KP, mKP and Boussinesq equations.

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