Cauchy problems related to integrable matrix hierarchies

G. F. Helminck

Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 2, pp. 251-270

Résumé

We discuss the solvability of two Cauchy problems in matrix pseudodifferential operators. The first is associated with a set of matrix pseudodifferential operators of negative order, a prominent example being the set of strict integral operator parts of products of a solution $(L,\{U_\alpha\})$ of the $\mathbf h[\partial]$-hierarchy, where $\mathbf h$ is a maximal commutative subalgebra of $gl_n(\mathbb{C})$. We show that it can be solved in the case of compatibility completeness of the adopted setting. The second Cauchy problem is slightly more general and relates to a set of matrix pseudodifferential operators of order zero or less. The key example here is the collection of integral operator parts of the different products of a solution $\{V_\alpha\}$ of the strict $\mathbf h[\partial]$-hierarchy. This system is solvable if two properties hold{:} the Cauchy solvability in dimension $n$ and the compatibility completeness. Both conditions are shown to hold in the formal power series setting.

Détail

Keywords: Cauchy problem, formal power series, integrable deformations, matrix pseudodifferential operators, $\mathbf h[\partial]$-hierarchy, strict $\mathbf h[\partial]$-hierarchy, zero-curvature equations.

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