Abstract
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.
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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 216, pp. 251–270 https://doi.org/10.4213/tmf10378.
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Helminck, G.F. Cauchy problems related to integrable matrix hierarchies. Theor Math Phys 216, 1124–1141 (2023). https://doi.org/10.1134/S0040577923080056
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DOI: https://doi.org/10.1134/S0040577923080056