Abstract
By using the inverse problem, we have found a system with two hierarchies, that is, the localized induction hierarchy and the hierarchy including the dispersionless equation. Recently, with the same system, we have found a fused hierarchy which is produced by fusing two hierarchies with help of the recursion formulae. In this short note, we would like to show that the mKdV hierarchy and the sine-Gordon hierarchy produce another fused hierarchy. It is recognized that the recursion operators of the mKdV hierarchy and the sine-Gordon hierarchy is the inverse operator in each other. However, it has been misunderstood that two hierarchies are not connected since the boundary conditions have not been carefully considered. If each member of equations in the mKdV hierarchy is written by tðx; tÞ nðx; tÞ 1⁄4 0; ðn 1⁄4 1; 2; . . .Þ ð1Þ
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