| Summary: | In this paper, a <inline-formula> <math display="inline"> <semantics> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> </semantics> </math> </inline-formula> spectral problem is proposed and a five-component equation that consists of two different mKdV equations is derived. A Darboux transformation of the five-component equation is presented relating to the gauge transformations between the Lax pairs. As applications of the Darboux transformations, interesting exact solutions, including soliton-like solutions and a solution that consists of rational functions of <inline-formula> <math display="inline"> <semantics> <msup> <mi>e</mi> <mi>x</mi> </msup> </semantics> </math> </inline-formula> and <i>t</i>, for the five-component equation are obtained.
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