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In this paper, a3 × 3spectral 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 ofe xand t, for the five-component equation are obtained.

A Five-Component Generalized mKdV Equation and Its Exact Solutions