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Abstract
A family of higher-order modified Korteweg–de Vries equations with variable coefficients (t-ho-mKdV) is introduced. A one-to-one correspondence between a real solution of these equations and a complex solution of the variable coefficient higher-order Korteweg–de Vries (t-ho-KdV) equations is established through a complex Miura transformation. An auto-Bäcklund transformation for these t-ho-mKdV equations is derived from that of the t-ho-KdV equations. The associated gauge transformations of the corresponding AKNS systems are presented. They enable one to construct a hierarchy of solutions of the t-ho-mKdV equations from a known hierarchy without solving the differential equations for the wave functions except the first one. A new family of higher-order evolution equations with an auto-Bäcklund transformation is also derived in connection with the gauge transformation of the t-ho-mKdV equations.
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