Some new integrable nonlinear evolution equations and Darboux transformation

Two classes of new nonlinear evolution equations are derived with the aid of a 4×4 matrix spectral problem with three potentials. A Darboux transformation for the first nontrivial nonlinear evolution equation is given based on the gauge transformation between the corresponding 4×4 matrix spectral problems. As an application of the Darboux transformation, some explicit solutions of the nontrivially nonlinear evolution equation are obtained.

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