We study analytically the characteristics of optical absorption and slow-light solitons in an asymmetrical fourlevel N configuration semiconductor quantum wells with the cross-coupling relaxation of longitudinal-optical phonons (CCRLOP). It is shown that, in the linear range, the electromagnetically induced transparency (EIT) depends on the coherence control of both the optical fields and the CCRLOP. A double EIT is obtained under a relatively strong optical field which is from the hole and antibonding states in the wide well. Especially, the double EIT becomes perfect under the condition of increasing the CCRLOP. In the nonlinear range, the CCRLOP has an important effect on both the amplitude and the group velocity of the solitons. The amplitude of solitons reveals parabolic changes which obtain a maximum value with the increase of CCRLOP. The group velocity of the solitons continuously slows down if there are fixed three-photon detunings. These results may have potential applications for all-optical switching and some optical information engineering in solid systems.

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