Abstract. An explicit Darboux transformation for the coupled Hirota equation is constructed with the help of a gauge transformation of the Ablowitz-Kaup-Newell-Segur (AKNS) system spectral problem. By using the Darboux transformation and the reduction technique, some N-soliton solutions and breather solutions for the coupled Hirota equation are obtained.

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Abstract. In this paper, quasiperiodic waves and asymptotic behavior for the nonisospectral and variable-coefficient KdV (nvcKdV) equation are considered. The Hirota bilinear method is extended to explicitly construct multiperiodic (quasiperiodic) wave solutions for the nvcKdV equation. And a limiting procedure is presented to analyze asymptotic behavior of the one-and two-periodic waves in details. The exact relations between the periodic wave solutions and the well-known soliton solutions are established. It is rigorously shown that the periodic wave solutions tend to the soliton solutions under a small amplitude limit.