| Summary: | The complex coupled integrable dispersionless (CID)system is an important system. It has been shown that complex CID equation is gauge equivalent to the Pohlmeyer–Lund–Regge model (a complex sine-Gordon equation) and can also be transformed to the complex short pulse equation. CID system can describe a current-fed string interacting with an external magnetic field. In this paper, we investigate the two-component complex coupled integrable dispersionless system. By using the Darboux transformation, we obtain a new type of solution (the breather-like soliton solution) with zero boundary condition and a large number of solutions (the breather-II type solution, bright–dark soliton solution, the resonance solutions and the semi-rational solutions) with non-zero boundary condition. In particular, these solutions are not available for the scalar complex integrable dispersionless system.
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