Darboux transformation of a semi-discrete coupled dispersionless integrable system

We present Darboux transformation (DT) of a semi-discrete coupled dispersionless (SDCD) integrable system. A Lax pair for the SDCD system is proposed and a Darboux transformation is employed on the solutions to the Lax pair and solutions to the SDCD system. Multisoliton solutions resulting from Darboux transformation are expressed in terms of determinants. We also compute explicit expressions of one- and two-soliton solutions with the help of a given seed solution. We apply an appropriate continuum limit to obtain multisoliton solutions of the continuous CD integrable system. By taking an appropriate choice of discrete variables, we show that the SDCD system is equivalent to semi-discrete sine-Gordon equation. From the soliton solution of SDCD system, semi-discrete kink solution of the semi-discrete sine-Gordon equation is obtained.