Cusp solitons to the long-short waves equation and the $\bar{\mathrm{\partial }}$-dressing method

The dressing method based on $3\times 3$ matrix $\bar{\mathrm{\partial }}$-problem is extended to study the long-short waves equation with the cases $\sigma =\pm 1$. The soliton solutions for the long-short wave equation for $\sigma =\pm 1$ are given by means of the properties of Cauchy matrix. It is shown that the long-short wave equation for $\sigma =1$ has the cusp solitons.