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

Abstract: The dressing method based on $3\times3$ matrix $\bar{\partial}$-problem is extended to study the long-short waves equation with the cases $\sigma=\pm1$. The soliton solutions for the long-short wave equation for $\sigma=\pm1$ 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.