THE $ \bar{\partial} $-DRESSING METHOD FOR THE TWO-DIMENSIONAL HARRY DYM EQUATION

The (2+1)-dimensional Harry Dym (HD) equation is solved via the $ \bar{\partial} $-dressing method in this paper. By introducing long derivatives $ E_x $, $ E_y $ and $ E_t $ and new expressions for the kernel functions $ K $ of $ \bar{\partial} $-problem, a type of general solution of the HD equation is obtained. Under the reality of the solution $ u $ of the HD equation, several classes of exact explicit solutions of the HD equation, including the solutions with functional parameters, line solitons and rational solutions, are constructed by the $ \bar{\partial} $-dressing method.