In this thesis, we examine the liquid flow and the free surface shape by using some methods of complex analysis during the early stage of dam breaking. In order to obtain the leading-order uniform solution, we use the asymptotic expansions with respect to a small parameter. The small parameter describe the initial duration of the stage. However, this solution which calls outer solution is not valid in a small vicinity of the intersection point between the vertical free surface and the horizontal rigid bottom. A local analysis of the outer solution helps to estimate the dimension of this vicinity. In the vicinity of the intersection point, local coordinates are stretched to solve the flow singularity and with the aid of these coordinates the leading order inner solution is obtained which describes the formation of the jet flow along the bottom.