Global wellposedness and limit behavior for the generalized finite-depth-fluid equation with small data in critical Besov spaces

In this paper, we study the Cauchy problem for the generalized finite-depth-fluid equation , where , k is an integer that is larger than 4. We obtain that it is globally wellposed if k⩾4 and the initial data in are sufficiently small, where and . Moreover, we show that its solution will converge to those of the generalized BO and KdV equations as the depth parameter δ→∞ and δ→0, respectively.