Abstract: Based on the local well-posedness of the solution, we discussed  the non-uniformly continuous dependence of the solutions to the Cauchy problem associated with the periodic water wave system on initial value in Besov spaces by using the method of approximate solutions. The results show that the local well-posedness of the solutions to this problem in the corresponding Besov spaces can not be obtained only by  contraction mapping principle.

Key words: non-uniform continuity, Besov space, periodic water wave system, approximate solution