Abstract: The properties of Cauchy problems are closely related with those of the initial values. The unique continuation properties of these problems are one of the important properties of the solution to the integrable system. Considered herein is the Cauchy problem associated with a class of seventh-order shallow water wave equations, which describe the propagation of weakly dispersive nonlinear long waves in the horizontal direction. The purpose here is to investigate the unique continuation property of the solutions to this Cauchy problem. Based on the complex variables technique and Paley-Wiener Theorem, it is proved that, if a sufficiently smooth solution to this Cauchy problem is supported compactly in a nontrivial time interval, then it vanishes identically.

Key words: Cauchy problem, seventh-order shallow water wave equations, compact support, unique continuation property