On a variational principle for the fractal Wu–Zhang system arising in shallow water

Abstract

The flow of the shallow water of the harbor can be described by the classic Wu–Zhang system. However when the the boundary (such as the coastline) is non-smooth, it becomes invalid. So we modify the classic Wu–Zhang system into its fractal form that can work under the non-smooth boundary by using He’s fractal derivatives. With the help of the semi-inverse method, we establish its fractal variational principle, which can not only provide the conservation laws in an energy form in the fractal space but also reveal the possible solution structures of the equation. The established variational principle in this work is expected to be helpful for finding conserved quantities for the equations, and to bring a light to the study of the fractal theoretical basis in geosciences.