A Fractal Modification of the Unsteady Korteweg–de Vries Model and Its Generalized Fractal Variational Principle and Diverse Exact Solutions

Under this work, we derive a new fractal unsteady Korteweg–de Vries model which can model the shallow water with the non-smooth boundary. The generalized fractal variational principle is constructed by employing the semi-inverse method and the fractal two-scale transform. In addition, we also investigate the abundant exact solutions by means of the sub-equation method. The impact of the fractal orders on the behaviors of the solutions is also discussed in detail. The obtained variational principle reveals the energy form of the conservation laws in the fractal space, and the obtained solutions can help the researchers to study the properties of the fractal solitary wave in the extremely small scale of time and space.