A fractal variational theory of the Broer-Kaup system in shallow water waves

Abstract

The Broer-Kaup equation is one of many equations describing some phenomena of shallow water wave. There are many errors in scientific research because of the existence of the non-smooth boundaries. In this paper, we generalize the Broer-Kaup equation to the fractal space and establish fractal variational formulations through the semi-inverse method. The acquired fractal variational formulation reveals conservation laws in an energy form in the fractal space and suggests possible solution structures of the morphology of the solitary waves.

Approximate analytic solution of the fractal Fisher’s equation via local fractional variational iteration method

“…where   is a fractal Lagrange multiplier, which can be identified by the fractal variational theory [28][29][30][31][32][33][34][35][36][37].…”

“…t   Following eq. ( 16), we have the stationary condition, which is given by [28][29][30][31][32][33][34][35][36][37]:…”

Fractional solitons: New phenomena and exact solutions

“…Therefore, fractional solitons have attracted increasing attention from both physics and oceanography. For example, shallow water waves [7,8] can describe the effects of waves in the ocean better than other mathematical models. Shallow water waves are fluctuations in the ocean with wavelengths much greater than the depth of the water (usually more than 25 times), and the dispersion of water waves is one of the key properties in many shallow water wave models, which has obvious memory property.…”

Fractal variational principle for an optimal control problem

“…Now, the two-scale fractal calculus has been widely applied to discontinuous mechanics. Ji-Huan He and colleagues suggested a fractal Chen–Lee–Liu equation for ultrashort pulses in optics 23 ; He and El-Dib gave a tutorial review on its properties; Liu, et al proposed for the first time ever an optimal model for charge transport 24 ; Liu et al gave an optimal model for charge transport 25 ; Ling and Wu established a fractal shallow water wave 26 ; He et al studied a fractal Toda system with great success 27 ; C.H. He et al studied the porous concrete and some new findings were found 28,29 ; and Zou and Liu obtained fractal resistance law for composites.…”