Variational approach for the fractal generalized Bogoyavlensky-Konopelchenko equation

This paper explores a fractal generalized Bogoyavlensky-Konopelchenko equation (gBK) defined by He's fractal derivative. Employing the variational aproach in conjunction with the two-scale fractal transformation has yielded the fractal wave solutions. Furthermore, we offer remarks on the variational formulation of the conventional gBK equation presented in previous literature. Notably, we have identified two novel solutions: the fractal periodic wave solution and the fractal bright soliton solution. These solutions have not been previously explored in the existing literature. The propagation behavior of these fractal wave solutions is vividly demonstrated through 3-D figures with diverse fractal dimensions and amplitudes. The findings of the present study contribute to the theoretical research on the gBK equation and provide valuable insights for future studies in the field of fractal differential equations.