Integrability, Hirota D-operator expression, multi solitons, breather wave, and complexiton of a generalized Korteweg-de Vries–Caudrey Dodd Gibbon equation

In this paper, we conducted an in-depth study of a generalized Korteweg-de Vries–Caudrey Dodd Gibbon (gKdV–CDG) equation modeling specific oceanic waves. Through the Bell polynomial approach (BPA), the Hirota $D$-operator expression of the gKdV–CDG equation was first constructed. An integrability test of the governing model was then carried out, and consequently, multi solitons were constructed using the Hirota method. Ultimately, using symbolic computations, breather and complexiton waves were derived from the gKdV–CDG equation by serving distinct ansatzes. A few representations positioned two- and three-dimensionally were provided to characterize the nonlinear wave's physical features. Based on the results, suitable methods were suggested to assess the height and width of nonlinear waves in the ocean.