This paper studies the integrability and nonlinear multi-wave solutions of a Korteweg–de Vries–Sawada–Kotera–Ramani-type (KdV–SKR-type) equation arising in shallow-water wave theory, which is important for modeling nonlinear wave interactions. The integrability of the equation is confirmed by Painlevé analysis, and its bilinear form is subsequently derived using the Bell polynomial (BP) method. Based on the resulting formulation, multi-soliton solutions are constructed via the simplified Hirota method, and the corresponding multi-lump solutions are obtained through the long-wave limit. Furthermore, a Pfaffian framework is developed to construct compact general $N$-soliton solutions and to systematically generate multi-lump waves within a unified algebraic structure. Various interaction phenomena, including resonant and breather waves, are also derived. The results confirm the integrable nature of the model and reveal rich nonlinear dynamics. The novelty of this work lies in the introduction of the Pfaffian formulation for this equation and the unified construction of its solutions, extending previous studies.

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.