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.

In this paper, diverse wave solutions for the newly introduced (3+1)-dimensional Painlevé-type evolution equation were derived using the improved generalized Riccati equation and generalized Kudryashov methods. This equation is now widely used in soliton theory, nonlinear wave theory, and plasma physics to study instabilities and the evolution of plasma waves. Using these methods, combined with wave transformation and homogeneous balancing techniques, we obtained concise and general wave solutions for the Painlevé-type equation. These solutions included rational exponential, trigonometric, and hyperbolic function solutions. Some of the obtained solutions for the Painlevé-type equation were plotted in terms of 3D, 2D, and contour graphs to depict the various exciting wave patterns that can occur. As the value of the amplitude increased in the investigated solutions, we observed the evolution of dark and bright solutions into rogue waves in the forms of Kuztnetsov-Ma breather and Peregrine-like solitons. Other exciting wave patterns observed in this work included the evolution of kink and multiple wave solitons at different time levels. We believe that the solutions obtained in this paper were concise and more general than existing ones and will be of great use in the study of solitons, nonlinear waves, and plasma physics.