A Korteweg–de Vries–Sawada–Kotera–Ramani-type equation: Its integrability and multi-wave solutions

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.