Nonlinear waves solutions and integrability for an extended Korteweg-de Vries equation in fluids

Abstract This study presents a comprehensive analysis of a higher-order extended Kortewegde Vries equation modeling nonlinear wave phenomena in fluid dynamics. Utilizing the Bell polynomial framework, we systematically derive the bilinear form of the equation and construct N -soliton solutions via the Hirota's method. Furthermore, breather wave solutions are obtained by virtue of the extended homoclinic test approach, and periodic wave solutions are obtained through the Riemann theta function, with their degeneration to soliton solutions rigorously demonstrated in the vanishing-amplitude limit. The model's integrability is established by deriving a Bäcklund transformation, an associated Lax pair, and an infinite hierarchy of conservation laws. These results provide new mathematical tools connecting periodic and localized wave structures in nonlinear dispersive media.