Evolution of nonlinear Rossby waves in a dynamically time-dependent background flow via a quartic Korteweg-de Vries model

This study systematically investigates the evolution of nonlinear Rossby waves within a dynamically time-dependent background flow, based on the quasi-geostrophic potential vorticity equation under the beta-plane approximation. By employing the Gardner-Morikawa transformation and the multiple-scale perturbation method, a variable-coefficient quartic Korteweg-de Vries (vfqKdV) equation is derived to characterize the amplitude dynamics. By introducing a fractional linear transformation, the fifth-degree polynomial form of the first integral is reduced to the standard Weierstrass equation. Under the weak nonlinearity approximation, where the quartic nonlinear effect is relatively weak, we construct exact periodic traveling wave solutions. Applying the degeneracy condition \varDelta = 0 further yields a solitary wave solution. This solution, obtained under the weak nonlinearity approximation, effectively captures the modulation of Rossby solitary waves by time-dependent background flows, providing a theoretical foundation for understanding the evolution of finite-amplitude Rossby waves in non-stationary geophysical flows.