Nonlinear evolution of Rossby solitary waves under time-dependent shear: A variable-coefficient Gardner model solved by physics-informed neural networks

Abstract

Constant-coefficient models inadequately capture how time-dependent shear flows alter the propagation of Rossby solitary waves, and traditional numerical methods face inherent difficulties with such variable-coefficient systems. We derive a variable-coefficient Gardner equation from quasi-geostrophic dynamics and solve it using physics-informed neural networks (PINNs) that integrate residual-adaptive refinement with a two-stage Adam and Limited-memory Broyden–Fletcher–Goldfarb–Shanno optimizer. For the constant-coefficient test case, the PINNs reproduce the exact two-soliton solution with a maximum absolute error of 5.53×10−3, a relative L2 error of 1.02×10−3, and a mean absolute error of 6.41×10−4. In the time-dependent regime where the quadratic nonlinearity coefficient m(t) periodically crosses zero, the PINNs reveal a complete steepening-and-recovery cycle. At the zero-crossing instant, quadratic nonlinearity is momentarily suppressed, triggering abrupt wave steepening together with a transient compression of the wave field in which the crest–trough separation narrows; as m(t) recovers, the waveform broadens again. This cyclic mechanism accounts for the intermittent amplitude fluctuations documented in the satellite altimetry. The amplitude-dependent phase speed c=c0+(m/3)A0+(n/6)A02, together with mean-flow Doppler shifting, quantitatively explains a 50% phase-speed increase from the eastern to western Pacific and the breaking of β-refracted fronts into isolated eddies. Our mesh-free PINN framework provides a robust solution strategy for variable-coefficient ocean wave problems, with clear potential for data assimilation and three-dimensional extensions.