Bifurcation and noise-induced transitions in weakly dissipative geophysical KdV equation

We investigate the dynamics of the geophysical Korteweg-de Vries (KdV) equation under deterministic, dissipative, and stochastic settings. The geophysical KdV equation is solved using a Fourier pseudo-spectral method combined with an exponential time-differencing fourth-order Runge-Kutta scheme (ETDRK4), which ensures high accuracy for dispersive operators. Traveling-wave reduction transforms the KdV equation into a two-dimensional autonomous system, enabling phase-plane and bifurcation analysis. A transcritical-like bifurcation is identified, where equilibrium stability exchanges between a center and a saddle. Extending the equation with a weak dissipative term and additive Gaussian white noise, we analyze the noise-induced transition from the stable focus towards the separatrix using the concept of Mean First Passage Time (MFPT). The MFPT is approximated via Kramers' formula and validated numerically through stochastic simulations, showing exponential sensitivity to noise amplitude. The results highlight how weak dissipation regularizes oscillatory modes into damped spirals while noise triggers transition events over effective potential barriers. This combined framework of spectral theory, bifurcation theory, and stochastic analysis provides new insights into the stability and transition mechanisms of geophysical dispersive waves.