On Stationary Solutions of the Reduced Gardner–Ostrovsky Equation - Obregon, Maria; Stepanyants, Yury (Discontinuity, Nonlinearity, and Complexity, 2026)
The detailed analysis of stationary solutions of the reduced Gardner– Ostrovsky (GO) equation is presented. The GO equation (ut + c0ux + αuux + α1u2ux + βuxxx )x = γu is the popular model for the description of large-amplitude internal oceanic waves affected by Earth’s rotation. Its reduced version in which the small-scale dispersion is neglected ( β = 0 ) is used when very long internal waves are considered. The equation is also applicable to other types of nonlinear waves in various media (plasma, optical media, relaxing media, etc.) when the large-scale dispersion ∼ γ plays a dominant role in comparison with the small-scale dispersion ∼ β. Balancing the nonlinear effect such dispersion gives rise to existence of stationary waves, both periodic and non-periodic. It is shown that only smooth periodic waves make physical sense. Systematic analysis of stationary solutions to the GO equation and their categorisation is presented.