Dynamical properties of ocean internal solitary waves: a study based on the modified intermediate long wave equation

This study presents a modified intermediate long wave (mILW) equation derived from the Navier–Stokes equations via multi-scale analysis and perturbation expansion, aimed at describing internal solitary waves (ISWs) in finite-depth, stratified oceans. Compared to the classical ILW model, the proposed mILW equation incorporates cubic nonlinearities and captures the dynamical behaviour of large-amplitude ISWs more accurately. The equation reduces to the modified Korteweg–de Vries equation and modified Benjamin–Ono equations in the shallow- and deep-water limits, respectively, thus providing a unified framework across varying depth regimes. Soliton solutions are constructed analytically using Hirota’s bilinear method, and numerical simulations investigate wave–wave interactions, including rogue waves and Mach reflection. Furthermore, a smooth tanh-type density profile is adopted to reflect realistic stratification. Associated vertical modal structures and vertical velocity fields are analysed, and higher-order statistics (skewness and kurtosis) are introduced to reveal the density dependence of wave asymmetry. The results offer new insights into the nonlinear dynamics of ISWs, with implications for ocean mixing, energy transport and submarine acoustics.