Modelling intermediate internal waves with currents and variable bottom

• Published in: Nonlinear analysis: real world applications Vol. 87; p. 104451
• Main Authors: Ivanov, Rossen, Ivanova, Lyudmila
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.02.2026
• Subjects: Dirichlet–Neumann operator; Intermediate long-wave equation; Internal waves; Shear current; Intermediate long-wave equation; Dirichlet–Neumann operator; Shear current; Internal waves
• ISSN: 1468-1218
• DOI: 10.1016/j.nonrwa.2025.104451

A model for internal interfacial waves between two layers of fluid in the presence of current and variable bottom is studied in the flat-surface approximation. Fluids are assumed to be incompressible and inviscid. Another assumption is that the upper layer is considerably deeper with a lower density than the lower layer. The fluid dynamics is presented in Hamiltonian form with appropriate Dirichlet–Neumann operators for the two fluid domains, and the depth-dependent current is taken into account. The well known integrable Intermediate Long Wave Equation (ILWE) is derived as an asymptotic internal waves model in the case of flat bottom. For a non-flat bottom the ILWE is with variable coefficients. Two limits of the ILWE lead to the integrable Benjamin–Ono and Korteweg-de Vries equations. Higher-order ILWE is obtained as well. •The problem of internal waves and currents over uneven bottom is analysed.•The dynamics in case with deep upper layer is given in Hamiltonian form.•The obtained model is the Intermediate Long Wave Equation (ILWE).•The integrability of ILWE is used in the study of the internal waves.•The KdV and the Benjamin-Ono equations are obtained as specific limits.