Integrable models for intermediate long internal waves with currents

Rossen Ivanov (Technological University Dublin)

Time:

11AM Wednesday, 19 November 2025

Location:

UCD Confucius Institute, Room 1.01

Internal water waves arise where there is a change in density stratification in a fluid, which may occur in an oceanographical context due to variations in temperature, salinity, or other fluctuations in the equations of state. We present a derivation of nonlinear integrable models for the propagation of interfacial internal waves, arising between two fluid layers of different densities (at the so-called pycnocline).

The models are developed from the Hamiltonian description of the problem employing techniques based on the Dirichlet-Neumann operators for the fluid domains.

In particular, we derive the integrable Intermediate Long Wave (ILW) equation and the Benjamin-Ono (BO) equation. The model incorporates underlying currents by permitting a sheared current in both fluid layers. We obtain the ILW and the BO equations for specific small-amplitude asymptotic regimes. The obtained integrable models have the advantage that they are exactly solvable and admit the so-called soliton solutions. The stability property of the solitons makes them suitable for detection and measurement. We show that the soliton characteristics are strongly affected by the parameters of the shear current.

Another aspect of the problem is the effect of a variable bottom on the internal wave propagation in the presence of stratification and underlying non-uniform currents. This effect could be studied by employing similar methods, resulting in model equations with depth-dependent variable coefficients. The soliton properties will be discussed in the case of small bottom variations.