An ideal compressible fluid is considered, with an equilibrium density being a given function of coordinates due to presence of some static external forces. The slow flows in such system, which do not disturb the density, are investigated with the help of the Hamiltonian formalism. The equations of motion of the system are derived for an arbitrary given topology of the vorticity field. The general form of the Lagrangian for frozen-in vortex lines is established. The local induction approximation for motion of slender vortex filaments in several inhomogeneous physical models is studied.

Two-dimensional free-surface potential flows of an ideal fluid over a strongly inhomogeneous bottom are investigated with the help of conformal mappings. Weakly-nonlinear and exact nonlinear equations of motion are derived by the variational method for arbitrary seabed shape parameterized by an analytical function. As applications of this theory, band structure of linear waves over periodic bottoms is calculated and evolution of a strong solitary wave running from a deep region to a shallow region is numerically simulated.

A specific, genuinely three-dimensional mechanism of rogue wave formation, in a late stage of the modulational instability of a perturbed Stokes deep-water wave, is recognized through numerical experiments. The simulations are based on fully nonlinear equations describing weakly three-dimensional potential flows of an ideal fluid with a free surface in terms of conformal variables. Spontaneous formation of zigzag patterns for wave amplitude is observed in a nonlinear stage of the instability. If initial wave steepness is sufficiently high (ka>0.06), these coherent structures produce rogue waves. The most tall waves appear in turns of the zigzags. For ka<0.06, the structures decay typically without formation of steep waves.

Finite-amplitude gravity water waves in Bragg resonance with a periodic one-dimensional topography are studied numerically using exact equations of motion for ideal potential free-surface flows. Spontaneous formation of highly nonlinear localized structures is observed in the numerical experiments. These coherent structures consisting of several nearly standing extreme waves are similar in many aspects to the Bragg solitons previously known in nonlinear optics.

It is demonstrated that a standard coupled-mode theory can successfully describe weakly nonlinear gravity water waves in Bragg resonance with a periodic one-dimensional topography. Analytical solutions for gap solitons provided by this theory are in reasonable agreement with numerical simulations of the exact equations of motion for ideal planar potential free-surface flows, even for strongly nonlinear waves. In numerical experiments, self-localized groups of nearly standing water waves can exist up to hundreds of wave periods. Generalizations of the model to the three-dimensional case are also derived.