Numerical study of a nonlocal model for water-waves with variable depth

We study numerically the propagation of solitary waves in a Hamiltonian nonlocal shallow water model for bidirectional wave propagation in channels of variable depth. The derivation uses small wave amplitude and small depth variation expansions for the Dirichlet–Neumann operator in the fluid domain, and in the long wave regime we simplify the nonlinear and bottom topography terms, while keeping the exact linear dispersion. Solitons are seen to propagate robustly in channels with rapidly varying bottom topography, and their speed is predicted accurately by an effective equation obtained by the homogenization theory of Craig et al. (2005) [7]. We also study the evolution from peaked initial conditions and give evidence for solitary waves with limiting peakon profiles at an apparent threshold before blow-up.

► Propose bidirectional nonlocal Hamiltonian shallow water model for varying bottom topography. ► Study soliton propagation over rapidly varying depth, confirm results of homogenization theory. ► Show evidence for limiting peaked solitons and blow-up. ► Compare with unidirectional Whitham equations.