Attracting multi-soliton configurations in the Serre–Green–Naghdi equations

Establish that multi-soliton configurations are attracting quasi-exact solutions of the Serre–Green–Naghdi equations, by proving the long-time asymptotic stability and attraction of multi-soliton states under localized perturbations.

Background

The Serre–Green–Naghdi (SGN) equations generalize the classical shallow-water equations by relaxing the hydrostatic approximation and admit solitary wave (soliton) solutions. In the weakly nonlinear regime, the SGN equations reduce asymptotically to the Korteweg–de Vries (KdV) equation, which is completely integrable and possesses exact multi-soliton solutions that act as attractors for localized initial data.

High-resolution numerical simulations in this paper demonstrate that SGN solitons exhibit behavior strikingly similar to KdV solitons, including robustness through double and triple collisions, recovery after frontal collisions and wall interactions, and persistence over variable topography, typically leaving only small residual wave wakes. These observations motivate the conjecture that multi-soliton configurations are attracting quasi-exact solutions for the SGN equations beyond the weakly nonlinear limit.