Dynamics of hydroelastic wavetrains with plate inertia

In the linear theory of hydroelastic waves, the inertial effects of thin plates are often neglected due to their minimal impact on the dispersion relation. This assumption also persists in many nonlinear studies. For instance, existing research that uses the nonlinear Schrödinger (NLS) equation to describe finite-amplitude responses near the critical velocity (the phase speed minimum) typically does not consider inertia. However, such a defocusing-type model fails to explain the fully localized responses observed experimentally in deep water. Our main contribution is demonstrating that incorporating plate inertia fundamentally alters the nonlinear hydroelastic wave dynamics. Specifically, when the areal density of the plates exceeds a critical threshold—determined by water density, gravity, and flexural rigidity—the deep-water NLS equation changes from a defocusing type to a focusing type. This transition enables solutions that decay in the far field. Through multiple-scale analyses, we also illustrate that inertia modifies the intensities of nonlinear wave–wave interactions. Our theoretical predictions align closely with numerical results derived from potential flow theory, and this agreement is further validated by a novel reformulation of time-dependent conformal mapping that takes the inertia term into account.