On the steadiness of symmetric solutions to two dimensional dispersive models

In this paper, we consider the steadiness of symmetric solutions to two dispersive models in shallow water and hyperelastic mechanics, respectively. These models are derived previously in the two-dimensional setting and can be viewed as the generalization of the Camassa-Holm and Kadomtsev-Petviashvili equations. For these two models, we prove that symmetry of classical solutions implies steadiness in the horizontal direction. We also confirm the such connection between symmetry and steadiness in weak formulation, which includes in particular the peaked solutions.