Transverse Instability of Small Periodic Waves for a Two-Dimensional Dullin–Gottwald–Holm Equation

In this work, we focus on a two-dimensional generalization of the Dullin–Gottwald–Holm equation with a weakly transverse effect for the horizontal velocity component, which models deformations of a hyperelastic compressible plate relative to a uniformly prestressed state. By means of the Lyapunov–Schmidt reduction, the small-amplitude periodic traveling waves are constructed. We further investigate the linearized problem with respect to two-dimensional perturbations which are either periodic or nonperiodic in the direction of propagation. In order to overcome the difficulties arising from the complicated linearized operator, we employ the perturbation method and Hamiltonian theory in our analysis and derive instability criteria in different transverse wavelength regimes.