Bilinear form and nonlinear waves of a (1+1)-dimensional generalized Boussinesq equation for the gravity waves over water surface

Abstract

Gravity waves are studied in, e.g., the oceanic and atmospheric sciences. In this paper, for the head-on interaction of oblique gravity wave profiles over the water surface, we investigate a (1+1)-dimensional generalized Boussinesq equation. Bilinear form is derived, and the higher-order rogue waves, breather and hybrid solutions in the determinant form are constructed via the Hirota bilinear method and Kadomtsev–Petviashvili hierarchy reduction. Influence of α, β and γ on the solutions is discussed, where α, β and γ are the nonzero coefficients in that equation. For the first-order rogue-wave solutions, we find that the wave widths and amplitudes are related to α and β, and the locations of the extreme points depend on α, β and γ. Structures of the higher-order rogue waves are also given. We construct three types of the first-order breathers, and the second-order breathers are obtained through the combinations of the first-order ones. α, β and γ are found to influence the amplitudes and locations of the first-order breathers. Hybrid solutions comprising the one rogue wave, the higher-order breathers and solitons are discussed graphically. Those results reveal the nonlinear properties of oblique gravity waves over the water surface, which can help the study on nonlinear waves in fluid mechanics, plasma physics and fiber optics.