High-order rogue waves of the generalized (3+1)-dimensional nonlinear wave in liquid with gas bubbles

We derive the general high-order rogue waves for the generalized (3+1)-dimensional nonlinear wave equation, which can be directed from a liquid containing gas bubbles. Based on the KP reduction method, the dynamical behaviors of these fundamental rogue waves, including line rogue waves and general first-order rogue waves, are given in different planes. In addition, the second-order rogue waves are also discussed. They consist of two interacting line waves and arise from the constant background, and then, they disappear into the constant background finally. Furthermore, the third-order rogue waves are shown in the (x, y) plane, which have different patterns with different values of parameters. Besides, we find that the third-order rogue waves are composed of three line waves interacting in the (x, t) plane.