Rogue waves of a (3+1)(3+1)-dimensional nonlinear evolution equation

• The bilinear forms and rational solutions of the (3 + 1)-d NEE.
• Fundamental rogue waves in the (3+1)-d NEE.
• Multi-rogue waves in (3 + 1)-d NEE.
• High-order rogue waves in (3 + 1)-d NEE.

General high-order rogue waves of a (3+1)(3+1)-dimensional Nonlinear Evolution Equation ((3+1)-d NEE) are obtained by the Hirota bilinear method, which are given in terms of determinants, whose matrix elements possess plain algebraic expressions. It is shown that the simplest (fundamental) rogue waves are line rogue waves which arise from the constant background with a line profile and then disappear into the constant background again. Two subclass of nonfundamental rogue waves are analyzed in details. By proper means of the regulations of free parameters, the dynamics of multi-rogue waves and high-order rogue waves have been illustrated in (x,t) plane and (y,z) plane by three dimensional figures.