Modulation instability and rogue waves for two and three dimensional nonlinear Klein–Gordon equation

We perform the modulation instability analysis of the 2D and 3D nonlinear Klein-Gordon equation. The instability region depends on dispersion and wavenumbers of the plane wave. The N-breathers of the nonlinear Klein-Gordon equation are constructed directly from its 2N-solitons obtained in history. The regularity conditions of breathers are established. The dynamic behaviors of breathers of the 2D nonlinear Klein-Gordon equation are consistent with modulation instability analysis. Furthermore, by means of the bilinear method together with improved long-wave limit technique, we obtain general high order rogue waves of the 2D and 3D nonlinear Klein-Gordon equation. In particular, the first- and second-order rogue waves and lumps of the 2D nonlinear Klein-Gordon equation are investigated by using their explicit expressions. We find that their dynamic behaviors are similar to the nonlinear Schrödinger equation. Finally, the first-order rational solutions are illustrated for the 3D nonlinear Klein-Gordon equation. It is demonstrated that the rogue waves of the 2D and 3D nonlinear Klein-Gordon equation always exist by choosing dispersion and wavenumber of plane waves.