Breathers, lump, M-shapes and other optical soliton interactions for the GRIN multimode optical fiber

The different types of optical soliton solutions are investigated for the 2-dimensional nonlinear Schrödinger equation with an instantaneous Kerr nonlinearity. This equation contains the beam propagation in a multimode optical fiber with a parabolic index profile. Different soliton solutions are examined, such as Breather waves, Lump waves, Mixed types, Periodic cross kink, Multiwave, M-shape, M-shape rational one, two kinks, and rogue waves. To obtain these optical solitons, we apply the Hirota bilinear transformation approach. These results are spectral, temporal, and spatial features of ultrashort light pulses that can be controlled in novel ways by this kind of nonlinear multimode optical fiber, which is gaining popularity. Moreover, the 3-dimensional, 2-dimensional, and contour plots are drawn for some solutions that show their physical interactions by selecting different parameters. These results are very helpful for further study of dynamical systems.