Nonlinear Coherent Structures of Alfv\'{e}n Wave in a Collisional Plasma.

The Alfv\'{e}n wave dynamics is investigated in the framework of Lagrangian two-fluid model in a cold magnetized collisional plasma in presence of finite electron inertia. In the quasi-linear limit, the dynamics of the nonlinear Alfv\'{e}n wave is shown to be governed by a modified Korteweg-de Vries Burgers (mKdVB) equation. In this mKdVB equation, the electron inertia is found to act as a source of dispersion and the electro-ion collision serves as a dissipation. In the long wavelength limit, we have also investigated wave modulation characteristics of the nonlinear Alfv\'{e}n wave. The dynamics of this modulated wave is shown to be governed by a damped nonlinear Schr\"{o}dinger equation (NLSE). These nonlinear equations are analysed by means of analytical and numerical simulation to elucidate the various aspects of the phase-space dynamics of the nonlinear wave. Results reveal that nonlinear Alfv\'{e}n wave exhibits shock, envelope and breather like structures. Numerical simulations also predict the formation of Alfv\'{e}nic rogue waves, rogue wave holes and giant breathers. These results could be useful for understanding the salient features of the Alfv\'{e}nic magnetic field structures from observational data in very low-$\beta $magnetized collisional plasmas in space and laboratory.