Stability analysis of nonlinear evolution patterns of modulational instability and chaos using one-dimensional Zakharov equations

In the present paper the long-term behavior of the nonlinear dynamical evolution of modulational instability is investigated by using a simplified model for one-dimensional Zakharov equations, which couples the electrostatic electron plasma wave and ion-acoustic wave propagation. The manuscript details on the occurrence of fixed points and fixed-point attractors for a suitable value of the wavenumber of perturbation through associated bifurcations, both for the adiabatic (nonlinear Schrödinger equation) and non-adiabatic cases for Zakharov equations. It is shown that these evolutions are quite sensitive to initial conditions, Fermi Pasta Ulam recurrence is broken up and a chaotic state develops for the non-adiabatic case. Regular patterns with a periodic sequence in space and time and spatiotemporal chaos with irregular localized patterns are formed in different regions of unstable wavenumbers, hence producing a self-organizing dynamical system. The results are consistent with those obtained by numerically solving Zakharov equations as previously reported and summarized in the present manuscript.