Nonlinear evolution of the modulational instability and chaos using one-dimensional Zakharov equations and a simplified model

R. P. Sharma (1) , K. Batra (1) , Alberto Verga (2)

1 Centre for Energy Studies
2 IRPHE - Institut de Recherche sur les Phénomènes Hors Equilibre

Nonlinear evolution of modulational instability by using the nonlinear Schrödinger equation in one dimension reveals a periodic reoccurrence of initial conditions. The nonlinear Schrödinger equation is the adiabatic limit of Zakharov equations, which couples the electrostatic electron plasma wave and ion-acoustic wave propagation. In the present paper nonlinear evolution of modulational instability is investigated by using one-dimensional Zakharov equations numerically. A simplified model is predicted that establishes the fact that the effect of relaxing the condition of adiabaticity is drastic on the nonlinear evolution patterns of modulational instability. These evolutions are quite sensitive to initial conditions, Fermi–Pasta–Ulam recurrence is broken up and a chaotic state develops. Next, quantitative methods like calculation of Lyapunov exponents and their variation with wave number is used to study spatial and temporally chaotic behavior. It is shown that regular patterns with a periodic sequence in space and time and spatiotemporal chaos with irregular localized patterns are formed in different regions of unstable wave numbers.

R. P. Sharma, K. Batra, Alberto Verga. Nonlinear evolution of the modulational instability and chaos using one-dimensional Zakharov equations and a simplified model. Physics of Plasmas, 2005, 12, n° 2, pp.1 - 7. ⟨10.1063/1.1850477⟩. ⟨hal-00014808⟩