Modulation Instability and Associated Dissipative Rogue Waves and Breathers in Ultra-Cold Neutral Plasma With Adiabatic Trapped Electron and Warm Ions

Nonlinear envelope dynamics of low-frequency ion-acoustic disturbances are examined in an ultra-cold, charge-neutral electron-ion plasma, in which the ions are modeled as a warm viscous fluid, and the electrons follow an adiabatically trapped degenerate equation of state. The study is devoted to clarifying how ion kinematic viscosity, together with electron degeneracy effects, control the onset and development of modulational instability and the emergence of strongly localized ion-acoustic envelopes, including breather-like configurations and rogue waves, in such ultra-cold environments. Starting from a fluid description of the plasma and adopting an adiabatic trapped model for the degenerate electrons, a dissipative nonlinear Schrödinger (DNLS) equation governing the slow evolution of the ion-acoustic carrier envelope is derived through a reductive perturbation technique (the derivative expansion method). Within this DNLS framework, the interplay between dispersion, cubic nonlinearity, and dissipation is analyzed, and the modulational instability characteristics are obtained by tracking the influence of the degenerate electron temperature T and the ion kinematic viscosity η 0 on the instability growth rate and associated stability domains of the carrier wave. To complement the analytical stability analysis, the DNLS equation is integrated numerically by means of a second-order split-step Fourier algorithm, which allows us to follow the spatiotemporal evolution of rational breather-type excitations on a finite background, including Akhmediev and Kuznetsov-Ma breathers, and the Peregrine soliton. The numerical results show that stronger ion kinematic viscosity substantially delays the development of modulationally unstable patterns and suppresses their peak amplitudes, whereas an increase in T enhances the effective nonlinearity and facilitates modulational growth. The overall picture that emerges provides a consistent description of dissipative breather and rogue wave dynamics in ultra-cold neutral plasmas and may be of relevance for interpreting localized electrostatic activity in space and astrophysical plasmas, ultra-dense laboratory configurations, and other high-energy-density plasma settings.