Electrostatic solitary waves in plasma have long been studied using nonlinear fluid models, bridging plasma dynamics to soliton theory, including the nonlinear Schr\{o}dinger framework for envelope solitons (breathers, freak waves) [1--3]. In this study, we investigate the nonlinear modulation of electron plasma (Langmuir) waves in an unmagnetized plasma consisting of stationary positive ions and two inertial electron populations (``cool`` and ``hot``), each subject to adiabatic thermal pressure. Starting from a multi-fluid/Poisson model, the governing equations are reduced to a nonlinear Schr\{o}dinger equation (NLSE) governing the slow envelope evolution [1--2], via a (Nayfeh/Newell type) multiple scales perturbation technique. Our analysis reveals a two-fold behavior: an electron-acoustic and an optic-like (Langmuir) mode coexist; this work focuses on the latter exclusively.

Explicit analytical expressions are obtained for the dispersion and nonlinearity coefficients, $P$ and $Q$, depending on the hot-to-cool equilibrium density ratio $\delta$, the cool-to-hot temperature ratio $\theta$, and the carrier wavenumber $k$. Focusing ($PQ > 0$) and defocusing ($PQ < 0$) regimes are identified, separated by a critical wavenumber $k_{\rm cr}(\delta, \theta)$.

A detailed parametric analysis reveals that increasing $\delta$ enhances nonlinear focusing, whereas increasing $\theta$ suppresses it. In the focusing regime, bright envelope solitons, breathers, and Peregrine-type rogue wave solutions are obtained as freak wave prototypes [2, 3], while dark/grey envelope structures arise in the defocusing domain [4].

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\textbf{References}

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[1] I. Kourakis and P.K. Shukla, \textit{Nonlinear Processes in Geophysics} \textbf{12}(3), 407 (2005).

[2] I.S. Elkamash, B. Reville, N. Lazarides, I. Kourakis, \textit{Chaos, Solitons \& Fractals}, \textbf{188}, 115531 (2024).

[3] G.P. Veldes, J. Borhanian, M. McKerr, V. Saxena, D.J. Frantzeskakis \& I. Kourakis, \textit{J. Optics} \textbf{15}, 064003 (2013).

[4] R. Murhaf \textit{et al.}, in preparation.