Modulational instability and exact solutions of nonlinear cubic complex Ginzburg-Landau equation of thermodynamically open and dissipative warm ion acoustic waves system

  • A. M. Abourabia
  • R. A. Shahein
Regular Article


Using the standard reductive perturbation technique, a nonlinear cubic complex Ginzburg-Landau equation (CGL3) is derived to study the modulational instability of ion acoustic waves (IAWs) in an unmagnetized plasma consisting of warm adiabatic ions and non-thermal electrons, which form the background. The CGL3 is exactly solved by using two methods: the separation of variables and the complex tanh function which produces four solutions; the results are compared and good agreement exists in most predictions. The CGL3 admits localized envelope (solitary wave) solutions of bright and dark types. We study the effects of the thermal conductivity of ions and non-thermally distributed electrons \( \beta\) on the modulational stability. The effect of ion thermal conductivity makes the frequency \( \omega\) be complex in the nonlinear dispersion relation. In terms of the first mode \( \omega_{{1}}^{}\) , the amplitude of the wave moving in the (+ ve) x -direction decreases when any one of the parameters \( \beta\) and the temperature ratio \( \sigma\) (= T i/T eff) increases (where Teff and Ti are the effective and ion temperatures, respectively). It is found that the whole k -\( \sigma\) plane transforms to an unstable region as \( \beta\) \( \approx\) 0.85 . Referring to the internal energy formula, the order of magnitude ratios of different kinds and contributions of the internal energy changes are calculated for various sets {\( \sigma\),\( \beta\)} . The obtained results encourage us to name the unstable system --which experiences loss and gain of ions escaping out or embedding themselves into the acoustic frequency domain, collision and recently experimentally due to ionization and deionization-- a non-equilibrium thermodynamic dissipative open IAW system.


Soliton Internal Energy Entropy Production Chaos Soliton Fractal Exact Travel Wave Solution 
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Copyright information

© Società Italiana di Fisica and Springer 2011

Authors and Affiliations

  • A. M. Abourabia
    • 1
  • R. A. Shahein
    • 1
  1. 1.Department of Mathematics, Faculty of ScienceMenoufiya UniversityShebin El-KomEgypt

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