Ion-acoustic Solitons in Solar Winds Plasma Out of Thermal Equilibrium

In this paper, by applying the reductive perturbation method to the plasma fluids equations and by using a non-Maxwellian distribution function which is labeled via an invariant spectralindex  and an independent parameter  as the potential degrees of freedom via perturbation, a generalized Korteweg-de Vries (KdV) equation is derived for the ion-acoustic solitons in Solar winds plasma, which involves near-equilibrium and out of thermal equilibrium states. Here, the spectralindex  describes the deviations from thermal equilibrium of plasma and itself is independent of the number of degrees of freedom of plasma. The near-equilibrium states where the spectral indices are distributed with the values of  are applied for the inner Heliosphere regions, and the far-equilibrium states which are described by the spectral indices as  that belongs to the Heliosheath regions. The analytical solution to the generalized KdV equation is calculated and its solitary wave solution is derived. Then, effects of the spectralindex , the potential degrees of freedom via perturbation , and the speed of pulse on the generalized dispersion coefficient () and generalized nonlinear coefficient () of KdV equation, and also on the structure of the ion-acoustic solitons are studied numerically.  
It is found that in the asymptotic limit of , it indicates a plasma in thermal equilibrium and the generalized KdV equation reduces to the standard KdV equation and its solitary wave solution. We show that the generalized dispersion coefficient  tends smoothly to the standard limit of  in the near-equilibrium states as , while it tends to zero in out of thermal equilibrium regions as . Furthermore, the generalized nonlinear coefficient  has negative large values in far-equilibrium states with ,while it tends smoothly to the standard limit of  in the case of an equilibrium plasma with . Moreover, the invariant spectral index has a critical value  in the far-equilibrium states, where for the generalized nonlinear coefficient  has positive values and for the generalized nonlinear coefficient  has negative values.
We found that in the vicinity of , corresponds to the escape state (where the transitions between near-equilibrium and far-equilibrium states happens), the variations of the coefficients  and  are considerable.
We also found that the generalized dispersion coefficient () and the generalized nonlinear coefficient () depend on the potential degrees of freedom via perturbation, but their dependences are not considerable.
Futhermore, depending on the values of the parameters  and , the occurrence of ion-acoustic solitons with both positive and negative potentials is possible. In the near-equilibrium states () only positive polarity solitons are possible, which is in consistence with the standard KdV theory. But, the occurrence of negative polarity solitons is predicted in the far-equilibrium states with .
Analyzing of the solitary wave profile shows that the amplitude and steepening of the ion-acoustic solitons grows in far-equilibrium states, labeled via indices . It is because of the existence of more fraction of suprathermal particles, which provide more effective interactions with the soliton and make it more prominent. Furthermore, propagation of a soliton with more speed results in a pulse with larger amplitude and narrower width, in consistence with the standard KdV theory.
Moreover, examining the results with the various degrees of freedom, shows that the amplitude and steepening of the ion-acoustic solitons decrease with an increase in the potential degrees of freedom via perturbation. It is to be noted that for a perturbed potential as in KdV theory, the potential degrees of freedom  has small values. 
Finally, we have analytically derived the amplitude () and the width () of the ion-acoustic solitons as functions of the spectral index  and the potential degrees of freedom .Then, numerical plotting of  and  with respect to  for various values of  has confirmed the mentioned results.