Solitons in a cold electron beam plasma

. Necessary and sufficient conditions for the existence of dissipative electron-acoustic solitons in a cold electron beam plasma with superthermal trapped electrons described by the Schamel equation are derived in this paper. Soliton solutions to the Schamel equation are constructed using formal analytical techniques which yield counter-intuitive conditions for the existence of these solutions. The existence conditions are derived in terms of system parameters and initial conditions. Computational experiments are used to validate the obtained results.


Introduction
Dissipative electron-acoustic solitary waves (EASWs) in a cold electron beam plasma with superthermal trapped electrons are investigated in [1]. It is shown in [1] that EASWs can be described by the Shamel equation [2]: where Φ is the linearized electrostatic potential; is the cold electron-neutral collision frequency; and are time and spatial coordinates; and are system parameters representing properties of the normalized electron plasma. Eq. (1) shows stronger nonlinearity (due to the Φ term) as compared to the paradigmatic KdV equation.
Eq. (1) is reduced to: In [1] by assuming the absence of the collision term 0 . It is shown in [1] that using the boundary conditions Φ → 0 and Φ → 0 at → ∞, Eq. (2) produces EASWs through the following solution:

Extended and narrowed differential equations
The concept of extended and narrowed differential equations is utilized to construct soliton solutions to Eq. (5) in this paper (this concept is discussed in detail in [3]). A short synopsis is presented below. Let us consider the following first-order ODE: Differentiation of Eq. (7) with respect to yields: Renaming the function to in Eq. (7) yields a second order ODE: It is proven in [3] that the solution to Eq. (8) coincides with the solution of Eq. (6) if and only if the initial condition satisfies the following constraint: Eq. (6) and (8) are referred to as the narrowed and extended equations respectively. In this paper, an extension of this technique is applied: the soliton solution of the narrowed equation is squared and differentiated -what yields the Schamel equation discussed in [1]. Then the condition analogous to Eq. (9) represents the existence condition for soliton solutions.

Soliton solution to the Schamel equation
Consider the following differential equation: Denoting = and differentiating Eq. (10) yields: Eq. (11) can be rewritten in respect to in the following form: 119 Combining Eqs. (10) and (12) results in the following polynomial: The values of , … , for which Eq. (11) is the extension of Eq. (10) are determined from Eq. (13): = 5 1 .
As observed in [4], the solution to Eq. (10) satisfies Eq. (11) if and only if the initial conditions satisfy the following constraint: Note that = 0 for the Schamel equation. Then special cases of Eq. (11) must be considered, because it follows from Eq. (15) that the parameters of Eq. (10) must satisfy the following relation:

Existence conditions for soliton solution to the Schamel equation
Following the assumptions in [1] we set = = 0. Without loss of generality let us assume that = − , > 0. Then, Eq. (19) is satisfied and the equation with respect to reads: with the initial condition ( ) = . Also, note that in this case = 0 and , read: As shown in [4] (Eqs. (49) and (61) in [4]) equation Eq. (20) admits the following soliton solution: Note that Eq. (22) is equivalent to: The function: is a solution to Eq. (5) if its initial conditions do satisfy = ±2 + √ . Note that Eq. (24) is a soliton solution and takes the same form as Eq. (3).

Computational experiments
It can be seen from Eq. (22) that soliton solution Eq. (24) to Eq. (5) has a singularity if > 0. Further, only the soliton solution without singularity is considered.
Let us consider the following partial differential equation, as described in [1]: Selecting the substitution = + 2 and denoting ( ) = ( + 2 ) = Φ ( , ) results in the following third-order ODE: if and only if initial conditions , of Eq. (27) do satisfy the following constraint: Soliton solution Eq. (28) is depicted in Fig. 1.
Returning to the original PDE (25), the soliton solution reads:

Conclusions
Soliton solutions to Schamel equation considered in [1] are constructed using the concept of narrowed and extended differential equations. Necessary and sufficient conditions for the existence of these solutions have been derived in the space of equation parameters and initial conditions. Ugnė Orinaitė is a Bachelor student at the Faculty of Mathematics and Natural Sciences in Kaunas University of Technology. U. Orinaite participates in the seminars and activities of "Nonlinear Systems Mathematical Research Centre" research group, accumulating experience in the problems and applications of AI-based techniques and algorithms. U. Orinaite has already gained experience in working with automatic analysis of concrete cracks, developing new mathematical algorithms for the detection of synchronization in real-world signals (heart rate variability and local earth magnetic field signals), creating new mathematical models to identify arterial pressure control circuits during physical exertion. U. Orinaite has published few papers and gave a presentation at International Conference.
Jūratė Ragulskienė has a Ph.D. degree in mechanical engineering from Vytautas Magnus University Agriculture Academy. She is Associate Professor at the Department of Mathematical Modelling of Kaunas University of Technology. Her research interests are mathematical modelling and numerical simulations.