^{1}

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The Sagdeev potential method is employed to compute the width of (Ion-acoustic) soliton propagated in a cold plasma. The computation indicates that the soliton width is a continuous function (of the Mach number M), which is expressed in terms of the Lambert Function. Despite the (fairly) complex form of the function, the numerical plotting makes sense about its changes.

Over the last decades, there has been a great deal of interest and significant progress in the study of nonlinear plasma theories and many of works and researches in plasma physics devote much attention to these theories. The nonlinear theories include a large number of effects and phenomena such as the nonlinear coherent structures as shock waves, solitary waves (solitons), vortices, etc.

The collective electrical and magnetic properties of plasmas could produce interactions that take the place of collisions and permit shocks and solitons to form. A shock wave is a sudden transition (a type of propagating disturbance) in the properties of a fluid medium (liquid or gas), involving a difference in flow velocity across a narrow (ideally, abrupt) transition. In high-energy density physics, nearly any experiment involves at least one shock wave. Such shock waves may be also produced by applying pressure to a surface or by creating a collision between two materials.

Nonlinear effects in plasmas occur when a large amplitude wave is excited by an external means. Soliton (shock) waves are formed as a result of a balance between the nonlinearity and dispersion (dissipation) of the medium. For example, the dispersion in the ion acoustic wave can be counter-balanced by nonlinearity and an ion acoustic soliton can propagate. From the mathematical point of view, solitons are the stationary solutions to the Korteweg-de Vries equation:

where

As mentioned, Equation (1) may arise in the study of such diverse physical systems as fluids and plasmas. In plasma ambient, Equation (1) may be due to the balancing between nonlinearity and dispersion and for this reason its study is of special interest.

One of the possible approaches to study of the nonlinear effects is the so-called reductive perturbation technique [

Another successful approach to study of the electrostatic solitons and shock waves has been the Sagdeev Potential (SP) or Pseudo Potential (PP) method. The SP is one particular notion that has become immensely important in soliton and shock research [

In the present work, our aim is to compute the soliton width (SW) for an ion acoustic solitary wave propagated in a cold plasma. The computation is based on analogy between pseudo-particle in PP well and real particle in conservative potential well. Knowing the angular frequency of (small) oscillation of real particle about its equilibrium position and comparing this with corresponding quantity in the plasma system, one attains a formula to compute the SW. To understand briefly, the SW is defined as spatial length corresponding to a (complete) spatial oscillation of psuedo-particle in the PP well; comparing this with the definition of period of temporal oscillation for real particle in the well, the computation is straightforward. Also, for better understanding of the changes, the graph of the width function is visualized, with the help of numerical plotting (computer algebra).

The work is organized as follows:

The model and the main calculation are presented in the next section, and conclusions are given in the last section.

We will consider a one dimensional model for propagation of ion acoustic wave in a cold plasma. The solitary wave is generated by spatial oscillation of electrostatic potential

This PP is subjected to the boundary condition

where

The form of the PP would determine whether soliton like solutions may exist or not. The conditions for the existence of solitary waves are:

where ^{1} and it is often called amplitude of the soliton (or shock). By virtue of above conditions, it is easily to show that the values of the Mach number are confined to the interval

It may be useful to plot the graph of the PP (2) versus its argument

The PP (2) satisfies the energy condition:

which is analogous to the principle of energy conservation

for a real particle of mass m moving in a (conservative) potential_{. }Hence, by the following replacements:

one can obtain similar formulas and results in the plasma system, as discussed below.

We know that for a real particle moving in the potential

and as a result, the period of oscillation motion T becomes

The natural question, then, is what conclusions can be drawn from this for our plasma system, more clearly, what quantities are given by similar formulas which are obtained by the above replacements in (5) and (6), that is

and

where we also use ^{2}.

It is easily to check that

We first remind the following definitions:

T: a time interval corresponding to one complete temporal oscillation of real particle (in the well).

On the other hand the width of the solitary wave is equal to the length that disturbance of the solitary wave takes place, or in SP terminology,

SW: a distance interval corresponding to one complete spatial oscillation of pseudo-particle (in the PP well), thus, it is natural to drawing the result

To use the above formula, we need to know the point

solving this equation for

where W is the Lambert function. Substituting

and using Equation (9), we obtain the final expression for the SW as function of the Mach Number, that is

Due to the presence of Lambert function, we have no clear idea of the behavior of the width function (13), then it is instructive to graph the function. Because of the transcendental form of the function, we have to use the numerical plotting and the corresponding graph is illustrated in

As mentioned, the soliton width is the length of a complete sweep of the pseudo-particle in the potential well and as is clear from the

the smaller one, then, the solitons corresponding to larger M have shorter width.

In order to have a quantitative measure of the changes of the width function, let us compute the ratio

where and

inserting these values in to the ratio, we obtain

This number is the average decreasing rate of the width per mach number, that is the change in the width divided by the change in Mach number. The negative sign indicates the decreasing nature of the function. As it is evident the amount of decreasing is relatively large.

In the above discussion, the width changes were described in terms of the Mach number itself, but, due to definition of the Mach number

1) A positive (negative) change in the soliton wave velocity

2) A positive (negative) change in the ratio (

In the end of this section, it is necessary to say that our ability to plot the width function

The shock and soliton waves occur most likely because of the nonlinear disturbances, namely discontinuities in various variables as energy density, pressure, temperature, etc, in plasmas and other mediums. In addition to amplitude and velocity of disturbance, the determination of spatial scales within the collision less shock or soliton may be of particular interest. For example, for a shock disturbance

with the initial condition

it can be shown [

In the case of ion acoustic wave which is mediated by electric potential in the plasma, the width of the shock or soliton may be particularly important in its relation to the width of the electrostatic potential drop across the shock.

In this work, the PP approach is employed to calculate the width of (one dimensional) ion acoustic solitary wave propagating in a cold plasma. The calculation shows that the soliton width is a continuous function of the Mach number and the function is expressed in terms of the Lambert function. Because of transcendental form of the width function, we have to use the numerical method (computer algebra) to graph of the function over the (short) allowable range of the Mach number (

Behrooz Maleko lkalami,Taimur Mohammadi,Khabat Ghamari, (2016) Ion Acoustic Soliton and the Lambert Function. Journal of Modern Physics,07,1345-1350. doi: 10.4236/jmp.2016.711120