## Abstract

Analytical solitary wave solution of the dust ion acoustic waves (DIAWs) is studied in the frame-work of Korteweg-de Vries (KdV), damped force Korteweg-de Vries (DFKdV), damped force modified Korteweg-de Vries (DFMKdV) and damped forced Zakharov-Kuznetsov (DFZK) equations in an unmagnetized collisional dusty plasma consisting of negatively charged dust grain, positively charged ions, Maxwellian distributed electrons and neutral particles. Using reductive perturbation technique (RPT), the evolution equations are obtained for DIAWs.

### Keywords

- solitary wave
- soliton
- KdV
- DKdV
- DFZK

## 1. Introduction

In the field of physics and applied mathematics research getting an exact solution of a nonlinear partial differential equation is very important. The elaboration of many complex phenomena in fluid mechanics, plasma physics, optical fibers, biology, solid-state physics, etc. is possible if analytical solutions can be obtained. Most of the differential equation arises in these field has no explicit solution as popularly known. This problem creates hindrances in the study of nonlinear phenomena and makes it time-consuming in the research of nonlinear models in the plasma and other science. However recent researches in nonlinear differential equations have seen the development of many approximate analytical solutions of partial and ordinary differential equations.

The history behind the discovery of soliton is not only interesting but also significant. In 1834 a Scottish scientist and engineer—John Scott-Russell first noticed the solitary water wave on the Edinburgh Glasgow Canal. In 1844 [1] in “Report on Waves” he accounted his examinations to the British Association. He wrote “I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished and after a chase of one or two miles I lost it in the windings of the channel. Such in the month of August 1834 was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.” He coined the word “solitary wave.” The solitary wave is called so because it often occurs as a single entity and is localized. The most important characteristics of solitary waves were unearthed after thorough study along with extensive wave-tank experiments. The following are the properties of solitary waves:

(a) These localized bell-shaped waves travel with enduring form and velocity. The speed of these waves are given by

John Scott-Russell’s study created a stir in the scientific community. His study not only initiated a debate with the prevailing knowledge of the theories of waves but also challenged the antecedent knowledge of waves. The previous study claimed that a periodic wave of finite amplitude and permanent shape are feasible only in deep water unlike Russell’s observation that the permanent profile is also possible in shallow water. Finally the stable form of solitary waves was received in scientific community with the aid of nonlinearity and dispersion. An ideal equilibrium between nonlinearity and dispersion can generate such waves.

Diederik Johannes Korteweg in 1895 [2] along with his PhD student Gustav De Vries obtained an equation from the primary equation of hydrodynamics. This equation explains shallow water waves where the existence of solitary waves was mathematically recognized. This equation is called KdV equation which is of the form

This chapter will discuss the analytical solitary wave solution of the KdV and KdV-like equations. In the study of nonlinear dispersive waves, these equations are generally seen. The KdV equation, a generic equation, is important in the study of weakly nonlinear long waves. This equation consists of a single humped wave characterized by several unique properties. The Soliton solutions of the KdV equation have been quite popular but it also not devoid of problems. The problems not only restrict to dispersion but also dissipation and interestingly these are not dominated by the KdV equation. The standard KdV equation fails to explain the development of small-amplitude solitary waves in case the particles collide in a plasma system. KdV equation with an additional damping term or the damped Korteweg-de Vries (DKdV) equation becomes handy in explaining this issue of elaborating the character of the wave. But in the presence of any critical physical situation (critical point) nonlinearity of the KdV equation disappears and the amplitude of the waves reaches infinity. To control this situation, a new nonlinear partial differential equation has to be derived that can explain the system at that critical point. This is known as the modified Korteweg-de Vries (MKdV) equation. In the presence of collisions, this equation is not also adequate and a damped MKdV equation is necessary. Also in the presence of force source term then the equation will be further modified and become DFKdV/DFMKdV.

## 2. The Korteweg-de Vries equation

Now we will derive the KdV equation from a classic plasma model, in which we consider a collision-free unmagnetized plasma consists of electrons and ions, in which ions are mobile and electrons obey the Maxwell distribution. The basic equation will be given as:

where the electrons obey Maxwell distribution, i.e.,

To write Eqs. (1)–(3) in dimensionless from we introduce the following dimensionless variables

where

To linearized (5)–(7), let us write the dependent variable as sum of equilibrium and perturbed parts, so that we write

neglecting the nonlinear term

which is the linearized form of Eq. (5).

Putting

Neglecting the nonlinear term from (10), we get

This is the linearized form of Eq. (6).

Putting

Hence Eqs. (9), (11), (12) are the linearized form of Eq. (5)–(7) respectively.

To get dispersion relation for low frequency wave let us assume that the perturbation is proportional to

So,

Putting these value in Eqs. (9), (11) and (12), we get,

Since the system (22)–(24) is a system of linear homogeneous equation so for nontrivial solutions we have

This is the **dispersion relation**.

For small

The phase velocity as

so that

In this case, we have

For long-wave as

The

Now the phase of the waves can be written as

Here

Let us consider a new stretched coordinates

where

where *x* and *t* are function of

We can express (5)–(7) in terms of

Substituting the Eqs. (31)–(34) in Eqs. (38)–(40) and collecting the lowest order

Integrating Eqs. (41)–(43) and all the variables tend to zero as

From Eq. (44)–(46) we get the phase velocity as

Substituting the Eqs.(31)–(34) in Eqs. (38)–(40) and collecting order

Differentiating Eq. (50) With respect to

Eq. (51) is known as **KdV equation**.

The steady-state solution of this KdV equation is obtained by transforming the independent variables

The steady state solution of the KdV Eq. (51) can be written as

where

## 3. Damped force KdV equation

Let us consider an unmagnetized collisional dusty plasma that contains cold inertial ions, stationary dusts with negative charge and Maxwellian electrons. The normalized ion fluid equations which include the equation of continuity, equation of momentum balance and Poisson’s equation, governing the DIAWs, are given by

where *j* = *i*,*e* for ion, electron),

### 3.1 Normalization

where

The normalized electron density is given by

### 3.2 Phase velocity and nonlinear evolution equation

We introduced the same stretched coordinates use in Eq.(31). The expansion of the dependent variables also considered as (32)–(34) with

Substituting (31)–(34) and (58)–(59) along with stretching coordinates into Eqs. (53)–(55) and equating the coefficients of lowest order of

Taking the coefficients of next higher order of **damped force KdV equation**

where

It has been noticed that the behavior of nonlinear waves changes significantly in the presence of external periodic force. It is paramount to note that the source term or forcing term due to the presence of space debris in plasmas may be of different kind, for example, Gaussian forcing term [4], hyperbolic forcing term [4], (in the form of

which is termed as **damped and forced KdV (DFKdV) equation**.

In absence of

where

In this case, it is well established that

is a conserved. For small values of

where

Differentiating Eq. (64) with respect to

Again,

Using Eq. (66) and (67) the expression of

Therefore, the solution of the Eq. (62) is

where

In Figure 3, the soliton solution of the damp force KdV equation is plotted from Eq. (65) for different values of the strength of the external periodic force

## 4. Damped KdV Burgers equation

To obtain damped KdV Burgers equation we considered an unmagnetized collisional dusty plasma which contains cold inertial ions, stationary dusts with negative charge and Maxwellian distributed electrons. The normalized ion fluid equations are as follows

where

Here normalization is taken as follows

### 4.1 Perturbation

To obtain damped KdV burger we introduced the same stretched coordinates use in Eq.(31). The expansion of the dependent variables are also considered same as (32)–(34) with

### 4.2 Phase velocity and nonlinear evolution equation

Substituting the above expansions (32)-(34) and (73)–(74) along with stretching coordinates (31) into Eqs. (69)–(71) and equating the coefficients of lowest order of

Taking the coefficients of next higher order of **DKdVB equation**

where

In absence of

where amplitude of the solitary waves

It is well established for the KdV equation that,

is a conserved quantity [7].

For small values of

where the amplitude

Differentiating Eq. (78) with respect to

where,

and

Substituting Eq. (81) and (82) into Eq. (80), we obtain

which is the Bernoulli’s equation, where

The solution of the Eq. (83) is

Therefore, the slow time dependence form of the ion acoustic solitary wave solution of the DKdVB Eq. (76) is given by (79)where.

## 5. Damped force MKdV equation

Let us consider an unmagnetized collisional dusty plasma that contains cold inertial ions, stationary dusts with negative charge and Maxwellian distributed electrons. The normalized ion fluid equations which include the equation of continuity, equation of momentum balance and Poisson’s equation, governing the DIAWs, are given by

where *j* = *i*,*e* for ion, electron),

where

The normalized

**Phase velocity and nonlinear evolution equation**

We introduced the same stretched coordinates use in Eq. (31). The expansion of the dependent variables also considered same as (32)–(34) and (58)–(59). Substituting (31)–(34) and (58)–(59) along with stretching coordinates into Eqs. (84)–(86) and equating the coefficients of lowest order of

with **DFKdV** equation

where

Now at the certain values, for example

and expand of the dependent variables same as Eqs. (32)–(34) with

Now substituting Eq. (32)–(34) and (91)–(93) into the basic Eqs. (84)–(86) and equating the coefficients of lowest order of

Equating the coefficients of next higher order of

Equating the coefficients of next higher order of

where

From Eq. (94)–(96), one can obtain the Phase velocity as

where

It has been noticed that the behavior of nonlinear waves changes significantly in the presence of external periodic force. For simplicity, we assume that

Such a form of this source function is observed in experimental situations or conditions for a particular device. Eq. (104) is termed as **damped force modified Korteweg-de Varies (DFMKdV)** equation.

In absence of **MKdV equation**.

The slow time dependence form of the ion acoustic waves solution of the DFMKdV Eq. (104) is given by,

where

The amplitude and width are as follows:

where

## 6. Damped force Zakharov-Kuznetsov equation

Let us consider a plasma model [9] consisting of cold ions, Maxwellian electrons in the presence of dust particles and the external static magnetic field

The normalized electron density is given by

where

Here the normalization is done as follows:

Here

To obtain the DFZK equation we introduce the new stretched coordinates as

where

Substituting the equations (112)-(119) into the system of Eqs. (106)-(110) equating the coefficient of

Equating the coefficient of

Considering the coefficient of

Comparing the coefficients of

Using the relationships (120)–(124), one can obtain the linear dispersion relation as

Expressing all the perturbed quantities in terms of **damped forced ZK equation** is obtained as

where

Choudhury et al. [5] studied analytical electron acoustic solitary wave (EASW) solution in the presence of periodic force for an unmagnetized plasma consisting of cold electron fluid, superthermal hot electrons and stationary ions. Motivated by the these works, here we consider the source term as

where

where

where,

The analytical solitary wave solution of the Eq. (134) as obtained in (68), is

where

## 7. Conclusions

It is clear from the structure of the solitary wave solution of the DFKdV, DFMKdV and DFZK that the soliton amplitude and width depends on the nonlinearity and dispersion of the evolution equations, which are the function of different plasma parameter involve in the consider plasma system. Also evident from the structure of the approximate analytical solution that the amplitude and the width of the soliton depends on the Mach number