**Exact Solutions Expressible in Hyperbolic and Jacobi Elliptic Functions of Some Important Equations of Ion-Acoustic Waves**

A. H. Khater<sup>1</sup> and M. M. Hassan<sup>2</sup>

<sup>1</sup> *Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef* <sup>2</sup> *Mathematics Department, Faculty of Science, Minia University, El-Minia Egypt*

### **1. Introduction**

18 Will-be-set-by-IN-TECH

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Many phenomena in physics and other fields are often described by nonlinear partial differential equations (NLPDEs). The investigation of exact and numerical solutions, in particular, traveling wave solutions, for NLPDEs plays an important role in the study of nonlinear physical phenomena. These exact solutions when they exist can help one to well understand the mechanism of the complicated physical phenomena and dynamical processes modeled by these nonlinear evolution equations (NLEEs). The ion-acoustic solitary wave is one of the fundamental nonlinear wave phenomena appearing in fluid dynamics [1] and plasma physics [2, 3]. It has recently became more interesting to obtain exact analytical solutions to NLPDEs by using appropriate techniques and symbolical computer programs such as Maple or Mathematica. The capability and power of these software have increased dramatically over the past decade. Hence, direct search for exact solutions is now much more viable. Several important direct methods have been developed for obtaining traveling wave solutions to NLEEs such as the inverse scattering method [3], the tanh-function method [4], the extended tanh-function method [5] and the homogeneous balance method [6]. We assume that the exact solution is expressed by a simple expansion *u*(*x*,*t*) = *U*(*ξ*) = ∑*<sup>N</sup> <sup>i</sup>*=<sup>0</sup> *AiF<sup>i</sup>* (*ξ*) where *Ai* are constants to be determined and the function *F*(*ξ*) is defined by the solution of an auxiliary ordinary differential equation (ODE). The tanh-function method is the well known method as a direct selection of the function *F*(*ξ*) = *tanh*( *ξ*). Recently, many exact solutions expressed by various Jacobi elliptic functions (JEFs) of many NLEEs have been obtained by Jacobi elliptic function expansion method [7-10], mapping method [11, 12], F-expansion method [13], extended F-expansion method [14], the generalized Jacobi elliptic function method [15] and other methods [16-20]. Various exact solutions were obtained by using these methods, including the solitary wave solutions, shock wave solutions and periodic wave solutions.

The main steps of the F-expansion method [13] are outlined as follows:

Step 1. Use the transformation *u*(*x*, *t*) = *u*(*ξ*); *ξ* = *k*(*x* − *ωt*) + *ξ*0, *ξ*<sup>0</sup> is an arbitrary constant, and reduce a given NLPDE, say in two independent variables,

$$F(\mathbf{u}\_{\prime}\mathbf{u}\_{\prime\prime}\mathbf{u}\_{\mathbf{x}\prime}\mathbf{u}\_{\prime\prime}\mathbf{u}\_{\prime\prime}\dots) = \mathbf{0},\tag{1.1}$$

The evolution of small but finite-amplitude solitary waves, studied by means of the Korteweg-de Vries (KdV) equation, is of considerable interest in plasma dynamics. In the study of multidimensional version two type of nonlinear waves are well known, the so called Kadomtsev-Petviashvilli (KP) equation and Zakharov - Kuzentsov (ZK) equation. Employing the reductive perturbation technique on the system of equations for hydrodynamics and the

<sup>69</sup> Exact Solutions Expressible in Hyperbolic

and Jacobi Elliptic Functions of Some Important Equations of Ion-Acoustic Waves

We construct several classes of exact JEF solutions of some nonlinear evolution equations of plasma physics by using the mapping method and the F-expansion method. The rest of this chapter is organized as follows: in section 2, we present the JEF solutions to the KdV equation, combined KdV - modified KdV equation. In section 3, we apply the F-expansion method to the Schamel- KdV equation. Moreover, using the ansatz solution (1.5) and the solutions of nonlinear ODE (1.6), many exact solutions of Schamel equation, ZK equation and modified

*ut* + *αuux* + *uxxx* = 0, models a variety of nonlinear phenomena, including ion acoustic waves in plasmas, dust acoustic solitary structures in magnetized dusty plasmas, and shallow water waves. On the

*ut* + *bu*2*ux* + *uxxx* = 0,

models the dust-ion acoustic waves, electromagnetic waves in size-quantized films, ion acoustic solitons, traffic flow problems, and in other applications. The KdV equation and the modified KdV equation are completely integrable equations that have multiple-soliton solutions and possess infinite conservation quantities. The KdV equation is the earliest soliton equation that was firstly derived by Korteweg and de Vries to model the evolution of shallow water wave in 1895. In the study of the KdV equation, traveling wave solution leads to periodic solution which is called cnoidal wave solution [22, 23]. Exact solutions of KdV equation have been studied extensively since they were first found. Solitary wave solutions and periodic wave solutions were obtained for the KdV and modified KdV equations [3, 7, 22]. The JEF solutions to two kinds of KdV equations with variables coefficients have been constructed by using the method of the auxiliary equation [19]. The reductive perturbation method [24] has been employed to derive the KdV equation for small but finite amplitude electrostatic ion-acoustic waves [23, 25, 26]. The basic equations describing the system in dimensionless variables is studied by El-Labany [26] and the KdV equation for the first-order

perturbed potential has been obtained using the reductive perturbation method.

where *α*, *β* and *δ* are constants. Equation (2.1) is widely used in various fields such as quantum field theory, dust-acoustic waves, ion acoustic waves in plasmas with a negative

*ut* <sup>+</sup> *<sup>α</sup>uux* <sup>+</sup> *<sup>β</sup>u*2*ux* <sup>+</sup> *<sup>δ</sup>uxxx* <sup>=</sup> 0, *<sup>β</sup>* �<sup>=</sup> 0. (2.1)

<sup>−</sup> *<sup>ω</sup>u*� <sup>+</sup> *<sup>α</sup>uu*� <sup>+</sup> *<sup>β</sup>u*2*u*� <sup>+</sup> *<sup>δ</sup>k*<sup>2</sup> *<sup>u</sup>*��� <sup>=</sup> 0. (2.2)

We consider the combined KdV and mKdV equation [22, 27, 28]

Let *u* = *u*(*ξ*), equation (2.1) transformed to the reduced equation

ion, solid-state physics and fluid dynamics.

dynamics of plasma waves to derive such equation.

fifth order KdV equation are given in sections 4, 5, 6.

**2. The KdV and modified KdV equations**

other hand, the modified KdV equation (mKdV)

The Korteweg de-Vries (KdV) equation

to the (ODE)

$$G(u, u', u'', \ldots) = 0, \quad u' = \frac{du}{d\tilde{\xi}}.\tag{1.2}$$

In general, the left hand side of Eq. (1.1) is a polynomial in *u* and its various derivatives. Step 2. The F-expansion method gives the solution of (1.1) in the form

$$\mu(\mathbf{x},t) = \mu(\xi) = \sum\_{i=0}^{N} a\_i F^i(\xi), \quad a\_N \neq 0,\tag{1.3}$$

where *ai* (*i* = 0, 1, 2, ..., *N*) are constants to be determined and *F*(*ξ*) satisfies the first order nonlinear ODE in the form

$$\left(F'(\xi)\right)^2 = q\_0 + q\_2 F^2(\xi) + q\_4 F^4(\xi),\tag{1.4}$$

where *q*0, *q*<sup>2</sup> and *q*<sup>4</sup> are constants and *N* in Eq. (1.3) is a positive integer that can be determined by balancing the nonlinear term(s) and the highest order derivatives in Eq. (1.1).

Step 3. Substituting the F-expansion (1.3) into (1.2) and using (1.4); setting each coefficient of the polynomial to zero yields a system of algebraic equations involving *a*0, *a*1, ...*aN*, *k* and *ω*. Step 4. Solving these equations, probably with the aid of Mathematica or Maple, then

*a*0, *a*1, ...*aN*, *k* and *ω* can be expressed by *q*0, *q*2, *q*4. Step 5. Substituting these results into F-expansion (1.3), then a general form of traveling wave solution of the NLPDE (1.1) can be obtained. Many solutions of equation (1.4) have been reported in [13, 14]. Substituting the values of *q*0, *q*2, *q*<sup>4</sup> and the corresponding JEF solution *F*(*ξ*) into the general form of solution, we may get several classes of exact solutions

of equations (1.1) involving JEFs. Also, we give a brief description of the mapping method to seek the traveling wave solutions of (1.1) in the form *u*(*x*, *t*) = *u*(*η*), *η* = *kx* − *ω t* + *η*0, *η*<sup>0</sup> is an arbitrary constant. Thus, Eq. (1.1) reduces to Eq. (1.2), whose solution can be express in the form

$$\mu(\eta) = \sum\_{i=0}^{n} A\_i f^i(\eta),\tag{1.5}$$

where *n* is a balancing number, *Ai* are constants to be determined and *f*(*η*) satisfies the nonlinear ODE

$$f^{\prime 2}(\eta) = 2 \, p \, f(\eta) + q \, f^2(\eta) + \frac{2}{3} r \, f^3(\eta). \tag{1.6}$$

Here *p*, *q* and *r* are constants. After substituting Eq. (1.5) into the ODE (1.2) and using Eq. (1.6), the constants *Ai*, *k* and *ω* may be determined. By using the solutions of auxiliary nonlinear equation (1.6), many JEF solutions of NLEEs have been obtained [19, 20].

The JEFs sn(*ξ*) = sn(*ξ*, *m*), cn(*ξ*) = cn( *ξ*, *m*) and dn(*ξ*) = dn(*ξ*, *m*) are double periodic and have the following properties:

$$\text{sn}^2(\mathfrak{f}) + \text{cn}^2(\mathfrak{f}) = 1, \quad \text{dn}^2(\mathfrak{f}) + m^2 \text{sn}^2(\mathfrak{f}) = 1.$$

In the limit *m* −→ 1, the JEFs degenerate to the hyperbolic functions, i.e.,

$$\mathrm{sn}(\mathfrak{J},1) \longrightarrow \mathrm{tanh}(\mathfrak{J}), \; \mathrm{cn}(\mathfrak{J},1) \longrightarrow \mathrm{sech}(\mathfrak{J}), \; \mathrm{dn}(\mathfrak{J},1) \longrightarrow \mathrm{sech}(\mathfrak{J}).$$

Detailed explanations about JEFs can be found in [21].

Some of the nonlinear models in fluids, plasma and dust plasma are described by canonical models and include the Korteweg-de Vries (KdV) and the modified KdV equations [22-25]. The evolution of small but finite-amplitude solitary waves, studied by means of the Korteweg-de Vries (KdV) equation, is of considerable interest in plasma dynamics. In the study of multidimensional version two type of nonlinear waves are well known, the so called Kadomtsev-Petviashvilli (KP) equation and Zakharov - Kuzentsov (ZK) equation. Employing the reductive perturbation technique on the system of equations for hydrodynamics and the dynamics of plasma waves to derive such equation.

We construct several classes of exact JEF solutions of some nonlinear evolution equations of plasma physics by using the mapping method and the F-expansion method. The rest of this chapter is organized as follows: in section 2, we present the JEF solutions to the KdV equation, combined KdV - modified KdV equation. In section 3, we apply the F-expansion method to the Schamel- KdV equation. Moreover, using the ansatz solution (1.5) and the solutions of nonlinear ODE (1.6), many exact solutions of Schamel equation, ZK equation and modified fifth order KdV equation are given in sections 4, 5, 6.

### **2. The KdV and modified KdV equations**

The Korteweg de-Vries (KdV) equation

2 Will-be-set-by-IN-TECH

In general, the left hand side of Eq. (1.1) is a polynomial in *u* and its various derivatives.

*N* ∑ *i*=0 *aiF<sup>i</sup>*

where *ai* (*i* = 0, 1, 2, ..., *N*) are constants to be determined and *F*(*ξ*) satisfies the first order

where *q*0, *q*<sup>2</sup> and *q*<sup>4</sup> are constants and *N* in Eq. (1.3) is a positive integer that can be determined

Step 3. Substituting the F-expansion (1.3) into (1.2) and using (1.4); setting each coefficient of the polynomial to zero yields a system of algebraic equations involving *a*0, *a*1, ...*aN*, *k* and *ω*. Step 4. Solving these equations, probably with the aid of Mathematica or Maple, then

Step 5. Substituting these results into F-expansion (1.3), then a general form of traveling wave solution of the NLPDE (1.1) can be obtained. Many solutions of equation (1.4) have been reported in [13, 14]. Substituting the values of *q*0, *q*2, *q*<sup>4</sup> and the corresponding JEF solution *F*(*ξ*) into the general form of solution, we may get several classes of exact solutions

Also, we give a brief description of the mapping method to seek the traveling wave solutions of (1.1) in the form *u*(*x*, *t*) = *u*(*η*), *η* = *kx* − *ω t* + *η*0, *η*<sup>0</sup> is an arbitrary constant. Thus, Eq.

> *n* ∑ *i*=0

where *n* is a balancing number, *Ai* are constants to be determined and *f*(*η*) satisfies the

Here *p*, *q* and *r* are constants. After substituting Eq. (1.5) into the ODE (1.2) and using Eq. (1.6), the constants *Ai*, *k* and *ω* may be determined. By using the solutions of auxiliary

The JEFs sn(*ξ*) = sn(*ξ*, *m*), cn(*ξ*) = cn( *ξ*, *m*) and dn(*ξ*) = dn(*ξ*, *m*) are double periodic and

sn2(*ξ*) + cn2(*ξ*) = 1, dn2(*ξ*) + *m*<sup>2</sup> sn2(*ξ*) = 1.

sn(*ξ*, 1) −→ tanh(*ξ*), cn(*ξ*, 1) −→ sech(*ξ*), dn(*ξ*, 1) −→ sech(*ξ*).

Some of the nonlinear models in fluids, plasma and dust plasma are described by canonical models and include the Korteweg-de Vries (KdV) and the modified KdV equations [22-25].

(*η*) = <sup>2</sup> *p f*(*η*) + *q f* <sup>2</sup>(*η*) + <sup>2</sup>

nonlinear equation (1.6), many JEF solutions of NLEEs have been obtained [19, 20].

In the limit *m* −→ 1, the JEFs degenerate to the hyperbolic functions, i.e.,

*Ai f <sup>i</sup>*

by balancing the nonlinear term(s) and the highest order derivatives in Eq. (1.1).

, *<sup>u</sup>*��, ...) = 0, *<sup>u</sup>*� <sup>=</sup> *du*

*dξ*

(*ξ*))<sup>2</sup> = *q*<sup>0</sup> + *q*2*F*2(*ξ*) + *q*4*F*4(*ξ*), (1.4)

. (1.2)

(*ξ*), *aN* �= 0, (1.3)

(*η*), (1.5)

<sup>3</sup> *r f* <sup>3</sup>(*η*). (1.6)

*G*(*u*, *u*�

Step 2. The F-expansion method gives the solution of (1.1) in the form

*u*(*x*, *t*) = *u*(*ξ*) =

(*F*�

(1.1) reduces to Eq. (1.2), whose solution can be express in the form

*f* �2

Detailed explanations about JEFs can be found in [21].

*u*(*η*) =

*a*0, *a*1, ...*aN*, *k* and *ω* can be expressed by *q*0, *q*2, *q*4.

to the (ODE)

nonlinear ODE in the form

of equations (1.1) involving JEFs.

have the following properties:

nonlinear ODE

$$
\mu\_t + \alpha \mu \mu\_x + \mu\_{\text{xxx}} = 0,
$$

models a variety of nonlinear phenomena, including ion acoustic waves in plasmas, dust acoustic solitary structures in magnetized dusty plasmas, and shallow water waves. On the other hand, the modified KdV equation (mKdV)

$$
\mu\_t + b\mu^2 \mu\_x + \mu\_{xxx} = 0,
$$

models the dust-ion acoustic waves, electromagnetic waves in size-quantized films, ion acoustic solitons, traffic flow problems, and in other applications. The KdV equation and the modified KdV equation are completely integrable equations that have multiple-soliton solutions and possess infinite conservation quantities. The KdV equation is the earliest soliton equation that was firstly derived by Korteweg and de Vries to model the evolution of shallow water wave in 1895. In the study of the KdV equation, traveling wave solution leads to periodic solution which is called cnoidal wave solution [22, 23]. Exact solutions of KdV equation have been studied extensively since they were first found. Solitary wave solutions and periodic wave solutions were obtained for the KdV and modified KdV equations [3, 7, 22]. The JEF solutions to two kinds of KdV equations with variables coefficients have been constructed by using the method of the auxiliary equation [19]. The reductive perturbation method [24] has been employed to derive the KdV equation for small but finite amplitude electrostatic ion-acoustic waves [23, 25, 26]. The basic equations describing the system in dimensionless variables is studied by El-Labany [26] and the KdV equation for the first-order perturbed potential has been obtained using the reductive perturbation method. We consider the combined KdV and mKdV equation [22, 27, 28]

$$
\alpha u\_t + \alpha u u\_x + \beta u^2 u\_x + \delta u\_{xxx} = 0, \quad \beta \neq 0. \tag{2.1}
$$

where *α*, *β* and *δ* are constants. Equation (2.1) is widely used in various fields such as quantum field theory, dust-acoustic waves, ion acoustic waves in plasmas with a negative ion, solid-state physics and fluid dynamics.

Let *u* = *u*(*ξ*), equation (2.1) transformed to the reduced equation

$$-\omega u' + \alpha u u' + \beta u^2 u' + \delta k^2 u'''' = 0.\tag{2.2}$$

where *a*<sup>0</sup> and *a*<sup>1</sup> are constants to be determined and *F*(*ξ*) is a solution of Eq. (1.4). Substituting Eq. (3.3) into Eq. (3.2) and equating the coefficients of the like powers of F to zero, yields a set

<sup>71</sup> Exact Solutions Expressible in Hyperbolic

<sup>1</sup> <sup>+</sup> <sup>12</sup>*δk*2*q*4]*a*<sup>1</sup> <sup>=</sup> 0,

Solving these algebraic equations, we gave a general form of traveling wave solutions of Eq.

<sup>1</sup> <sup>+</sup> <sup>6</sup>*δk*2*a*0*q*4]*a*<sup>1</sup> <sup>=</sup> 0,

<sup>0</sup> <sup>+</sup> <sup>4</sup>*δk*2*q*2]*a*<sup>1</sup> <sup>=</sup> 0,

*F*(*ξ*) 2

(*x* + <sup>16</sup>*α*<sup>2</sup>

(*x* + <sup>16</sup>*α*<sup>2</sup>

(*x* + <sup>16</sup>*α*<sup>2</sup>

<sup>75</sup>*<sup>β</sup> t*) + *ξ*<sup>0</sup>

<sup>75</sup>*<sup>β</sup> t*) + *ξ*<sup>0</sup>

<sup>75</sup>*<sup>β</sup> t*) + *ξ*<sup>0</sup>

<sup>75</sup>*<sup>β</sup> t*) + *ξ*<sup>0</sup>

<sup>75</sup>*<sup>β</sup> t*) + *ξ*<sup>0</sup>

<sup>2</sup>

<sup>2</sup>

<sup>2</sup>

<sup>2</sup>

<sup>2</sup>

, *βδ* < 0,

, *βδ* > 0,

(3.4)

. (3.5)

, *βδ* < 0, (3.6)

, *βδ* > 0, (3.7)

, *βδ* > 0, (3.8)

(3.9)

<sup>0</sup> <sup>+</sup> *<sup>δ</sup>k*2*q*2]*a*<sup>0</sup> <sup>=</sup> 0.

[*βa*<sup>2</sup>

and Jacobi Elliptic Functions of Some Important Equations of Ion-Acoustic Waves

<sup>1</sup> <sup>+</sup> <sup>3</sup>*βa*0*a*<sup>2</sup>

[−*<sup>ω</sup>* <sup>+</sup> <sup>2</sup>*αa*<sup>0</sup> <sup>+</sup> <sup>3</sup>*βa*<sup>2</sup>

[−*<sup>ω</sup>* <sup>+</sup> *<sup>α</sup>a*<sup>0</sup> <sup>+</sup> *<sup>β</sup>a*<sup>2</sup>

 1 ± −2*q*<sup>4</sup> *q*2

When *<sup>q</sup>*<sup>0</sup> <sup>=</sup> 1, *<sup>q</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> <sup>−</sup> *<sup>m</sup>*2, *<sup>q</sup>*<sup>4</sup> <sup>=</sup> *<sup>m</sup>*2, solutions of Eq. (1.4) is *<sup>F</sup>*(*ξ*) = sn*<sup>ξ</sup>* , we have

5

5

5

*α* 5

> 2*α* 5

2*α*

√−<sup>6</sup>*δβ*(*m*<sup>2</sup>+1)

If *<sup>q</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>m</sup>*2, *<sup>q</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup>*m*<sup>2</sup> <sup>−</sup> 1, *<sup>q</sup>*<sup>4</sup> <sup>=</sup> <sup>−</sup>*m*2, *<sup>F</sup>*(*ξ*) = cn*ξ*, thus yields the exact solutions of Eq.

2*α*

<sup>√</sup>6*δβ*(2*m*<sup>2</sup>−1)

If *<sup>q</sup>*<sup>0</sup> <sup>=</sup> *<sup>m</sup>*<sup>2</sup> <sup>−</sup> 1, *<sup>q</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup> <sup>−</sup> *<sup>m</sup>*2, *<sup>q</sup>*<sup>4</sup> <sup>=</sup> <sup>−</sup>1, the solution of Eq (1.4) is *<sup>F</sup>*(*ξ*) = dn*ξ*. So, we obtained

2*α*

<sup>√</sup>6*δβ*(2−*m*<sup>2</sup>)

Many types of JEF solutions of Eq. (3.1) are given [30]. As *m* −→ 1, Eqs. (3.6)-(3.8) degenerate

√−<sup>3</sup>*δβ* (*<sup>x</sup>* <sup>+</sup> <sup>16</sup>*α*<sup>2</sup>

The solitary wave solutions (3.9) in terms of tanh are equivalent to the solutions given in [31]. The JEF solutions of (3.1) may be describe various features of waves and may be helpful in

<sup>√</sup>6*δβ* (*<sup>x</sup>* <sup>+</sup> <sup>16</sup>*α*<sup>2</sup>

*<sup>u</sup>* <sup>=</sup> <sup>4</sup>*α*<sup>2</sup> 25*β*<sup>2</sup>

Therefore, we obtained in [30] the JEF solutions of Eq. (3.1) as follows:

 <sup>2</sup>*m*<sup>2</sup> *<sup>m</sup>*<sup>2</sup>+<sup>1</sup> sn

 <sup>2</sup>*m*<sup>2</sup> <sup>2</sup>*m*<sup>2</sup>−<sup>1</sup> cn

<sup>2</sup>

<sup>2</sup>−*m*<sup>2</sup> dn

<sup>1</sup> <sup>±</sup> tanh

<sup>1</sup> <sup>±</sup> <sup>√</sup>2 sech

[*αa*<sup>2</sup>

of algebraic equations for *a*0, *a*1, *k* and *ω*:

*<sup>u</sup>*<sup>1</sup> <sup>=</sup> <sup>4</sup>*α*<sup>2</sup> 25*β*<sup>2</sup> 1 ±

*<sup>u</sup>*<sup>2</sup> <sup>=</sup> <sup>4</sup>*α*<sup>2</sup> 25*β*<sup>2</sup> 1 ±

*<sup>u</sup>*<sup>3</sup> <sup>=</sup> <sup>4</sup>*α*<sup>2</sup> 25*β*<sup>2</sup> 1 ±

the exact solutions of Eq. (3.1) in the form

*<sup>u</sup>*<sup>4</sup> <sup>=</sup> <sup>4</sup>*α*<sup>2</sup> 25*β*<sup>2</sup> 

*<sup>u</sup>*<sup>5</sup> = <sup>4</sup>*α*<sup>2</sup> 25*β*<sup>2</sup> 

understanding the problems in ion acoustic waves.

(3.1)

(3.1)

to

Balancing *u*��� with *u*2*u*� yields *N* = 1, so the *F*-expansion method gives

$$
\mu(\mathbf{x}, t) = a\_0 + a\_1 F(\mathcal{G}). \tag{2.3}
$$

Substituting (2.3) into (2.2) and equating the coefficients of like powers of *F*(*ξ*) to zero, we obtain a set of algebraic equations. Solving these algebraic equations, we obtain the exact solutions of (2.1) as follows:

When *<sup>q</sup>*<sup>0</sup> <sup>=</sup> 1, *<sup>q</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> <sup>−</sup> *<sup>m</sup>*2, *<sup>q</sup>*<sup>4</sup> <sup>=</sup> *<sup>m</sup>*2, solutions of Eq. (1.4) is *<sup>F</sup>*(*ξ*) = sn*<sup>ξ</sup>* , we have

$$u = -\frac{a}{2\beta} \pm k \sqrt{\frac{-6m^2\delta}{\beta}} \operatorname{sn} \left( k(\mathbf{x} + (\frac{a^2 + 4\beta\delta k^2(m^2 + 1)}{4\beta})t) + \mathfrak{f}\_0 \right), \tag{2.4}$$

If *<sup>q</sup>*<sup>0</sup> <sup>=</sup> *<sup>m</sup>*<sup>2</sup> <sup>−</sup> 1, *<sup>q</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup> <sup>−</sup> *<sup>m</sup>*2, *<sup>q</sup>*<sup>4</sup> <sup>=</sup> <sup>−</sup>1, the solution of Eq (1.4) is *<sup>F</sup>*(*ξ*) = dn*ξ*. Thus, we obtain the periodic wave solutions of Eq. (2.1)

$$u = -\frac{a}{2\beta} \pm k \sqrt{\frac{6\delta}{\beta}} \operatorname{dn}(k(\mathbf{x} + \frac{a^2 - 4\beta\delta k^2(2 - m^2)}{4\beta}t) + \xi\_0),\tag{2.5}$$

Selecting the values of the *q*0, *q*<sup>2</sup> and *q*<sup>4</sup> of equation (1.4) and the corresponding function *F*, we can construct various JEF solutions of (2.1). Other JEF solutions are omitted here for simplicity. If we put *α* = 0 in (2.4), we get the periodic solution of the modified KdV equation which coincides with that given by Liu et al. [7]. Moreover, the solutions (2.5) to equation (2.1) given in [28] are recovered. With *m* −→ 1 in (2.4) , (2.5), the solitary wave solutions to (2.1) given in [7, 27, 28] are also recovered.

We notice that the solutions of the KdV equation cannot obtain from (2.4) and (2.5) as *β* = 0. In this case, the general form of cnoidal wave solutions of the KdV equation are given by

$$u(\mathbf{x},t) = -\frac{3\omega q\_4}{\alpha q\_2} F^2(\xi), \quad \tilde{\xi} = \sqrt{\frac{\omega}{4\delta q\_2}} \left(\mathbf{x} - \omega t\right) + \tilde{\xi}\_0. \tag{2.6}$$

Thus we can obtain abundant cnoidal wave solutions of the KdV equation in terms of JEFs. Some periodic wave solutions of the KdV equation and modified KdV equation have been studied in [7,23, 28]. As *m* −→ 1, these solutions will degenerate into the corresponding solitary wave solutions.

### **3. The JEF solutions of Schamel- KdV equation**

We consider the Schamel- KdV equation [29, 30]

$$(u\_t + (au^{1/2} + \beta u)u\_x + \delta u\_{xxx} = 0, \quad \beta \neq 0 \tag{3.1}$$

where *α*, *β* and *δ* are constants and *u* is the wave potential. In order to find the periodic wave solution of (3.1), we use the transformations *u* = *v*2, *v*(*x*, *t*) = *V*(*ξ*); *ξ* = *k*(*x* − *ωt*) + *ξ*0, then (2.7) becomes

$$-\omega V V' + (aV^2 + \beta V^3)V' + \delta k^2 [V V''^\prime + 3V^\prime V'^\prime] = 0. \tag{3.2}$$

The balancing procedure implies that *N* = 1. Therefore, the *F*-expansion method gives the solution

$$V(\mathbf{x}, t) = V(\boldsymbol{\xi}) = a\_0 + a\_1 F(\boldsymbol{\xi}),\tag{3.3}$$

4 Will-be-set-by-IN-TECH

Substituting (2.3) into (2.2) and equating the coefficients of like powers of *F*(*ξ*) to zero, we obtain a set of algebraic equations. Solving these algebraic equations, we obtain the exact

If *<sup>q</sup>*<sup>0</sup> <sup>=</sup> *<sup>m</sup>*<sup>2</sup> <sup>−</sup> 1, *<sup>q</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup> <sup>−</sup> *<sup>m</sup>*2, *<sup>q</sup>*<sup>4</sup> <sup>=</sup> <sup>−</sup>1, the solution of Eq (1.4) is *<sup>F</sup>*(*ξ*) = dn*ξ*. Thus, we obtain

Selecting the values of the *q*0, *q*<sup>2</sup> and *q*<sup>4</sup> of equation (1.4) and the corresponding function *F*, we can construct various JEF solutions of (2.1). Other JEF solutions are omitted here for simplicity. If we put *α* = 0 in (2.4), we get the periodic solution of the modified KdV equation which coincides with that given by Liu et al. [7]. Moreover, the solutions (2.5) to equation (2.1) given in [28] are recovered. With *m* −→ 1 in (2.4) , (2.5), the solitary wave solutions to (2.1)

We notice that the solutions of the KdV equation cannot obtain from (2.4) and (2.5) as *β* = 0. In this case, the general form of cnoidal wave solutions of the KdV equation are given by

Thus we can obtain abundant cnoidal wave solutions of the KdV equation in terms of JEFs. Some periodic wave solutions of the KdV equation and modified KdV equation have been studied in [7,23, 28]. As *m* −→ 1, these solutions will degenerate into the corresponding

In order to find the periodic wave solution of (3.1), we use the transformations *u* = *v*2,

The balancing procedure implies that *N* = 1. Therefore, the *F*-expansion method gives the

<sup>−</sup> *<sup>ω</sup>V V*� + (*αV*<sup>2</sup> <sup>+</sup> *<sup>β</sup>V*3)*V*� <sup>+</sup> *<sup>δ</sup>k*2[*VV*��� <sup>+</sup> <sup>3</sup>*V*�

 *ω* 4*δq*<sup>2</sup>

*ut* + (*αu*1/2 <sup>+</sup> *<sup>β</sup>u*)*ux* <sup>+</sup> *<sup>δ</sup>uxxx* <sup>=</sup> 0, *<sup>β</sup>* �<sup>=</sup> <sup>0</sup> (3.1)

*V*(*x*, *t*) = *V*(*ξ*) = *a*<sup>0</sup> + *a*1*F*(*ξ*), (3.3)

*F*2(*ξ*), *ξ* =

*<sup>k</sup>*(*<sup>x</sup>* + ( *<sup>α</sup>*<sup>2</sup> <sup>+</sup> <sup>4</sup>*βδk*2(*m*<sup>2</sup> <sup>+</sup> <sup>1</sup>)

*<sup>α</sup>*<sup>2</sup> <sup>−</sup> <sup>4</sup>*βδk*2(<sup>2</sup> <sup>−</sup> *<sup>m</sup>*2)

When *<sup>q</sup>*<sup>0</sup> <sup>=</sup> 1, *<sup>q</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> <sup>−</sup> *<sup>m</sup>*2, *<sup>q</sup>*<sup>4</sup> <sup>=</sup> *<sup>m</sup>*2, solutions of Eq. (1.4) is *<sup>F</sup>*(*ξ*) = sn*<sup>ξ</sup>* , we have

*<sup>β</sup>* dn(*k*(*<sup>x</sup>* <sup>+</sup>

*u*(*x*, *t*) = *a*<sup>0</sup> + *a*1*F*(*ξ*). (2.3)

<sup>4</sup>*<sup>β</sup>* )*t*) + *<sup>ξ</sup>*<sup>0</sup>

<sup>4</sup>*<sup>β</sup> <sup>t</sup>*) + *<sup>ξ</sup>*0), (2.5)

(*x* − *ωt*) + *ξ*0. (2.6)

*V*��] = 0. (3.2)

, (2.4)

Balancing *u*��� with *u*2*u*� yields *N* = 1, so the *F*-expansion method gives

solutions of (2.1) as follows:

*<sup>u</sup>* <sup>=</sup> <sup>−</sup> *<sup>α</sup>*

the periodic wave solutions of Eq. (2.1)

given in [7, 27, 28] are also recovered.

solitary wave solutions.

solution

*<sup>u</sup>* <sup>=</sup> <sup>−</sup> *<sup>α</sup>*

<sup>2</sup>*<sup>β</sup>* <sup>±</sup> *<sup>k</sup>*

<sup>2</sup>*<sup>β</sup>* <sup>±</sup> *<sup>k</sup>*

*<sup>u</sup>*(*x*, *<sup>t</sup>*) = <sup>−</sup>3*ωq*<sup>4</sup>

**3. The JEF solutions of Schamel- KdV equation**

where *α*, *β* and *δ* are constants and *u* is the wave potential.

*v*(*x*, *t*) = *V*(*ξ*); *ξ* = *k*(*x* − *ωt*) + *ξ*0, then (2.7) becomes

We consider the Schamel- KdV equation [29, 30]

*α q*<sup>2</sup>

−6*m*2*<sup>δ</sup> <sup>β</sup>* sn 

 6*δ* where *a*<sup>0</sup> and *a*<sup>1</sup> are constants to be determined and *F*(*ξ*) is a solution of Eq. (1.4). Substituting Eq. (3.3) into Eq. (3.2) and equating the coefficients of the like powers of F to zero, yields a set of algebraic equations for *a*0, *a*1, *k* and *ω*:

$$\begin{aligned} [\beta a\_1^2 + 12\delta k^2 q\_4] a\_1 &= 0, \\ [\alpha a\_1^2 + 3\beta a\_0 a\_1^2 + 6\delta k^2 a\_0 q\_4] a\_1 &= 0, \\ [-\omega + 2\alpha a\_0 + 3\beta a\_0^2 + 4\delta k^2 q\_2] a\_1 &= 0, \\ [-\omega + \alpha a\_0 + \beta a\_0^2 + \delta k^2 q\_2] a\_0 &= 0. \end{aligned} \tag{3.4}$$

Solving these algebraic equations, we gave a general form of traveling wave solutions of Eq. (3.1)

$$\mu = \frac{4a^2}{25\beta^2} \left[ 1 \pm \sqrt{\frac{-2q\_4}{q\_2}} F(\xi) \right]^2. \tag{3.5}$$

Therefore, we obtained in [30] the JEF solutions of Eq. (3.1) as follows: When *<sup>q</sup>*<sup>0</sup> <sup>=</sup> 1, *<sup>q</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> <sup>−</sup> *<sup>m</sup>*2, *<sup>q</sup>*<sup>4</sup> <sup>=</sup> *<sup>m</sup>*2, solutions of Eq. (1.4) is *<sup>F</sup>*(*ξ*) = sn*<sup>ξ</sup>* , we have

$$\mu\_1 = \frac{4a^2}{25\beta^2} \left[ 1 \pm \sqrt{\frac{2m^2}{m^2+1}} \text{sn} \left( \frac{2a}{5\sqrt{-6\beta(m^2+1)}} (\mathbf{x} + \frac{16a^2}{75\beta}\mathbf{t}) + \xi\_0 \right) \right]^2, \quad \beta\delta < 0,\tag{3.6}$$

If *<sup>q</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>m</sup>*2, *<sup>q</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup>*m*<sup>2</sup> <sup>−</sup> 1, *<sup>q</sup>*<sup>4</sup> <sup>=</sup> <sup>−</sup>*m*2, *<sup>F</sup>*(*ξ*) = cn*ξ*, thus yields the exact solutions of Eq. (3.1)

$$\mu\_2 = \frac{4a^2}{25\tilde{\rho}^2} \left[ 1 \pm \sqrt{\frac{2m^2}{2m^2 - 1}} \text{cn} \left( \frac{2a}{5\sqrt{6\beta(2m^2 - 1)}} (\mathbf{x} + \frac{16a^2}{75\tilde{\rho}}t) + \tilde{\xi}\_0 \right) \right]^2, \quad \beta\delta > 0,\tag{3.7}$$

If *<sup>q</sup>*<sup>0</sup> <sup>=</sup> *<sup>m</sup>*<sup>2</sup> <sup>−</sup> 1, *<sup>q</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup> <sup>−</sup> *<sup>m</sup>*2, *<sup>q</sup>*<sup>4</sup> <sup>=</sup> <sup>−</sup>1, the solution of Eq (1.4) is *<sup>F</sup>*(*ξ*) = dn*ξ*. So, we obtained the exact solutions of Eq. (3.1) in the form

$$\mu\_3 = \frac{4a^2}{25\beta^2} \left[ 1 \pm \sqrt{\frac{2}{2 - m^2}} \text{dn} \left( \frac{2a}{5\sqrt{6\beta(2 - m^2)}} (\mathbf{x} + \frac{16a^2}{75\beta}\mathbf{t}) + \xi\_0 \right) \right]^2, \quad \beta\delta > 0,\tag{3.8}$$

Many types of JEF solutions of Eq. (3.1) are given [30]. As *m* −→ 1, Eqs. (3.6)-(3.8) degenerate to

$$\begin{split} u\_{4} &= \frac{4a^{2}}{25\beta^{2}} \Biggl[ 1 \pm \tanh\left(\frac{\alpha}{5\sqrt{-3\delta\beta}}(\mathbf{x} + \frac{16a^{2}}{75\beta}t) + \xi\_{0}\right) \Biggr]^{2}, \quad \beta\delta < 0, \\ u\_{5} &= \frac{4a^{2}}{25\beta^{2}} \Biggl[ 1 \pm \sqrt{2} \operatorname{sech}\left(\frac{2\alpha}{5\sqrt{6\delta\beta}}(\mathbf{x} + \frac{16a^{2}}{75\beta}t) + \xi\_{0}\right) \Biggr]^{2}, \quad \beta\delta > 0, \end{split} \tag{3.9}$$

The solitary wave solutions (3.9) in terms of tanh are equivalent to the solutions given in [31]. The JEF solutions of (3.1) may be describe various features of waves and may be helpful in understanding the problems in ion acoustic waves.

*<sup>u</sup>*6(*x*, *<sup>t</sup>*) = <sup>225</sup>*δ*<sup>2</sup> *<sup>k</sup>*<sup>4</sup>

Consider the modified KP equation

The equation

[43].

in the form

the modified ZK equation:

*u* = *m*

*u* = *m*

 6*ω* (*m*<sup>2</sup>+1)*<sup>β</sup>* sn(

6*ω*

(2−*m*<sup>2</sup>)*<sup>β</sup>* dn(

In the following we apply the mapping method to the ZK equation

In this case, we have *n* = 1. Thus Eq. (5.3) has a solution in the form

4

and Jacobi Elliptic Functions of Some Important Equations of Ion-Acoustic Waves

auxiliary equation (1.6) to find the solutions of equation (4.9) (see [39]).

**5. The ZK equation and modified ZK equation**

 1 −

The KdV equation in two dimensions, known as Kadomtsev Petviashivili (KP) equation [32], was derived for ion-acoustic waves in a non magnetized plasma by Kako and Rowlands [33]. Therefore the modified KP equation containing a square root nonlinearity is very attractive model for the study of ion-acoustic waves in plasma and dusty plasma [34- 36]. Extensive work has been devoted to the study of nonlinear waves associated with the dust ion-acoustic waves, particularly the dust ion-acoustic solitary and shock waves in dusty plasmas in which dust particles are stationary and provide only the neutrality [37]. The KP equation is derived [38] for the propagation of nonlinear waves in warm dusty plasmas with variable dust charge, two-temperature ions and nonthermal electrons by using the reductive perturbation theory.

<sup>73</sup> Exact Solutions Expressible in Hyperbolic

where *α* and *β* are constants. The modified KP equation (4.9) for ion-acoustic waves in a multi species plasma consisting of non-isothermal electrons have been derived by Chakraborty and Das [34]. We applied the mapping method with the ansatz solution (4.3) and the solutions of

is the modified ZK in (2+1) dimensions which is a model for acoustic plasma waves [40, 41]. The ZK equation was first derived for describing weakly nonlinear ion- acoustic waves in a strongly magnetized lossless plasma in two dimension [41]. The ZK equation and modified ZK equation possess traveling wave structures [28, 42]. Peng [42] studied the exact solutions of ZK equation by using extended mapping method. Various types of solutions of Schamel-KdV equation and modified ZK equation arising in plasma and dust plasma are presented in

We apply the *F*-expansion method to the modified ZK equation. Thus, Eq. (5.1) has a solution

*u*(*ξ*) = *a*<sup>0</sup> + *a*<sup>1</sup> *F*(*ξ*), *ξ* = *k*(*x* + *ly* − *ω t*) + *ξ*0. Substituting this equation into Eq. (5.1), we obtain the following classes of exact solutions of

<sup>−</sup>*<sup>ω</sup>*

*ω*

*u*(*η*) = *A*<sup>0</sup> + *A*<sup>1</sup> *f*(*η*), *η* = *kx* + *ly* − *ω t* + *η*0.

 tanh (*kx*−<sup>4</sup> *<sup>δ</sup>k*<sup>3</sup> *<sup>t</sup>*+*η*0) <sup>1</sup><sup>+</sup> sech(*kx*−<sup>4</sup> *<sup>δ</sup>k*<sup>3</sup> *<sup>t</sup>*+*η*<sup>0</sup> )

(*ut* + *αu*1/2*ux* + *βuxxx*)*<sup>x</sup>* + *δuyy* = 0, (4.9)

*ut* + *βu*2*ux* + *uxxx* + *uyyx* = 0, (5.1)

(*m*<sup>2</sup>+1)(1+*l*<sup>2</sup>) (*x* + *ly* − *ω t* + *ξ*0)),

(2−*m*<sup>2</sup>)(1+*l*<sup>2</sup>) (*<sup>x</sup>* <sup>+</sup> *ly* <sup>−</sup> *<sup>ω</sup> <sup>t</sup>* <sup>+</sup> *<sup>ξ</sup>*0)).

*ut* + *αu ux* + *uxxx* + *uyyx* = 0. (5.3)

2 2

. (4.8)

(5.2)

### **4. Schamel equation and modified KP equation**

The equation describing ion-acoustic waves in a cold-ion plasma where electrons do not behave isothermally during their passage of the wave is

$$
\mu\_t + \mu^{1/2} u\_x + \delta u\_{xxx} = 0. \tag{4.1}
$$

Schamel [29] derived this equation and a simple solitary wave solution having a sech4 profile was obtained. Therefore the Schamel equation (4.1) containing a square root nonlinearity is very attractive model for the study of ion-acoustic waves in plasmas and dusty plasmas. In order to find the periodic wave solution of (4.1), we use the transformations *<sup>u</sup>* <sup>=</sup> *<sup>v</sup>*2, *<sup>v</sup>*(*x*, *<sup>t</sup>*) = *<sup>V</sup>*(*η*); *<sup>η</sup>* <sup>=</sup> *kx* <sup>−</sup> *<sup>ω</sup> <sup>t</sup>* <sup>+</sup> *<sup>η</sup>*0,, then (4.1) becomes

$$-\omega V \, V' + kV^2 V' + \delta k^3 [V V'^\prime + \mathfrak{B} V' V''] = 0. \tag{4.2}$$

According to the mapping method, we assume that Eq. (4.2) has the following solution:

$$V(\eta) = A\_0 + A\_1 f(\eta),\tag{4.3}$$

where *A*<sup>0</sup> and *A*<sup>1</sup> are constants to be determined and *f*(*η*) satisfies Eq. (1.6). Substitution of Eq. (4.3) into Eq. (4.2) and selecting the values of *p*, *q* and *r*, we have the solutions of Eq. (4.1) which was given in [20] as follows:

**Case 1.** *<sup>p</sup>* <sup>=</sup> 2, *<sup>q</sup>* <sup>=</sup> <sup>−</sup><sup>4</sup> (<sup>1</sup> <sup>+</sup> *<sup>m</sup>*2), *<sup>r</sup>* <sup>=</sup> <sup>6</sup> *<sup>m</sup>*2. In this case, we have *<sup>f</sup>*(*η*) = sn2*η*. Thus the periodic wave solutions of Eq. (4.1) are

$$\begin{aligned} u\_1(\mathbf{x}, t) &= 100\delta^2 k^4 \left[ 1 + m^2 \pm \sqrt{1 - m^2 + m^4} - 3 \, m^2 \, \text{sn}^2 \, \eta \right]^2, \\ \eta &= k\mathbf{x} \mp 16 \, \delta k^3 \sqrt{1 - m^2 + m^4} \, t + \eta\_0. \end{aligned} \tag{4.4}$$

**Case 2.** *<sup>p</sup>* <sup>=</sup> <sup>−</sup>(1−*m*<sup>2</sup>)<sup>2</sup> <sup>2</sup> , *<sup>q</sup>* <sup>=</sup> <sup>2</sup>(<sup>1</sup> <sup>+</sup> *<sup>m</sup>*2), *<sup>r</sup>* <sup>=</sup> <sup>−</sup><sup>3</sup> <sup>2</sup> . The solutions of Eq. (1.6) are *f*(*η*) = (*<sup>m</sup>* cn*<sup>η</sup>* <sup>±</sup> dn*η*)2. Thus the exact solutions of Eq. (4.1) are

$$\begin{split} u\_2(\mathbf{x}, t) &= \frac{25\delta^2 k^4}{4} \left[ -2(1 + m^2) \pm \sqrt{1 + 14m^2 + m^4} + 3 \left( m \operatorname{cn} \eta \pm \operatorname{dn} \eta \right)^2 \right]^2, \\ \eta = k\mathbf{x} &\mp 4\,\delta k^3 \sqrt{1 + 14m^2 + m^4}t + \eta\_0. \end{split} \tag{4.5}$$

**Case 3.** *p* = *<sup>m</sup>*<sup>2</sup> <sup>2</sup> , *<sup>q</sup>* <sup>=</sup> <sup>2</sup>(*m*<sup>2</sup> <sup>−</sup> <sup>2</sup>), *<sup>r</sup>* <sup>=</sup> <sup>3</sup> *<sup>m</sup>*<sup>2</sup> <sup>2</sup> . The solutions of Eq. (1.6) are *<sup>f</sup>*(*η*) = *<sup>m</sup>* sn*<sup>η</sup>* <sup>1</sup><sup>±</sup> dn*<sup>η</sup>* 2 . So, we obtained the exact solutions of Eq. (4.1)in the form

$$\begin{aligned} u\_3(\mathbf{x}, t) &= \frac{25\delta^2 k^4}{4} \left[ 2(2 - m^2) \pm \sqrt{16 - 16m^2 + m^4} - 3 \, m^4 \left( \frac{\operatorname{sn} \eta}{1 \pm \operatorname{dn} \eta} \right)^2 \right]^2, \\ \eta &= k \mathbf{x} \mp 4 \, \delta k^3 \sqrt{16 - 16m^2 + m^4} \, t + \eta\_0. \end{aligned} \tag{4.6}$$

There are several exact solutions for the Eq. (4.1) which are omitted here for simplicity. As *m* → 1, these solutions reduce to the solitary wave solutions

$$\begin{aligned} u\_4(\mathbf{x}, t) &= 900 \delta^2 \, k^4 \operatorname{sech}^4(\mathbf{k}\mathbf{x} - 16 \, \delta k^3 \, t + \eta\_0), \\ u\_5(\mathbf{x}, t) &= 100 \delta^2 \, k^4 \left[ 2 - 3 \operatorname{sech}^2(\mathbf{k}\mathbf{x} + 16 \, \delta k^3 \, t + \eta\_0) \right]^2. \end{aligned} \tag{4.7}$$

$$u\_{\theta}(\mathbf{x},t) = \frac{225\delta^2 k^4}{4} \left[ 1 - \left( \frac{\tanh\left(kx - 4\delta k^3 t + \eta\_0\right)}{1 + \mathrm{sech}\left(kx - 4\delta k^3 t + \eta\_0\right)} \right)^2 \right]^2. \tag{4.8}$$

The KdV equation in two dimensions, known as Kadomtsev Petviashivili (KP) equation [32], was derived for ion-acoustic waves in a non magnetized plasma by Kako and Rowlands [33]. Therefore the modified KP equation containing a square root nonlinearity is very attractive model for the study of ion-acoustic waves in plasma and dusty plasma [34- 36]. Extensive work has been devoted to the study of nonlinear waves associated with the dust ion-acoustic waves, particularly the dust ion-acoustic solitary and shock waves in dusty plasmas in which dust particles are stationary and provide only the neutrality [37]. The KP equation is derived [38] for the propagation of nonlinear waves in warm dusty plasmas with variable dust charge, two-temperature ions and nonthermal electrons by using the reductive perturbation theory. Consider the modified KP equation

$$(\mu\_t + \alpha u^{1/2} u\_x + \beta u\_{xxx})\_x + \delta u\_{yy} = 0,\tag{4.9}$$

where *α* and *β* are constants. The modified KP equation (4.9) for ion-acoustic waves in a multi species plasma consisting of non-isothermal electrons have been derived by Chakraborty and Das [34]. We applied the mapping method with the ansatz solution (4.3) and the solutions of auxiliary equation (1.6) to find the solutions of equation (4.9) (see [39]).

### **5. The ZK equation and modified ZK equation**

The equation

6 Will-be-set-by-IN-TECH

The equation describing ion-acoustic waves in a cold-ion plasma where electrons do not

Schamel [29] derived this equation and a simple solitary wave solution having a sech4 profile was obtained. Therefore the Schamel equation (4.1) containing a square root nonlinearity is very attractive model for the study of ion-acoustic waves in plasmas and dusty plasmas.

In order to find the periodic wave solution of (4.1), we use the transformations

where *A*<sup>0</sup> and *A*<sup>1</sup> are constants to be determined and *f*(*η*) satisfies Eq. (1.6).

<sup>2</sup> , *<sup>q</sup>* <sup>=</sup> <sup>2</sup>(<sup>1</sup> <sup>+</sup> *<sup>m</sup>*2), *<sup>r</sup>* <sup>=</sup> <sup>−</sup><sup>3</sup>

<sup>−</sup> <sup>2</sup>(<sup>1</sup> <sup>+</sup> *<sup>m</sup>*2) <sup>±</sup> <sup>√</sup>

1 + 14*m*<sup>2</sup> + *m*<sup>4</sup> *t* + *η*0.

<sup>2</sup>(<sup>2</sup> <sup>−</sup> *<sup>m</sup>*2) <sup>±</sup> <sup>√</sup>

<sup>16</sup> − <sup>16</sup>*m*<sup>2</sup> + *<sup>m</sup>*<sup>4</sup> *<sup>t</sup>* + *<sup>η</sup>*0.

<sup>−</sup> *<sup>ω</sup>V V*� <sup>+</sup> *k V*2*V*� <sup>+</sup> *<sup>δ</sup>k*3[*VV*��� <sup>+</sup> <sup>3</sup>*V*�

According to the mapping method, we assume that Eq. (4.2) has the following solution:

Substitution of Eq. (4.3) into Eq. (4.2) and selecting the values of *p*, *q* and *r*, we have the

**Case 1.** *<sup>p</sup>* <sup>=</sup> 2, *<sup>q</sup>* <sup>=</sup> <sup>−</sup><sup>4</sup> (<sup>1</sup> <sup>+</sup> *<sup>m</sup>*2), *<sup>r</sup>* <sup>=</sup> <sup>6</sup> *<sup>m</sup>*2. In this case, we have *<sup>f</sup>*(*η*) = sn2*η*. Thus the

<sup>1</sup> <sup>+</sup> *<sup>m</sup>*<sup>2</sup> <sup>±</sup> <sup>√</sup>

<sup>1</sup> − *<sup>m</sup>*<sup>2</sup> + *<sup>m</sup>*<sup>4</sup> *<sup>t</sup>* + *<sup>η</sup>*0.

*ut* + *u*1/2 *ux* + *δ uxxx* = 0. (4.1)

*V*(*η*) = *A*<sup>0</sup> + *A*<sup>1</sup> *f*(*η*), (4.3)

<sup>1</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*<sup>4</sup> <sup>−</sup> <sup>3</sup> *<sup>m</sup>*<sup>2</sup> sn2 *<sup>η</sup>*

<sup>1</sup> <sup>+</sup> <sup>14</sup>*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*<sup>4</sup> <sup>+</sup> <sup>3</sup> (*<sup>m</sup>* cn*<sup>η</sup>* <sup>±</sup> dn*η*)<sup>2</sup>

<sup>2</sup> . The solutions of Eq. (1.6) are

<sup>16</sup> <sup>−</sup> <sup>16</sup>*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*<sup>4</sup> <sup>−</sup> <sup>3</sup> *<sup>m</sup>*<sup>4</sup>

. So, we obtained the exact solutions of Eq. (4.1)in the form

There are several exact solutions for the Eq. (4.1) which are omitted here for simplicity. As

*<sup>u</sup>*5(*x*, *<sup>t</sup>*) = <sup>100</sup>*δ*<sup>2</sup> *<sup>k</sup>*<sup>4</sup> [<sup>2</sup> <sup>−</sup> 3 sech2(*kx* <sup>+</sup> <sup>16</sup> *<sup>δ</sup>k*<sup>3</sup> *<sup>t</sup>* <sup>+</sup> *<sup>η</sup>*0)]2.

*<sup>u</sup>*4(*x*, *<sup>t</sup>*) = <sup>900</sup>*δ*<sup>2</sup> *<sup>k</sup>*<sup>4</sup> sech4(*kx* <sup>−</sup> <sup>16</sup> *<sup>δ</sup>k*<sup>3</sup> *<sup>t</sup>* <sup>+</sup> *<sup>η</sup>*0),

*V*��] = 0. (4.2)

2 ,

> 2 ,

<sup>2</sup> . The solutions of Eq. (1.6) are *f*(*η*) =

 sn*η* <sup>1</sup><sup>±</sup> dn*<sup>η</sup>* 2 2 , (4.4)

(4.5)

(4.6)

(4.7)

**4. Schamel equation and modified KP equation**

behave isothermally during their passage of the wave is

*<sup>u</sup>* <sup>=</sup> *<sup>v</sup>*2, *<sup>v</sup>*(*x*, *<sup>t</sup>*) = *<sup>V</sup>*(*η*); *<sup>η</sup>* <sup>=</sup> *kx* <sup>−</sup> *<sup>ω</sup> <sup>t</sup>* <sup>+</sup> *<sup>η</sup>*0,, then (4.1) becomes

solutions of Eq. (4.1) which was given in [20] as follows:

*u*1(*x*, *t*) = 100*δ*<sup>2</sup> *k*<sup>4</sup>

*<sup>η</sup>* <sup>=</sup> *kx* <sup>∓</sup> <sup>16</sup> *<sup>δ</sup>k*<sup>3</sup> <sup>√</sup>

4 

(*<sup>m</sup>* cn*<sup>η</sup>* <sup>±</sup> dn*η*)2. Thus the exact solutions of Eq. (4.1) are

<sup>2</sup> , *<sup>q</sup>* <sup>=</sup> <sup>2</sup>(*m*<sup>2</sup> <sup>−</sup> <sup>2</sup>), *<sup>r</sup>* <sup>=</sup> <sup>3</sup> *<sup>m</sup>*<sup>2</sup>

*m* → 1, these solutions reduce to the solitary wave solutions

periodic wave solutions of Eq. (4.1) are

*<sup>u</sup>*2(*x*, *<sup>t</sup>*) = <sup>25</sup>*δ*<sup>2</sup> *<sup>k</sup>*<sup>4</sup>

*<sup>η</sup>* <sup>=</sup> *kx* <sup>∓</sup> <sup>4</sup> *<sup>δ</sup>k*<sup>3</sup> <sup>√</sup>

2

*<sup>u</sup>*3(*x*, *<sup>t</sup>*) = <sup>25</sup>*δ*<sup>2</sup> *<sup>k</sup>*<sup>4</sup>

*<sup>η</sup>* <sup>=</sup> *kx* <sup>∓</sup> <sup>4</sup> *<sup>δ</sup>k*<sup>3</sup> <sup>√</sup>

4

**Case 2.** *<sup>p</sup>* <sup>=</sup> <sup>−</sup>(1−*m*<sup>2</sup>)<sup>2</sup>

**Case 3.** *p* = *<sup>m</sup>*<sup>2</sup>

 *m* sn*η* <sup>1</sup><sup>±</sup> dn*<sup>η</sup>*

*f*(*η*) =

$$
\mu\_t + \beta \mu^2 \mu\_x + \mu\_{\text{xxx}} + \mu\_{yyx} = 0,\tag{5.1}
$$

is the modified ZK in (2+1) dimensions which is a model for acoustic plasma waves [40, 41]. The ZK equation was first derived for describing weakly nonlinear ion- acoustic waves in a strongly magnetized lossless plasma in two dimension [41]. The ZK equation and modified ZK equation possess traveling wave structures [28, 42]. Peng [42] studied the exact solutions of ZK equation by using extended mapping method. Various types of solutions of Schamel-KdV equation and modified ZK equation arising in plasma and dust plasma are presented in [43].

We apply the *F*-expansion method to the modified ZK equation. Thus, Eq. (5.1) has a solution in the form

$$
\mu(\xi) = a\_0 + a\_1 F(\xi), \quad \xi = k(\mathfrak{x} + ly - \omega \, t) + \xi\_0.
$$

Substituting this equation into Eq. (5.1), we obtain the following classes of exact solutions of the modified ZK equation:

$$\begin{split} \mu &= m \sqrt{\frac{6\omega}{(m^2+1)\beta}} \operatorname{sn}(\sqrt{\frac{-\omega}{(m^2+1)(1+l^2)}} \left( \mathbf{x} + l\mathbf{y} - \omega \, t + \xi\_0 \right)), \\ \mu &= m \sqrt{\frac{6\omega}{(2-m^2)\beta}} \operatorname{dn}(\sqrt{\frac{\omega}{(2-m^2)(1+l^2)}} \left( \mathbf{x} + l\mathbf{y} - \omega \, t + \xi\_0 \right)). \end{split} \tag{5.2}$$

In the following we apply the mapping method to the ZK equation

$$
\mu\_l + \mathfrak{a}\mathfrak{u}\,\mathfrak{u}\_\mathfrak{x} + \mathfrak{u}\_{\mathfrak{xx}\mathfrak{x}} + \mathfrak{u}\_{\mathfrak{yy}\mathfrak{x}} = \mathbf{0}.\tag{5.3}
$$

In this case, we have *n* = 1. Thus Eq. (5.3) has a solution in the form

$$
\mu(\eta) = A\_0 + A\_1 f(\eta), \quad \eta = k\mathfrak{x} + ly - \omega \, t + \eta\_0.
$$

where *β*, *c*<sup>3</sup> and *c*<sup>5</sup> are constants. Here, we review the exact traveling wave solutions of equation (6.1) using exact solutions of the auxiliary equation (1.5) and applied the mapping

<sup>75</sup> Exact Solutions Expressible in Hyperbolic

Substituting equation (6.2) into (6.1) and equating the coefficients of like powers of *f* to zero, yields a system of algebraic equations for *A*0, *A*1, *k* and *ω* and then solve it. Therefore, the

<sup>∓</sup> *<sup>k</sup>*<sup>2</sup>

<sup>10</sup>*c*<sup>5</sup> *t*

 *m* sn*η* <sup>1</sup>±dn*<sup>η</sup>*

 sn*η* <sup>1</sup>±cn*<sup>η</sup>*

There are several other JEFs of Eq. (6.1) which are omitted here for simplicity. When *m* −→ 1,

<sup>−</sup><sup>90</sup> *<sup>c</sup>*<sup>5</sup> 4*β* 

> + *η*0.

*m* cn *η* ± dn *η*

2 ,

2 ,

*<sup>x</sup>* <sup>+</sup> (128*m*<sup>4</sup>−128*m*<sup>2</sup>+23) *<sup>c</sup>*<sup>2</sup>

−*c*<sup>3</sup> 5*c*<sup>5</sup>

, *η* = ±

*<sup>x</sup>* <sup>+</sup> (23*m*<sup>4</sup>−128*m*<sup>2</sup>+128) *<sup>c</sup>*<sup>2</sup>

<sup>200</sup>*c*<sup>5</sup> (1+*m*<sup>2</sup>)<sup>2</sup> *<sup>t</sup>*

<sup>200</sup>*c*<sup>5</sup> (*m*<sup>2</sup>−2)<sup>2</sup> *<sup>t</sup>*

<sup>200</sup>*c*<sup>5</sup> (1−2*m*<sup>2</sup>)<sup>2</sup> *<sup>t</sup>*

(*x* +

 *c*<sup>3</sup> 10*c*<sup>5</sup> *x* +

*ut* + (*α* + *β up*) *upux* + *uxxx* + *δuyyx* = 0, (6.9)

*<sup>x</sup>* <sup>+</sup> (23*m*<sup>4</sup>+82*m*<sup>2</sup>+23) *<sup>c</sup>*<sup>2</sup>

3)

solutions of the modified fifth order KdV equation (6.1) was given in [39] as follows:

5*k*<sup>4</sup> (*m*<sup>4</sup>+14*m*<sup>2</sup>+1)+*c*<sup>2</sup>

√−<sup>10</sup> *<sup>β</sup> <sup>c</sup>*<sup>5</sup>

√−<sup>10</sup> *<sup>β</sup> <sup>c</sup>*<sup>5</sup>

√−<sup>10</sup> *<sup>β</sup> <sup>c</sup>*<sup>5</sup>

sech2( 1 2

2

Finally, we can construct various types of exact and explicit solutions of the generalized ZK

by using suitable method and using an appropriate transformation. Also, we can study the exact solution of the generalized KdV equation (*δ* = 0) which studied by many authors [22, 23, 31]. The generalized ZK equation was first derived for describing weakly nonlinear ion-acoustic waves in strongly magnetized lossless plasma in two dimensions and governs the behavior of weakly nonlinear ion-acoustic waves in plasma comprising cold ions and

)+*c*3)

−10*βc*<sup>5</sup>

and Jacobi Elliptic Functions of Some Important Equations of Ion-Acoustic Waves

*u*(*η*) = *A*<sup>0</sup> + *A*<sup>1</sup> *f*(*η*), *η* = *kx* − *ω t* + *η*0, (6.2)

*m* cn *η* ± dn *η*

2 ,

3

3

3

4 *c*<sup>2</sup> 3 25*c*<sup>5</sup>  + *η*0.

*t*) + *η*0), (6.7)

+ *η*0. (6.8)

23 *c*<sup>2</sup> 3 200*c*<sup>5</sup> *t* 

 + *η*0,

 + *η*0. 2 ,

(6.3)

(6.4)

(6.5)

(6.6)

method. Thus, Eq. (6.1) has the solutions in the form

*<sup>u</sup>*<sup>1</sup> <sup>=</sup> <sup>±</sup> (<sup>10</sup> *<sup>k</sup>*<sup>2</sup> *<sup>c</sup>*5(1+*m*<sup>2</sup> √

*<sup>x</sup>* <sup>+</sup> (15*c*<sup>2</sup>

*<sup>u</sup>*<sup>2</sup> <sup>=</sup> <sup>±</sup> <sup>3</sup>*c*<sup>3</sup> 2(1+*m*<sup>2</sup>)

*<sup>u</sup>*<sup>3</sup> <sup>=</sup> <sup>±</sup> <sup>3</sup>*m*<sup>2</sup> *<sup>c</sup>*<sup>3</sup> 2(*m*<sup>2</sup>−2)

*<sup>η</sup>* <sup>=</sup> <sup>±</sup> *<sup>c</sup>*<sup>3</sup>

*<sup>u</sup>*<sup>4</sup> <sup>=</sup> <sup>±</sup> <sup>3</sup>*c*<sup>3</sup> 2(1−2*m*<sup>2</sup>)

> <sup>−</sup>*c*<sup>3</sup> 10*c*5(1−2*m*<sup>2</sup>)

> > 3*c*<sup>3</sup> −10*βc*<sup>5</sup>

 tanh *η* 1 ± sech *η*

We notice that Eq. (6.7) is the solution given by Example 2 in Ref. [47].

 <sup>−</sup>*c*<sup>3</sup> 10*c*5(1+*m*<sup>2</sup>)

10*c*5(2−*m*<sup>2</sup>)

*η* = *k* 

If we choose *A*<sup>0</sup> = 0, equation (6.3) takes the form

*η* = ±

Moreover, we have obtained the exact solutions

*η* = ±

then (6.4)-(6.6) become the solitary wave solutions

*<sup>u</sup>*<sup>5</sup> <sup>=</sup> <sup>±</sup>

3*c*<sup>3</sup>

−<sup>10</sup> *<sup>β</sup> <sup>c</sup>*<sup>5</sup>

*u*<sup>6</sup> = ∓

equation

2

Substituting this equation into Eq. (5.3) to determine *A*0, *A*1, *k*, *ω* and using the solutions of auxiliary equation (1.6), we obtained the following classes of exact solutions of the ZK equation [39]:

$$\begin{split} u\_1(\mathbf{x}, y, t) &= \frac{\omega}{k\,\mathbf{a}} + \frac{4(1 + m^2)(l^2 + k^2)}{\mathbf{a}} - \frac{12m^2(l^2 + k^2)}{\mathbf{a}} \operatorname{sn}^2(k\mathbf{x} + ly - \omega\, t + \eta\_0), \\ u\_2(\mathbf{x}, y, t) &= \frac{\omega}{k\,\mathbf{a}} + \frac{4(m^2 - 2)(l^2 + k^2)}{\mathbf{a}} + \frac{12(l^2 + k^2)}{\mathbf{a}} \operatorname{dn}^2(k\mathbf{x} + ly - \omega\, t + \eta\_0), \end{split} \tag{5.4}$$

$$\mu\_3 = \frac{\omega}{\hbar a} - \frac{2(1+m^2)(l^2+k^2)}{a} + \frac{3(l^2+k^2)}{a} [m \operatorname{cn}(\eta) \pm \operatorname{dn}(\eta)]^2,\tag{5.5}$$

$$\ln \mu\_{4}(\mathbf{x}, y, t) = \mathop{\rm tr\!}\_{\mathbf{k}, \mathbf{u}} -\frac{2(1 + m^{2})(l^{2} + k^{2})}{\mathfrak{a}} - \frac{3(1 - m^{2})(l^{2} + k^{2})}{\mathfrak{a}} \left( \frac{\mathfrak{C}\mathfrak{n}(k\mathbf{x} + ly - \omega t + \eta\_{0})}{1 \pm \mathfrak{s}\mathfrak{n}(k\mathbf{x} + ly - \omega t + \eta\_{0})} \right)^{2} . \tag{5.6}$$

When *m* −→ 1, some of these solutions degenerate as solitary wave solutions of ZK equation. The solutions (5.3) are coincide with the solutions given in [44].

Recently, some properties of the quantum ion-acoustic waves were also investigated in dense quantum plasmas by studying the quantum hydrodynamical equations in different conditions, which includes the quantum Zakharov Kuznetsov equation, the extended quantum Zakharov Kuznetsov equation, and the quantum Zakharov system [45]. The three-dimensional extended quantum Zakharov Kuznetsov (QZK) equation [46] was investigated in dense quantum plasmas which arises from the dimensionless hydrodynamics equations describing the nonlinear propagation of the quantum ion-acoustic waves. The three-dimensional extended QZK equation was given in [46]

$$
\Phi\_l + (A\Phi + B\Phi^2)\Phi\_\mathbf{x} + C\Phi\_{zzz} + D\left(\Phi\_{\mathbf{x}xz} + \Phi\_{yyz}\right) = 0,\tag{5.7}
$$

where *A*, *B*, *C* and *D* are constants. This equation has the following JEF solutions (see [45, 46]):

$$\Phi\_1 = -\frac{A}{2\mathcal{B}} + mk\sqrt{\frac{-6E}{B}} \operatorname{sn}(k(\mathbf{x} + ly + \gamma \mathbf{z} - \omega \, t + \eta\_0)), \quad \omega = -\frac{4BE^2(1+m^2) + A^2}{4B}, \quad BE < 0,\tag{5.8}$$

$$\Phi\_2 = -\frac{A}{2\mathcal{B}} + mk\sqrt{\frac{6E}{B}} \operatorname{cn}(k(\mathbf{x} + ly + \gamma \mathbf{z} - \omega \, t + \eta\_0)), \quad \omega = -\frac{4BE^2(1-2m^2) + A^2}{4B}, \quad BE > 0,\tag{5.9}$$

with *E* = *Cγ*<sup>2</sup> + *D*(1+ *l* <sup>2</sup>). Moreover, many types of analytical solutions of the extended QZK equation are constructed in terms of some powerful ansatze, which include doubly periodic wave solutions, solitary wave solutions, kink-shaped wave solutions, rational wave solutions and singular solutions [46].

### **6. The modified fifth order KdV equation**

Higher order KdV equations have many applications in different fields of mathematical physics. For example the fifth-order KdV equations can be derived in fluid dynamics and in magneto-acoustic waves in plasma and its exact solutions was given in [47-51]. The higher-order KdV equation can be derived for magnetized plasmas by using the reductive perturbation technique. Traveling wave solutions of Kawahara equation and modified Kawahara equation have been studied [9, 48, 49]. Moreover, the solitary wave solutions of nonlinear equations with arbitrary odd-order derivatives were studied by many authors [47, 51].

Consider the modified fifth order KdV equation

$$
\mu\_l + \beta \mu^2 u\_x + c\_3 u\_{xxx} + c\_5 u\_{xxxxx} = 0,\tag{6.1}
$$

where *β*, *c*<sup>3</sup> and *c*<sup>5</sup> are constants. Here, we review the exact traveling wave solutions of equation (6.1) using exact solutions of the auxiliary equation (1.5) and applied the mapping method. Thus, Eq. (6.1) has the solutions in the form

$$
\mu(\eta) = A\_0 + A\_1 f(\eta), \quad \eta = k\mathfrak{x} - \omega \, t + \eta\_{0\prime} \tag{6.2}
$$

Substituting equation (6.2) into (6.1) and equating the coefficients of like powers of *f* to zero, yields a system of algebraic equations for *A*0, *A*1, *k* and *ω* and then solve it. Therefore, the solutions of the modified fifth order KdV equation (6.1) was given in [39] as follows:

$$\begin{split} \mu\_{1} &= \pm \frac{(10k^{2}c\_{5}(1+m^{2})+c\_{3})}{\sqrt{-10kc\_{5}}} \mp k^{2} \sqrt{\frac{-90c\_{5}}{4\beta}} \left( m \operatorname{cn} \,\eta \pm \operatorname{dn} \,\eta \right)^{2}, \\ \eta &= k \left[ \mathbf{x} + \frac{(15c\_{5}^{2}k^{4}(m^{4}+14m^{2}+1)+c\_{3}^{2})}{10c\_{5}}t \right] + \eta\_{0}. \end{split} \tag{6.3}$$

If we choose *A*<sup>0</sup> = 0, equation (6.3) takes the form

8 Will-be-set-by-IN-TECH

Substituting this equation into Eq. (5.3) to determine *A*0, *A*1, *k*, *ω* and using the solutions of auxiliary equation (1.6), we obtained the following classes of exact solutions of the ZK

<sup>2</sup>+*k*<sup>2</sup>)

<sup>2</sup>+*k*<sup>2</sup>) *α*

Φ*<sup>t</sup>* + (*A*Φ + *B* Φ2)Φ*<sup>x</sup>* + *C* Φ*zzz* + *D* (Φ*xxz* + Φ*yyz*) = 0, (5.7)

<sup>2</sup>). Moreover, many types of analytical solutions of the extended QZK

*ut* + *βu*2*ux* + *c*3*uxxx* + *c*5*uxxxxx* = 0, (6.1)

<sup>2</sup>+*k*<sup>2</sup>)

<sup>2</sup>+*k*<sup>2</sup>)

*<sup>α</sup>* sn2( *kx* <sup>+</sup> *ly* <sup>−</sup> *<sup>ω</sup> <sup>t</sup>* <sup>+</sup> *<sup>η</sup>*0),

*<sup>α</sup>* [*<sup>m</sup>* cn( *<sup>η</sup>*) <sup>±</sup> dn( *<sup>η</sup>*)]2, (5.5)

2

<sup>4</sup> *<sup>B</sup>* , *BE* < 0,

<sup>4</sup> *<sup>B</sup>* , *BE* > 0,

(5.8)

 cn( *kx*+*ly*−*<sup>ω</sup> <sup>t</sup>*+*η*<sup>0</sup> ) <sup>1</sup>±sn( *kx*+*ly*−*<sup>ω</sup> <sup>t</sup>*+*η*<sup>0</sup> ) (5.4)

. (5.6)

*<sup>α</sup>* dn2( *kx* <sup>+</sup> *ly* <sup>−</sup> *<sup>ω</sup> <sup>t</sup>* <sup>+</sup> *<sup>η</sup>*0),

<sup>2</sup>+*k*<sup>2</sup>) *<sup>α</sup>* <sup>−</sup> <sup>12</sup>*m*<sup>2</sup>(*<sup>l</sup>*

<sup>2</sup>+*k*<sup>2</sup>) *<sup>α</sup>* <sup>+</sup> <sup>12</sup>(*<sup>l</sup>*

<sup>2</sup>+*k*<sup>2</sup>) *<sup>α</sup>* <sup>+</sup> <sup>3</sup>(*<sup>l</sup>*

<sup>2</sup>+*k*<sup>2</sup>)

*<sup>α</sup>* <sup>−</sup> <sup>3</sup>(1−*m*<sup>2</sup>)(*<sup>l</sup>*

When *m* −→ 1, some of these solutions degenerate as solitary wave solutions of ZK equation.

Recently, some properties of the quantum ion-acoustic waves were also investigated in dense quantum plasmas by studying the quantum hydrodynamical equations in different conditions, which includes the quantum Zakharov Kuznetsov equation, the extended quantum Zakharov Kuznetsov equation, and the quantum Zakharov system [45]. The three-dimensional extended quantum Zakharov Kuznetsov (QZK) equation [46] was investigated in dense quantum plasmas which arises from the dimensionless hydrodynamics equations describing the nonlinear propagation of the quantum ion-acoustic waves. The

where *A*, *B*, *C* and *D* are constants. This equation has the following JEF solutions (see [45, 46]):

equation are constructed in terms of some powerful ansatze, which include doubly periodic wave solutions, solitary wave solutions, kink-shaped wave solutions, rational wave solutions

Higher order KdV equations have many applications in different fields of mathematical physics. For example the fifth-order KdV equations can be derived in fluid dynamics and in magneto-acoustic waves in plasma and its exact solutions was given in [47-51]. The higher-order KdV equation can be derived for magnetized plasmas by using the reductive perturbation technique. Traveling wave solutions of Kawahara equation and modified Kawahara equation have been studied [9, 48, 49]. Moreover, the solitary wave solutions of nonlinear equations with arbitrary odd-order derivatives were studied by many authors [47,

*<sup>B</sup>* sn( *<sup>k</sup>*(*<sup>x</sup>* <sup>+</sup> *ly* <sup>+</sup> *<sup>γ</sup><sup>z</sup>* <sup>−</sup> *<sup>ω</sup> <sup>t</sup>* <sup>+</sup> *<sup>η</sup>*0)), *<sup>ω</sup>* <sup>=</sup> <sup>−</sup>4*BEk*<sup>2</sup> (1+*m*<sup>2</sup>)+*A*<sup>2</sup>

*<sup>B</sup>* cn( *<sup>k</sup>*(*<sup>x</sup>* <sup>+</sup> *ly* <sup>+</sup> *<sup>γ</sup><sup>z</sup>* <sup>−</sup> *<sup>ω</sup> <sup>t</sup>* <sup>+</sup> *<sup>η</sup>*0)), *<sup>ω</sup>* <sup>=</sup> <sup>−</sup>4*BEk*<sup>2</sup> (1−2*m*<sup>2</sup>)+*A*<sup>2</sup>

equation [39]:

<sup>Φ</sup><sup>1</sup> <sup>=</sup> <sup>−</sup> *<sup>A</sup>*

<sup>Φ</sup><sup>2</sup> <sup>=</sup> <sup>−</sup> *<sup>A</sup>*

51].

<sup>2</sup>*<sup>B</sup>* + *m k*

<sup>2</sup>*<sup>B</sup>* + *mk*

with *E* = *Cγ*<sup>2</sup> + *D*(1+ *l*

and singular solutions [46].

<sup>−</sup>6*<sup>E</sup>*

<sup>6</sup>*<sup>E</sup>*

**6. The modified fifth order KdV equation**

Consider the modified fifth order KdV equation

*u*1(*x*, *y*, *t*) = *<sup>ω</sup>*

*u*2(*x*, *y*, *t*) = *<sup>ω</sup>*

*u*4(*x*, *y*, *t*) = *<sup>ω</sup>*

*u*<sup>3</sup> = *<sup>ω</sup>*

*<sup>k</sup> <sup>α</sup>* <sup>+</sup> <sup>4</sup>(1+*m*<sup>2</sup>)(*<sup>l</sup>*

*<sup>k</sup> <sup>α</sup>* <sup>+</sup> <sup>4</sup>(*m*<sup>2</sup>−2)(*<sup>l</sup>*

*<sup>k</sup> <sup>α</sup>* <sup>−</sup> <sup>2</sup>(1+*m*<sup>2</sup>)(*<sup>l</sup>*

*<sup>k</sup> <sup>α</sup>* <sup>−</sup> <sup>2</sup>(1+*m*<sup>2</sup>)(*<sup>l</sup>*

The solutions (5.3) are coincide with the solutions given in [44].

three-dimensional extended QZK equation was given in [46]

$$\begin{split} \mu\_{2} &= \pm \frac{3c\_{3}}{2(1+m^{2})\sqrt{-10\beta}c\_{5}} \left( m \operatorname{cn} \eta \pm \operatorname{dn} \eta \right)^{2}, \\ \eta &= \pm \sqrt{\frac{-c\_{3}}{10c\_{5}(1+m^{2})}} \left[ \mathbf{x} + \frac{(23m^{4} + 82m^{2} + 23)c\_{3}^{2}}{200c\_{5}(1+m^{2})^{2}} t \right] + \eta\_{0}. \end{split} \tag{6.4}$$

Moreover, we have obtained the exact solutions

$$\begin{split} \mu\_{3} &= \pm \frac{3m^{2}c\_{3}}{2(m^{2}-2)\sqrt{-10\beta\epsilon\_{5}}} \left(\frac{m\sin\eta}{1\pm\text{d}\Pi\eta}\right)^{2}, \\ \eta &= \pm \sqrt{\frac{c\_{3}}{10\varepsilon\left(2-m^{2}\right)}} \left[\mathbf{x} + \frac{(23m^{4}-128m^{2}+128)}{200\varepsilon(m^{2}-2)^{2}}t\right] + \eta\_{0'} \end{split} \tag{6.5}$$
 
$$\begin{split} \mu\_{4} &= \pm \frac{3c\_{3}}{2(1-2m^{2})\sqrt{-10\beta\epsilon\_{5}}} \left(\frac{\mathbf{x}\eta}{1\pm\text{d}\Pi\eta}\right)^{2}, \\ \eta &= \pm \sqrt{\frac{-c\_{3}}{10\varepsilon\left(1-2m^{2}\right)}} \left[\mathbf{x} + \frac{(128m^{4}-128m^{2}+23)}{200\varepsilon(1-2m^{2})^{2}}t\right] + \eta\_{0'}. \end{split} \tag{6.6}$$

There are several other JEFs of Eq. (6.1) which are omitted here for simplicity. When *m* −→ 1, then (6.4)-(6.6) become the solitary wave solutions

$$u\_5 = \pm \frac{3c\_3}{\sqrt{-10\beta c\_5}} \operatorname{sech}^2(\frac{1}{2}\sqrt{\frac{-c\_3}{5c\_5}}(x + \frac{4c\_3^2}{25c\_5}t) + \eta\_0),\tag{6.7}$$

$$u\_6 = \mp \frac{3c\_3}{2\sqrt{-10\beta c\_5}} \left(\frac{\tanh \eta}{1 \pm \mathrm{sech}\,\eta}\right)^2, \quad \eta = \pm \sqrt{\frac{c\_3}{10c\_5}} \left[x + \frac{23}{200c\_5}t\right] + \eta\_0. \tag{6.8}$$

We notice that Eq. (6.7) is the solution given by Example 2 in Ref. [47]. Finally, we can construct various types of exact and explicit solutions of the generalized ZK equation

$$\left(\mu\_t + \left(\mu + \beta \,\,\mu^p\right)\,\mu^p\mu\_x + \mu\_{\text{xxx}} + \delta\mu\_{yyx} = 0\right) \tag{6.9}$$

by using suitable method and using an appropriate transformation. Also, we can study the exact solution of the generalized KdV equation (*δ* = 0) which studied by many authors [22, 23, 31]. The generalized ZK equation was first derived for describing weakly nonlinear ion-acoustic waves in strongly magnetized lossless plasma in two dimensions and governs the behavior of weakly nonlinear ion-acoustic waves in plasma comprising cold ions and

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$$
\mu\_t + c\_1 \mu \,\mu\_x + c\_2 \,\mu\_{\text{XX}} + \delta \mu\_{\text{XX}\text{XX}} = 0.
$$

This equation appears in the theory of shallow water waves with surface tension and the theory of magneto-acoustic waves in plasmas [9]. Wazwaz [55] studied soliton solutions of fifth-order KdV equation. We can use a suitable method to construct the exact solutions of some special types of nonlinear evolution equations aries in plasma physics such as Liouville, sine-Gordon and sinh-Poisson equations.

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**5** 

 *India* 

**Acoustic Wave** 

*Department of Physics, Tezpur University, Napaam, Tezpur, Assam* 

An acoustic wave basically is a mechanical oscillation of pressure that travels through a medium like solid, liquid, gas, or plasma in a periodic wave pattern transmitting energy from one point to another in the medium [1-2]. It transmits sound by vibrating organs in the ear that produce the sensation of hearing and hence, it is also called acoustic signal. This is well-known that air is a fluid. Mechanical waves in air can only be longitudinal in nature; and therefore, all sound waves traveling through air must be longitudinal waves originating in the transmission form of compression and rarefaction from vibrating matter in the medium. The propagation of sound in absence of any material medium is always impossible. Therefore, sound does not travel through the vacuum of outer space, since there is nothing to carry the vibrations from a source to a receiver. The nature of the molecules making up a substance determines how well or how rapidly the substance will carry sound waves. The two characteristic variables affecting the propagation of acoustic waves are (1) the inertia of the constituent molecules and (2) the strength of molecular interaction. Thus, hydrogen gas, with the least massive molecules, will carry a sound wave at 1,284.00 ms-1 when the gas temperature is 00 C [1]. More massive helium gas molecules have more inertia and carry a sound wave at only 965.00 ms-1 at the same temperature. A solid, however, has molecules that are strongly attached, so acoustic vibrations are passed rapidly from molecule to molecule. Steel, for an instant example, is highly elastic, and sound will move rapidly through a steel rail at 5,940.00 ms-1 at the same temperature. The temperature of a medium influences the phase speed of sound through it. The gas molecules in warmer air thus have a greater kinetic energy than those of cooler air. The molecules of warmer air therefore transmit an acoustic impulse from molecule to molecule more rapidly. More precisely, the speed of a sound wave increases by 0.60 ms-1 for each Celcius degree rise in temperature

Acoustic waves, or sound waves, are defined generally and specified mainly by three characteristics: wavelength, frequency, and amplitude. The wavelength is the distance from the top of one wave's crest to the next (or, from the top of one trough to the next). The frequency of a sound wave is the number of waves that pass a point each second [1]. Sound waves with higher frequencies have higher pitches than sound waves with lower frequencies and vice versa. Amplitude is the measure of energy in a sound wave and affects volume. The greater the amplitude of an acoustic wave, the louder the sound and vice versa. An acoustic wave is what makes humans and other animals able to hear. A person's ear perceives the vibrations of an acoustic wave and interprets it as sound [1]. The outer ear, the visible part, is shaped like a funnel that collects sound waves and sends them into the ear

**1. Introduction** 

above 00 C.

P. K. Karmakar

waves in dusty plasma consisting of cold dust particles and two-temperature isothermal ions, *Phys. Plasmas* 6 (1999) 4542-4547.

