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

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) = ∑ i=0 AiF i(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) + ξ0, ξ0 is an arbitrary constant, and reduce a given NLPDE, say in two independent variables,


Introduction
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)=∑ N i=0 A i F i (ξ) where A i 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][8][9][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][17][18][19][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 0 is an arbitrary constant, and reduce a given NLPDE, say in two independent variables, to the (ODE) 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 where a i (i = 0, 1, 2, ..., N) are constants to be determined and F(ξ) satisfies the first order nonlinear ODE in the form where q 0 , q 2 and q 4 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 , ...a N , k and ω.
Step 4. Solving these equations, probably with the aid of Mathematica or Maple, then a 0 , a 1 , ...a N , 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 4 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 0 is an arbitrary constant.Thus, Eq.
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][23][24][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.

The KdV and modified KdV equations
The Korteweg de-Vries (KdV) equation 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) 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] 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

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Exact Solutions Expressible in Hyperbolic and Jacobi Elliptic Functions of Some Important Equations of Ion-Acoustic Waves www.intechopen.com Balancing u ′′′ with u 2 u ′ yields N = 1, so the F-expansion method gives u(x, t)=a 0 + a 1 F(ξ).
(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 ) Thus, we obtain the periodic wave solutions of Eq. (2.1) Selecting the values of the q 0 , q 2 and q 4 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 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.

The JEF solutions of Schamel-KdV equation
We consider the Schamel-KdV equation [29,30] 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 The balancing procedure implies that N = 1.Therefore, the F-expansion method gives the solution V(x, t)=V(ξ)=a 0 + a 1 F(ξ), ( where a 0 and a 1 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: Solving these algebraic equations, we gave a general form of traveling wave solutions of Eq. (3.1) Therefore, we obtained in [30] the JEF solutions of Eq. (3.1) as follows: When If If q 0 = m 2 − 1, q 2 = 2 − m 2 , q 4 = −1, the solution of Eq (1.4) is F(ξ)=dnξ.So, we obtained the exact solutions of Eq. (3.1) in the form Many types of JEF solutions of Eq. (3.1) are given [30].As m −→ 1, Eqs. (3.6)-(3.8)degenerate to , βδ < 0, 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.

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

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 Schamel [29] derived this equation and a simple solitary wave solution having a sech 4 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 u = v 2 , v(x, t)=V(η); η = kx − ω t + η 0 ,, then (4.1) becomes According to the mapping method, we assume that Eq. ( 4.2) has the following solution: where A 0 and A 1 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,w eh a v et h e solutions of Eq. (4.1) which was given in [20] as follows: In this case, we have f (η)=sn 2 η.Thus the periodic wave solutions of Eq. (4.1) are , q = 2(1 + m 2 ), r = −3 2 .T h es o l u t i o n so fE q .(1.6) are f (η)= (m cnη ± dnη) 2 .Thus the exact solutions of Eq. (4.1) are . 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 m → 1, these solutions reduce to the solitary wave solutions 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][35][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 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]).

The ZK equation and modified ZK equation
The equation 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 Substituting this equation into Eq.(5.1), we obtain the following classes of exact solutions of the modified ZK equation: 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 73 Exact Solutions Expressible in Hyperbolic and Jacobi Elliptic Functions of Some Important Equations of Ion-Acoustic Waves 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]: ) ) (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] where A, B, C and D are constants.This equation has the following JEF solutions (see [45,46]): , BE < 0, , BE > 0, (5.8) with E = Cγ 2 + D(1 + l 2 ).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].

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][48][49][50][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

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Acoustic Waves -From Microdevices to Helioseismology www.intechopen.comwhere β, c 3 and c 5 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 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: If we choose A 0 = 0, equation (6.3) takes the form Moreover, we have obtained the exact solutions 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 u t +(α + β u p ) u p u x + u xxx + δu yyx = 0, (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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Exact Solutions Expressible in Hyperbolic and Jacobi Elliptic Functions of Some Important Equations of Ion-Acoustic Waves www.intechopen.comhot isothermal electrons in the presence of a uniform magnetic field.Eq. (6.9) includes considerable interesting equations, such as KdV equation, mKdV equation, ZK equation and mZK equation.Exact traveling wave solutions for the generalized ZK equation with higher-order nonlinear terms have obtained in [52][53][54].Moreover, we can use the symbolic computations and apply the mapping method with the ansatz solution (1.5) to find the several classes of traveling wave solutions of the fifth order KdV equation