Bright and dark soliton solutions for a new fifth-order nonlinear integrable equation with perturbation terms

Houria Triki; Abdul-Majid Wazwaz

doi:10.1016/j.jksus.2011.08.004

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ORIGINAL ARTICLE

24 (

); 295-299

doi:

10.1016/j.jksus.2011.08.004

Bright and dark soliton solutions for a new fifth-order nonlinear integrable equation with perturbation terms

Houria Triki^a, Abdul-Majid Wazwaz^b,*

a Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, Annaba 23000, Algeria

b Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA

*Corresponding author. Tel.: +1 773 298 3397 wazwaz@sxu.edu (Abdul-Majid Wazwaz)

Received: 2011-7-20, Accepted: 2011-8-14, Published: 2011-07-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Available online 22 August 2011

Abstract

In this paper, we use the solitary wave ansatz method to carry out the integration of a new fifth-order nonlinear equation having perturbation terms. The perturbation terms that are considered are the first-order dispersion term, the power law nonlinearity term, and the fifth- order dispersion term. Both bright and dark soliton solutions are obtained. The physical parameters in the soliton solutions: amplitude, inverse width, and velocity are obtained as functions of the dependent model coefficients. The conditions of the existence of the derived solitons are presented.

Keywords

Bright and dark solitons

K(m, n) equation

Solitary wave ansatz

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1 Introduction

Recent advances in the study of the wave dynamics in nonlinear systems have led to the construction of new integrable models exhibiting a rich variety of exact solutions with interesting properties. A new class of nonlinear partial differential equations (NLPDEs) with constant and variable coefficients that model a lot of physical phenomena has been formulated depending on the physical situation. In parallel, some other NLPDEs are generalized which offer a rich knowledge on the propagation behavior of waves in nonlinear systems of all kinds. Examples of many equations that have been generalized are the K(m, n) equation (Rosenau and Hyman, 1993), the B(m, n) equation (Zhang, 2006; Biswas, 2009b), the ZK(m, n, k) equation (Inc, 2007), the CKG(m, n, k) (Yan, 2007) and so on.

The problem of finding exact solutions for nonlinear models is of great importance, both in mathematical and physical point of view. Based on the exact solutions, we can accurately analyze the properties of the propagating waves in a given nonlinear system. Several kinds of closed form solutions, including compactons, peakons, solitons, cuspons, similaritons, etc., have been obtained under specific conditions. In particular, exact solutions of soliton-type are of general physical interest, because the soliton approach is universal in different fields of modern physics. Notice that the soliton pulses are observed in many physics areas, such as fluid mechanics, plasma physics, optical fibers, hydrodynamics, biology, solid state physics, etc. It is worth noting that, the existence of such shape-preserving waves in the nonlinear systems is the result of exact balance among dispersion (or diffraction) and the nonlinearity effects.

Recently, in the mathematical physics literature has appeared a large amount of new powerful methods to calculate exact solutions to NLPDEs. Among these modern methods of integrability, we can cite the coupled amplitude-phase formalism (Palacios et al., 1999; Du et al., 1995), the hyperbolic tangent method (Malfliet, 1992), Hirota bilinear method (Nakkeeran, 2002; Wazwaz, 2010), the sub-ODE method (Li and Wang, 2007; Triki and Wazwaz, 2009), the solitary wave ansatz method (Biswas and Milovic, 2010; Saha et al., 2009; Triki and Wazwaz, 2009; Biswas, 2008a,b, 2009a ) and other methods as well. The solitary wave ansatz has recently been applied successfully to a wide range of NLPDEs (Biswas and Milovic, 2010; Saha et al., 2009; Triki and Wazwaz, 2009; Biswas, 2008a,b, 2009a ).

Very recently, Wazwaz (2011a,b) introduced a new fifth-order nonlinear integrable evolution equation in the form

(1.a)

u_{ttt} - u_{txxxx} - {(u_{x} u_{t})}_{xx} - 4 {(u_{x} u_{xt})}_{x} = 0,

and a new generalized fifth-order nonlinear integrable equation

(1.b)

u_{ttt} - u_{txxxx} - α {(u_{x} u_{t})}_{xx} - β {(u_{x} u_{xt})}_{x} = 0,

and derived the multiple soliton solutions for each equation by using the Hereman–Nuseri method.

In this work we prove that a family of this new evolution equation having perturbation terms in the form

(1)

u_{ttt} - {au}_{txxxx} - b {(u_{x} u_{t})}_{xx} - c {(u_{x} u_{xt})}_{x} = α u_{x} + γ u^{m} u_{x} + δ u_{xxxxx}

possesses exact bright and dark soliton solutions under certain parametric conditions. It should be noted that the finding of a new model supporting soliton-type solutions is a very interesting result and is also helpful for future research.

Here, in (2), a, b, c, α, γ and δ are all constants. The three perturbation terms on the right-hand side of Eq. (2) represent the first-order dispersion term, the power law nonlinearity term, and the fifth-order dispersion term, respectively. Also, noting that the parameter m indicates the power law nonlinearity. On setting a = 1, b = c = 4, and α = γ = δ = 0, Eq. (2) reduces to the model Eq. (1.a).

2 Bright solitons

In order to obtain the bright soliton solution to (2), the solitary wave ansatz is assumed as (Triki and Wazwaz, 2009; Biswas, 2008a; Biswas, 2009a )

(2)

u (x, t) = \frac{A}{\cosh^{p} τ}

where

(3)

τ = B (x - vt)

Here A is the soliton amplitude, B is the inverse width of the soliton and v is the soliton velocity. The unknown index p will be determined during the course of derivation of the solution of this Eq. (2).

From (3) and (4), we find

(4)

u_{ttt} = p^{3} {Av}^{3} B^{3} \frac{\tanh τ}{\cosh^{p} τ} - p (p + 1) (p + 2) {Av}^{3} B^{3} \frac{\tanh τ}{\cosh^{p + 2} τ}

(5)

u_{txxxx} = p^{5} {vAB}^{5} \frac{\tanh τ}{\cosh^{p} τ} - 2 p (p + 1) (p + 2) (p^{2} + 2 p + 2) {vAB}^{5} \frac{\tanh τ}{\cosh^{p + 2} τ} + p (p + 1) (p + 2) (p + 3) (p + 4) {vAB}^{5} \frac{\tanh τ}{\cosh^{p + 4} τ},

(6)

{(u_{x} u_{t})}_{xx} = - 4 p^{4} {vA}^{2} B^{4} \frac{1}{\cosh^{2 p} τ} + 2 p^{2} (4 p^{2} + 5 p + 2) {vA}^{2} B^{4} \frac{1}{\cosh^{2 p + 2} τ} - 2 p^{2} (p + 1) (2 p + 3) {vA}^{2} B^{4} \frac{1}{\cosh^{2 p + 4} τ},

(7)

{(u_{x} u_{xt})}_{x} = - 2 p^{4} {vA}^{2} B^{4} \frac{1}{\cosh^{2 p} τ} + p^{2} (4 p^{2} + 5 p + 2) {vA}^{2} B^{4} \frac{1}{\cosh^{2 p + 2} τ} - p^{2} (p + 1) (2 p + 3) {vA}^{2} B^{4} \frac{1}{\cosh^{2 p + 4} τ},

(8)

u_{x} = - pAB \frac{\tanh τ}{\cosh^{p} τ},

(9)

u^{m} u_{x} = - {pA}^{m + 1} B \frac{\tanh τ}{\cosh^{p (m + 1)} τ},

and

(10)

u_{xxxxx} = - p^{5} {AB}^{5} \frac{\tanh τ}{\cosh^{p} τ} + 2 p (p + 1) (p + 2) (p^{2} + 2 p + 2) {AB}^{5} \frac{\tanh τ}{\cosh^{p + 2} τ} - p (p + 1) (p + 2) (p + 3) (p + 4) {AB}^{5} \frac{\tanh τ}{\cosh^{p + 4} τ} .

Substituting (5)–(11) into (2) gives

(11)

(p^{3} v^{3} {AB}^{3} - {ap}^{5} {vAB}^{5} + δ p^{5} {AB}^{5} + α pAB) \frac{\tanh τ}{\cosh^{p} τ} + {- p (p + 1) (p + 2) {Av}^{3} B^{3} + 2 (av - δ) p (p + 1) (p + 2) (p^{2} + 2 p + 2) {AB}^{5}} \frac{\tanh τ}{\cosh^{p + 2} τ} - (av - δ) p (p + 1) (p + 2) (p + 3) (p + 4) {AB}^{5} \frac{\tanh τ}{\cosh^{p + 4} τ} + 2 (2 b + c) p^{4} {vA}^{2} B^{4} \frac{1}{\cosh^{2 p} τ} - (2 b + c) p^{2} (4 p^{2} + 5 p + 2) {vA}^{2} B^{4} \frac{1}{\cosh^{2 p + 2} τ} + (2 b + c) p^{2} (p + 1) (2 p + 3) {vA}^{2} B^{4} \frac{1}{\cosh^{2 p + 4} τ} + γ {pA}^{m + 1} B \frac{1}{\cosh^{(m + 1) p} τ} = 0 .

Now, from (12), equating the exponents (m + 1)p and p + 2 leads to

(12)

(m + 1) p = p + 2,

which gives

(13)

p = \frac{2}{m} .

From (12) setting the coefficients of tanh τ/cosh^p+j τ and 1/cosh^2p+j τ functions to zero where j = 0, 2, 4, since these are linearly independent functions, gives the following algebraic equations:

(14)

p^{3} v^{3} {AB}^{3} - {ap}^{5} {vAB}^{5} + δ p^{5} {AB}^{5} + α pAB = 0,

(15)

\begin{matrix} - p (p + 1) (p + 2) {Av}^{3} B^{3} + 2 (av - δ) p (p + 1) (p + 2) \\ \times (p^{2} + 2 p + 2) {AB}^{5} + γ {pA}^{m + 1} B = 0, \end{matrix}

(16)

- (av - δ) p (p + 1) (p + 2) (p + 3) (p + 4) {AB}^{5} = 0,

(17)

2 (2 b + c) p^{4} {vA}^{2} B^{4} = 0,

(18)

- (2 b + c) p^{2} (4 p^{2} + 5 p + 2) {vA}^{2} B^{4} = 0,

(19)

(2 b + c) p^{2} (p + 1) (2 p + 3) {vA}^{2} B^{4} = 0 .

Solving the above system yields

(20)

A = {[- \frac{(m + 1) (m + 2) α}{2 γ}]}^{\frac{1}{m}}, γ \neq 0,

(21)

v = \frac{δ}{a}, a \neq 0,

(22)

B = \frac{ma}{2 δ} \sqrt{- \frac{α a}{δ}},

and

(23)

c = - 2 b .

Eq. (21) shows that solitons will exist for

(24)

α γ < 0,

if m is an even integer. However, if m is an odd integer there is no such restriction but the soliton will be pointing downwards.

From (23) we clearly see that the solitons will exist for

(25)

δ α a < 0 .

Thus, finally, the bright soliton solution to the new fifth-order nonlinear evolution equation with perturbation terms (2) is given by

(26)

u (x, t) = \frac{A}{\cosh^{\frac{2}{m}} [B (x - vt)]},

where the amplitude A and the width B are given by (21) and (23) respectively while the velocity of the soliton is given by (22). Finally the constraint relation between the nonlinear dispersion parameters c and b is displayed in (24).

3 Dark solitons

In this section the search is going to be for the topological 1-soliton solution to the new fifth-order nonlinear evolution equation given by (2). To start off, the hypothesis is given by (Triki and Wazwaz, 2009)

(27)

u (x, t) = A \tanh^{p} τ,

where

(28)

τ = B (x - vt),

where in (28) and (29), A and B are free parameters and v is the velocity of the wave. Also, the unknown exponent p will be determined during the course of the derivation of the soliton solution to (2). Therefore from (28), we get

(29)

u_{ttt} = - {pAB}^{3} v^{3} [(p - 1) (p - 2) \tanh^{p - 3} τ - {2 p^{2} + (p - 1) (p - 2)} \tanh^{p - 1} τ + {2 p^{2} + (p + 1) (p + 2)} \tanh^{p + 1} τ - (p + 1) (p + 2) \tanh^{p + 3} τ],

(30)

{(u_{x} u_{t})}_{xx} = - 2 p^{2} A^{2} B^{4} v [(p + 1) (2 p + 3) \tanh^{2 p + 4} τ + (p - 1) (2 p - 3) \tanh^{2 p - 4} τ - 2 (4 p^{2} + 5 p + 2) \tanh^{2 p + 2} τ - 2 (4 p^{2} - 5 p + 2) \tanh^{2 p - 2} τ + 2 (6 p^{2} + 1) \tanh^{2 p} τ],

(31)

{(u_{x} u_{xt})}_{x} = - p^{2} A^{2} B^{4} v [(p + 1) (2 p + 3) \tanh^{2 p + 4} τ + (p - 1) (2 p - 3) \tanh^{2 p - 4} τ - 2 (4 p^{2} + 5 p + 2) \tanh^{2 p + 2} τ - 2 (4 p^{2} - 5 p + 2) \tanh^{2 p - 2} τ + 2 (6 p^{2} + 1) \tanh^{2 p} τ],

(32)

u_{x} = pAB (\tanh^{p - 1} τ - \tanh^{p + 1} τ),

(33)

u^{m} u_{x} = {pA}^{m + 1} B (\tanh^{p (m + 1) - 1} τ - \tanh^{p (m + 1) + 1} τ),

(34)

u_{txxxx} = - {pAB}^{5} v \{(p - 1) (p - 2) (p - 3) (p - 4) \tanh^{p - 5} τ - (p + 1) (p + 2) (p + 3) (p + 4) \tanh^{p + 5} τ - (p - 1) (p - 2) {2 p^{2} + 2 {(p - 2)}^{2} + (p - 3) (p - 4)} \tanh^{p - 3} τ + (p + 1) (p + 2) {2 p^{2} + 2 {(p + 2)}^{2} + (p + 3) (p + 4)} \tanh^{p + 3} τ + [2 (p - 1) (p - 2) {p^{2} + {(p - 2)}^{2}} + 4 p^{4} + p {(p - 1)}^{2} (p - 2) + p {(p + 1)}^{2} (p + 2)] \tanh^{p - 1} τ - [2 (p + 1) (p + 2) {p^{2} + {(p + 2)}^{2}} + 4 p^{4} + p {(p - 1)}^{2} (p - 2) + p {(p + 1)}^{2} (p + 2)] \tanh^{p + 1} τ\},

and

(35)

u_{xxxxx} = {pAB}^{5} \{(p - 1) (p - 2) (p - 3) (p - 4) \tanh^{p - 5} τ - (p + 1) (p + 2) (p + 3) (p + 4) \tanh^{p + 5} τ - (p - 1) (p - 2) {2 p^{2} + 2 {(p - 2)}^{2} + (p - 3) (p - 4)} \tanh^{p - 3} τ + (p + 1) (p + 2) {2 p^{2} + 2 {(p + 2)}^{2} + (p + 3) (p + 4)} \tanh^{p + 3} τ + [2 (p - 1) (p - 2) {p^{2} + {(p - 2)}^{2}} + 4 p^{4} + p {(p - 1)}^{2} (p - 2) + p {(p + 1)}^{2} (p + 2)] \tanh^{p - 1} τ - [2 (p + 1) (p + 2) {p^{2} + {(p + 2)}^{2}} + 4 p^{4} + p {(p - 1)}^{2} (p - 2) + p {(p + 1)}^{2} (p + 2)] \tanh^{p + 1} τ\} .

Now substituting (30)–(36) into (2) gives

(36)

- {pAB}^{3} v^{3} [(p - 1) (p - 2) \tanh^{p - 3} τ - {2 p^{2} + (p - 1) (p - 2)} \tanh^{p - 1} τ + {2 p^{2} + (p + 1) (p + 2)} \tanh^{p + 1} τ - (p + 1) (p + 2) \tanh^{p + 3} τ] + (av - δ) {pAB}^{5} {(p - 1) (p - 2) (p - 3) (p - 4) \tanh^{p - 5} τ - (p + 1) (p + 2) (p + 3) (p + 4) \tanh^{p + 5} τ - (p - 1) (p - 2) {2 p^{2} + 2 {(p - 2)}^{2} + (p - 3) (p - 4)} \tanh^{p - 3} τ + (p + 1) (p + 2) {2 p^{2} + 2 {(p + 2)}^{2} + (p + 3) (p + 4)} \tanh^{p + 3} τ + [2 (p - 1) (p - 2) {p^{2} + {(p - 2)}^{2}} + 4 p^{4} + p {(p - 1)}^{2} (p - 2) + p {(p + 1)}^{2} (p + 2)] \tanh^{p - 1} τ - [2 (p + 1) (p + 2) {p^{2} + {(p + 2)}^{2}} + 4 p^{4} + p {(p - 1)}^{2} (p - 2) + p {(p + 1)}^{2} (p + 2)] \tanh^{p + 1} τ} + (2 b + c) p^{2} A^{2} B^{4} v [(p + 1) (2 p + 3) \tanh^{2 p + 4} τ + (p - 1) (2 p - 3) \tanh^{2 p - 4} τ - 2 (4 p^{2} + 5 p + 2) \tanh^{2 p + 2} τ - 2 (4 p^{2} - 5 p + 2) \tanh^{2 p - 2} τ + 2 (6 p^{2} + 1) \tanh^{2 p} τ] - α pAB (\tanh^{p - 1} τ - \tanh^{p + 1} τ) - γ {pA}^{m + 1} B (\tanh^{p (m + 1) - 1} τ - \tanh^{p (m + 1) + 1} τ) = 0 .

From (37), equating the exponents p(m + 1) + 1 and p + 3 gives

(37)

p (m + 1) + 1 = p + 3,

so that

(38)

p = \frac{2}{m} .

It needs to be noted that the same value of p is yielded when the exponents p(m + 1) −1 and p + 1 are equated with each other.

Now from (37) the linearly independent functions are tanh^p+j τ for j = ±1, ±3, ±5 and tanh^2p+k τ for k = 0, ±2, ±4. Hence setting their respective coefficients to zero yields the following set of equations:

(39)

\begin{matrix} - {pAB}^{3} v^{3} (p - 1) (p - 2) \\ - (av - δ) {pAB}^{5} (p - 1) (p - 2) {2 p^{2} + 2 {(p - 2)}^{2} \\ + (p - 3) (p - 4)} = 0, \end{matrix}

(40)

\begin{matrix} + {pAB}^{3} v^{3} {2 p^{2} + (p - 1) (p - 2)} - α pAB \\ + (av - δ) {pAB}^{5} [2 (p - 1) (p - 2) {p^{2} + {(p - 2)}^{2}} \\ + 4 p^{4} + p {(p - 1)}^{2} (p - 2) + p {(p + 1)}^{2} (p + 2)] = 0, \end{matrix}

(41)

\begin{matrix} - {pAB}^{3} v^{3} {2 p^{2} + (p + 1) (p + 2)} + α pAB - γ {pA}^{m + 1} B \\ - (av - δ) {pAB}^{5} [2 (p + 1) (p + 2) {p^{2} + {(p + 2)}^{2}} + 4 p^{4} \\ + p {(p - 1)}^{2} (p - 2) + p {(p + 1)}^{2} (p + 2)] = 0, \end{matrix}

(42)

\begin{matrix} {pAB}^{3} v^{3} (p + 1) (p + 2) + (av - δ) {pAB}^{5} (p + 1) (p + 2) \\ \times {2 p^{2} + 2 {(p + 2)}^{2} + (p + 3) (p + 4)} \\ + γ {pA}^{m + 1} B = 0, \end{matrix}

(43)

(av - δ) {pAB}^{5} (p - 1) (p - 2) (p - 3) (p - 4) = 0

(44)

- (av - δ) {pAB}^{5} (p + 1) (p + 2) (p + 3) (p + 4) = 0,

(45)

(2 b + c) p^{2} A^{2} B^{4} v (p + 1) (2 p + 3) = 0,

(46)

(2 b + c) p^{2} A^{2} B^{4} v (p - 1) (2 p - 3) = 0,

(47)

- 2 (2 b + c) p^{2} A^{2} B^{4} v (4 p^{2} + 5 p + 2) = 0,

(48)

- 2 (2 b + c) p^{2} A^{2} B^{4} v (4 p^{2} - 5 p + 2) = 0,

(49)

2 (2 b + c) p^{2} A^{2} B^{4} v (6 p^{2} + 1) = 0 .

By solving (44)–(50), we get

(50)

v = \frac{δ}{a}

and

(51)

c = - 2 b .

By substituting (51) into (40), we obtain

(52)

- {pAB}^{3} v^{3} (p - 1) (p - 2) = 0 .

To solve (53), we have considered the following two cases:

3.1

3.1 Case 1: p = 1

From (39), this yields

(53)

m = 2 .

Further substitution of p = 1 into (41)–(43), respectively, gives

(54)

A = {(\frac{- 3 α}{γ})}^{\frac{1}{m}},

and

(55)

B = \frac{a}{δ} \sqrt{\frac{α a}{2 δ}},

which shows that solitons will exist for

(56)

a α δ > 0 .

3.2

3.2 Case 2: p = 2

From (39), this yields

(57)

m = 1 .

By substituting p = 2 into (41)–(43), respectively, we obtain

(58)

A = {(- \frac{3 α}{2 γ})}^{\frac{1}{m}},

and

(59)

B = \frac{a}{2 δ} \sqrt{\frac{α a}{2 δ}},

which shows that solitons will exist for

(60)

a α δ > 0 .

Lastly, we can determine the dark soliton solution for the new evolution equation with perturbation terms (2) when we substitute (51), (55) and (56) in (28) with the respective constraint (54) for the first case of solution or we substitute (51), (59) and (60) in (28) with the respective constraint (58) for the second case of solution as

(61)

u (x, t) = A \tanh^{\frac{2}{m}} [B (x - vt)],

which exists provided that aαδ > 0 and c = −2b.

4 Conclusion

This paper obtains the bright and dark soliton solutions of a new fifth-order nonlinear equation in presence of perturbation terms including the power law nonlinearity. The solitary wave ansatz method is employed to integrate the considered equation. Parametric conditions for the existence of the soliton solutions are found. In view of the analysis, we see that the examined equation is an interesting model for soliton-type solutions. In addition, we note that the solitary wave ansatz method is an efficient method for constructing exact soliton solutions for such nonlinear evolution equation that includes perturbation terms.

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Introduction

Bright solitons

Dark solitons

Case 1: p = 1
Case 2: p = 2

Conclusion

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[1] Biswas A., . 1-soliton solution of the K(m, n) equation with generalized evolution. Phys. Lett. A. 2008;372:4601-4602.
[Google Scholar]

[2] Biswas A., . 1-Soliton solution of (1+2) dimensional nonlinear Schrödinger’s equation in dual-power law media. Phys. Lett. A. 2008;372:5941-5943.
[Google Scholar]

[3] Biswas A., . Solitary wave solution for the generalized Kawahara equation. Appl. Math. Lett.. 2009;22:208-210.
[Google Scholar]

[4] Biswas A., . 1-soliton solution of the B(m, n) equation with generalized evolution. Commun. Nonlinear Sci. Numer. Simul.. 2009;14:3226-3229.
[Google Scholar]

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