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formerly Central European Journal of Physics

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Volume 16, Issue 1

Issues

Volume 13 (2015)

Solitary wave solutions of two KdV-type equations

Khalil Salim Al-Ghafri
  • Corresponding author
  • Ibri College of Applied Sciences, Ministry of Higher Education, P.O. Box: 14 Ibri, Postal Code: 516, Oman
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Published Online: 2018-05-30 | DOI: https://doi.org/10.1515/phys-2018-0043

Abstract

The present paper investigates the solitary wave solutions of the nonlinear evolution equations with power nonlinearties. The study has been carried out for two examples of KdV-type equations, namely, the nonlinear dispersive equation and the generalised KdV equation. To achieve our goal, we have applied the projective Riccati equation method. As a result, many exact solutions in the form of solitary wave solutions and combined formal solitary wave solutions are obtained

Keywords: Solitary wave solutions; projective Riccati equation; the nonlinear dispersive equation; the generalised KdV equation

PACS: 02.30.Jr; 02.70.Wz; 04.20.Jb; 05.45.Yv

1 Introduction

Recently, the nonlinear evolution equations (NEEs) with power nonlinearties have been under a considerable concentration since they are widely used to describe complex phenomena in various fields of science, such as fluid mechanics, plasma physics, solid state physics and optical fibres, etc. Thus, the search for explicit exact solutions, in particular, solitary wave solutions, is very important, as it allows to understand these nonlinear phenomena and to provide valuable reference for other related research. Many powerful and efficient methods have been developed, such as Backlund transformation method [1, 2], Hirota’s direct method [3], tanh method[4, 5], extended tanh method [6, 7, 8], sine-cosine method [9, 10, 11], F-expansion method [12, 13, 14, 15], (G′/G)-expansion method [16, 17, 18], exp-function method [19, 20, 21], rational function method [22], first integral method [23, 24], auxiliary equation method [25, 26, 27, 28], and so on. Furthermore, the fractional differential equations have been found to govern some physical models in the natural science. Such equations have fractional derivatives with arbitrary order and different kinds in physical applications. Thus, the use of fractional calculus tools may depend on the type of fractional derivatives. For more information on the fractional calculus, see [29, 30, 31].

In the last three decades, one of the more effective methods that has been developed to seek many kinds of exact solutions to NEEs is the projective Riccati equation method. For example, in 1992, Conte and Musette [32] introduced this method to find more new solitary wave solutions to NEEs that can be expressed as polynomial in two elementary functions which satisfy a projective Riccati equation. Thereafter, many studies have been carried out to generalise the projective Riccati equation method and to obtain a variety of exact solutions to nonlinear partial diffrenetial equations. For more details, see [33, 34, 35, 36, 37, 38]. Recently, Kumar and Chand [39] have examined the exact traveling wave solutions of some nonlinear evolution equations. They assumed that the projective Riccati equations are given in the form

f(ξ)=pf(ξ)g(ξ),g(ξ)=R+pg2(ξ)rf(ξ),g2(ξ)=p[R2rf(ξ)+r2+δRf2(ξ)],(1)

where p, R and r are constants and δ = ±1. It is found that the system of Eq. (1) admits solutions in terms of hyperbolic functions when p = −1 while the system possess solutions in terms of trigonometric functions once p = 1.

In this article, we utilise the projective Riccati system (1) with p = −1 and R = 1 to seek various solitary wave solutions for two KdV-type equations with power nonlinearties. In Section (2), we describe the projective Riccati equation method and its technique for solving NEEs. Then, we apply the method to solve the nonlinear dispersive equation in Section (3) and the generalised KdV equation in Section (4). Finally, in Section (5) we present our discussion and conclusion.

2 Analysis of the method

Consider a nonlinear partial differential equation (NLPDE) for u(x, t) in the form

P(u,ut,ux,uxx,uxt,utt,uxxx)=0,(2)

where P is a polynomial in its arguments. The principle of the generalised projective Riccati method will be demonstrated as follows.

Since we seek for travelling wave solutions, we introduce the wave variable

u(x,t)=U(ξ),ξ=xct,(3)

and this transforms the NLPDE (2) to the following ordinary differential equation (ODE)

Q(U,U,U,U,)=0,(4)

where prime denotes the derivative with respect to ξ. Then, integrate (4), if possible, to reduce the order of differentiation.

Now, assume that the solution of (4) can be expressed in the finite series form

U(ξ)=i=0maifi(ξ)+j=1mbjfj1(ξ)g(ξ),(5)

where a0, ai, bj, (i, j = 1, 2, …, m) are constants to be determined. The parameter m, which is a positive integer, can be determined by balancing the highest order derivative terms with the nonlinear terms in (4).

The variables f(ξ), g(ξ) satisfy the projective Riccati equations

f(ξ)=f(ξ)g(ξ),g(ξ)=1g2(ξ)rf(ξ),g2(ξ)=12rf(ξ)+(r2+ϵ)f2(ξ),(6)

where ϵ = ±1 and r is arbitrary constant. The set of equations (6) is found to accept the following solutions

f1(ξ)=αβcosh(ξ)+γsinh(ξ)+αr,g1(ξ)=βsinh(ξ)+γcosh(ξ)βcosh(ξ)+γsinh(ξ)+αr,(7)

when ϵ = 1: α, β, γ satisfy γ2 = α2 + β2 and when ϵ = −1: α, β, γ satisfy β2 = α2 + γ2.

f2(ξ)=1sinh(ξ)+r,g2(ξ)=cosh(ξ)sinh(ξ)+r,(8)

demands ϵ = 1.

f3(ξ)=1cosh(ξ)+r,g3(ξ)=sinh(ξ)cosh(ξ)+r,(9)

implies ϵ = −1.

The substitution of (5) along with (6) into (4) generates a polynomial in fi(ξ) gj(ξ). Equating each coefficient of fi(ξ) gj(ξ) in this polynomial to zero, yields a set of algebraic equations for ai, bj. Solving this system of equations, we can obtain many exact solutions of (2) according to (7)-(9).

In the following two sections, we will apply this method for the nonlinear dispersive equation and the generalised KdV equation.

3 The nonlinear dispersive equation

Let us consider the nonlinear dispersive equation of the form

un(un)t+a(u3n)x+bun(un)xxx=0,n>0,(10)

where a, b are constants and the dependent variable u(x, t) represents the wave profile. The parameter n is a positive integer with n > 0. The effect of nonlinearity of the middle term of (10) on the physical structure of the resulting solutions has been previously studied [40, 41, 42]. It is found that the exponent of the middle term manipulates the existence of compactons, solitons and periodic solutions.

In order to deal with the complex form of (10), we use the transformation (3) and then suppose that

v=Un.(11)

Hence, we arrive at

cv(v)+3av2(v)+bv(v)=0.(12)

Dividing both sides by v and integrating once with respect to ξ, we obtain

kcv+32av2+bv=0,(13)

where k is the integration constant. Now, we assume that the solutions of (13) can be expressed by

v(ξ)=i=0maifi+j=1mbjfj1g,(14)

where f = f(ξ), g = g(ξ) satisfy (6). The homogeneous balance between the highest order derivative term and nonlinear term in (13) leads to m = 2. So we have

v(ξ)=a0+a1f+a2f2+b1g+b2fg.(15)

Substituting (15) along with (6) into (13) and collecting all the terms with the same power of figj together, equating each coefficient to zero, yields a set of algebraic equations. Solving these equations, leads to

  • Case 1

    a0=c4b3a,a2=4bϵa,k=c216b26a,r=0,a1=b1=b2=0.(16)

  • Case 2

    a0=cb3a,a2=2bϵa,b2=±2bϵa,k=c2b26a,r=0,a1=b1=0.(17)

  • Case 3

    a0=cb3a,a1=±2bϵa,k=c2b26a,r=±ϵ,a2=b1=b2=0.(18)

  • Case 4

    a0=cb3a,a1=2bra,a2=2b(r2+ϵ)a,b2=±2br2+ϵa,k=c2b26a,b1=0.(19)

According to the above results combined with (7)-(9), (11) and (15) we can obtain the following families of solitary wave solutions to the nonlinear dispersive equation.

From Case 1 we reach the solutions

u1={c4b3a4bα2ϵa[βcosh(ξ)+γsinh(ξ)]2}1/n,(20)

when ϵ = 1: α, β, γ satisfy γ2 = α2 + β2 and when ϵ = −1: α, β, γ satisfy β2 = α2 + γ2.

u2={c4b3a4ba[sinh(ξ)]2}1/n,ϵ=1.(21)

u3={c4b3a+4ba[cosh(ξ)]2}1/n,ϵ=1.(22)

From Case 2 we obtain

u4={cb3a2bα2ϵa[βcosh(ξ)+γsinh(ξ)]2±2bϵaα[βsinh(ξ)+γcosh(ξ)][βcosh(ξ)+γsinh(ξ)]2}1/n,(23)

when ϵ = 1: α, β, γ satisfy γ2 = α2 + β2 and when ϵ = −1: α, β, γ satisfy β2 = α2 + γ2.

u5={cb3a2ba[1±cosh(ξ)][sinh(ξ)]2}1/n,ϵ=1.(24)

u6={cb3a+2ba[1±isinh(ξ)][cosh(ξ)]2}1/n,ϵ=1.(25)

From Case 3 we arrive at

u7={cb3a±2bαϵa[βcosh(ξ)+γsinh(ξ)±αϵ]}1/n,(26)

when ϵ = 1: α, β, γ satisfy γ2 = α2 + β2 and when ϵ = −1: α, β, γ satisfy β2 = α2 + γ2.

u8={cb3a±2iba[sinh(ξ)±i]}1/n,ϵ=1.(27)

u9={cb3a±2ba[cosh(ξ)±1]}1/n,ϵ=1.(28)

From Case 4 we obtain

u10={cb3a+2brαa[βcosh(ξ)+γsinh(ξ)+αr]2b(r2+ϵ)α2a[βcosh(ξ)+γsinh(ξ)+αr]2±2br2+ϵaα[βsinh(ξ)+γcosh(ξ)][βcosh(ξ)+γsinh(ξ)+αr]2}1/n,(29)

when ϵ = 1: α, β, γ satisfy γ2 = α2 + β2 and when ϵ = −1: α, β, γ satisfy β2 = α2 + γ2.

u11={cb3a+2bra[sinh(ξ)+r]2br2+1a[r2+1±cosh(ξ)][sinh(ξ)+r]2}1/n,ϵ=1.(30)

u12={cb3a+2bra[cosh(ξ)+r]2br21a[r21±sinh(ξ)][cosh(ξ)+r]2}1/n,ϵ=1.(31)

Remarks

  1. Case 2 can be recovered from Case 4 when r = 0 and therefore the solutions (29) - (31) are converted to (23) - (25).

  2. Case 3 can be recovered from Case 4 if r=±ϵ, so the solutions (29) - (31) reduce to (26) - (28).

  3. Once β = 0 in γ2 = α2 + β2 then γ = α. Therefore, solutions (20), (23), (26) and (29) change into solutions (21), (24), (27) and (30), respectively.

  4. If γ = 0 in β2 = α2 + γ2 so β = α and hence solutions (20), (23), (26) and (29) are transformed into solutions (22), (25), (28) and (31), respectively.

To shed light on the behaviour of solitary wave solutions, we shall present the profile of some obtained solutions of the nonlinear dispersive equation in Figures 14. For example, the solutions u2 and u5 display the singular waves while the solutions u3 and u12 show the bell-shaped solitary waves. The graphs are depicted with a = 1, b = 1, n = 2 and c = 6 for the solutions u2 and u3 whereas c = 4 for the solutions u5 and u12. The value of constant r in the solution u12 is taken to be 2.

Graphical representation of solution (21)
Figure 1

Graphical representation of solution (21)

Graphical representation of solution (22)
Figure 2

Graphical representation of solution (22)

Graphical representation of solution (24)
Figure 3

Graphical representation of solution (24)

Graphical representation of solution (31)
Figure 4

Graphical representation of solution (31)

4 The generalised KdV equation

In this section we study the generalised KdV equation with two power nonlinearities [43] given by

ut+(aunbu2n)ux+uxxx=0,n>0,(32)

where a, b are constants. This equation describes the propagation of nonlinear long acoustic-type waves. Once the amplitude is not supposed to be small, Eq. (32) serves as an approximate model for the description of weak dispersive effects on the propagation of nonlinear waves along a characteristic direction [43]. It can be noted that for n = 1, (32) represents the standard Gardner equation, or the combined KdV-mKdV equation.

The use of wave transformation (3) reduces (32), after integrating once, to an ODE in the form

cU+an+1Un+1b2n+1U2n+1+U=0,(33)

where integration constant is considered zero. It is more convenient to replace the exponents of nonlinear terms by integers in order to seek for a closed form solution. Accordingly assuming that U = v1/n, (33) reduces to

Av2+Bv3Cv4Dv2+vv=0,(34)

where

A=cn,B=ann+1,C=bn2n+1,D=n1n.(35)

Similarly, we suppose that (34) has solutions in the form of (14). Then, by the homogeneous balance principle we have thus assumed that (34) has the following solution

v(ξ)=a0+a1f+b1g,(36)

where f = f(ξ), g = g(ξ) satisfy (6). Substituting (36) with (6) into (34) and setting the coefficients of figj (i, j = 1, 2, …) to zero generates a system of algebraic equations. Solving this system of equations, gives rise to

  • Case 1

    a1=(n+1)(n+2)ran2,c=1n2,a2b=(n+1)(n+2)2r2(2n+1)(r2+ϵ)n2,a0=b1=0.(37)

  • Case 2

    a0=2(n+1)(n+2)an2,b1=±a0,c=4n2,a2b=4(n+1)(n+2)2(2n+1)n2,r=0,a1=0.(38)

  • Case 3

    a0=(n+1)(n+2)2an2,b1=±a0,c=1n2,a2b=(n+1)(n+2)2(2n+1)n2,r=ϵ,a1=0.(39)

  • Case 4

    a0=(n+1)(n+2)2an2,b1=±a0,a1=±a0ϵ,c=1n2,a2b=(n+1)(n+2)2(2n+1)n2,r=0.(40)

  • Case 5

    a0=(n+1)(n+2)2an2,b1=±a0,a1=±a0r2+ϵ,c=1n2,a2b=(n+1)(n+2)2(2n+1)n2.(41)

  • Case 6

    a0=(n+1)(n+2)2an2,b1=±a0,a1=±a0n+1nnϵ3(n2),c=1n2,a2b=(n+1)(n+2)2(2n+1)n2,r=±(2n1)nnϵ3(n2).(42)

Inserting these results along with (7)-(9) into (36), various solitary wave solutions of the generalised KdV equation can be created as follows.

From Case 1 we obtain the solutions

u1={(n+1)(n+2)rαan2[βcosh(ξ)+γsinh(ξ)+αr]}1/n,(43)

when ϵ = 1: α, β, γ satisfy γ2 = α2 + β2 and when ϵ = −1: α, β, γ satisfy β2 = α2 + γ2.

u2={(n+1)(n+2)ran2[sinh(ξ)+r]}1/n,ϵ=1.(44)

u3={(n+1)(n+2)ran2[cosh(ξ)+r]}1/n,ϵ=1.(45)

From Case 2 we arrive at the solutions

u4={2(n+1)(n+2)an2[1±βsinh(ξ)+γcosh(ξ)βcosh(ξ)+γsinh(ξ)]}1/n,(46)

when ϵ = 1: α, β, γ satisfy γ2 = α2 + β2 and when ϵ = −1: α, β, γ satisfy β2 = α2 + γ2.

u5={2(n+1)(n+2)an2[1±coth(ξ)]}1/n,ϵ=1.(47)

u6={2(n+1)(n+2)an2[1±tanh(ξ)]}1/n,ϵ=1.(48)

From Case 3 we find the solutions

u7={(n+1)(n+2)2an2×[1±βsinh(ξ)+γcosh(ξ)βcosh(ξ)+γsinh(ξ)±αϵ]}1/n,(49)

when ϵ = 1: α, β, γ satisfy γ2 = α2 + β2 and when ϵ = −1: α, β, γ satisfy β2 = α2 + γ2.

u8={(n+1)(n+2)2an2[1±cosh(ξ)sinh(ξ)±i]}1/n,ϵ=1.(50)

u9={(n+1)(n+2)2an2[1±sinh(ξ)cosh(ξ)±1]}1/n,ϵ=1.(51)

From Case 4 we acquire the solutions

u10={(n+1)(n+2)2an2[1±αϵβcosh(ξ)+γsinh(ξ)±βsinh(ξ)+γcosh(ξ)βcosh(ξ)+γsinh(ξ)]}1/n,(52)

when ϵ = 1: α, β, γ satisfy γ2 = α2 + β2 and when ϵ = −1: α, β, γ satisfy β2 = α2 + γ2.

u11={(n+1)(n+2)2an2[1±csch(ξ)±coth(ξ)]}1/n,ϵ=1.(53)

u12=(n+1)(n+2)2an21±isech(ξ)±tanh(ξ)1/n,ϵ=1.(54)

From Case 5 we obtain the solutions

u13={(n+1)(n+2)2an2[1±αr2+ϵβcosh(ξ)+γsinh(ξ)+αr±βsinh(ξ)+γcosh(ξ)βcosh(ξ)+γsinh(ξ)+αr]}1/n,(55)

when ϵ = 1: α, β, γ satisfy γ2 = α2 + β2 and when ϵ = −1: α, β, γ satisfy β2 = α2 + γ2.

u14={(n+1)(n+2)2an2[1±r2+1sinh(ξ)+r±cosh(ξ)sinh(ξ)+r]}1/n,ϵ=1.(56)

u15={(n+1)(n+2)2an2[1±r21cosh(ξ)+r±sinh(ξ)cosh(ξ)+r]}1/n,ϵ=1.(57)

From Case 6 we come by the solutions

u16={(n+1)(n+2)2an2[1±(n+1)nnϵ3(n2)×αβcosh(ξ)+γsinh(ξ)±α(2n1)nnϵ3(n2)±βsinh(ξ)+γcosh(ξ)βcosh(ξ)+γsinh(ξ)±α(2n1)nnϵ3(n2)]}1/n,(58)

when ϵ = 1: α, β, γ satisfy γ2 = α2 + β2 and when ϵ = −1: α, β, γ satisfy β2 = α2 + γ2.

u17={(n+1)(n+2)2an2[1±i(n+1)nn3(n2)×1sinh(ξ)±i(2n1)nn3(n2)±cosh(ξ)sinh(ξ)±i(2n1)nn3(n2)]}1/n,ϵ=1.(59)

u18={(n+1)(n+2)2an2[1±(n+1)nn3(n2)×1cosh(ξ)±(2n1)nn3(n2)±sinh(ξ)cosh(ξ)±(2n1)nn3(n2)]}1/n,(60)

Remarks

  1. Case 3 can be recaptured from Case 5 when r=±ϵ and therefore the solutions (55) - (57) are converted to (49) - (51).

  2. Case 4 can be retrieved from Case 5 if r = 0, so the solutions (55) - (57) reduce to (52) - (54).

  3. Case 6 can be brought back from Case 5 once r=±(2n1)nnϵ3(n2) and as a result the solutions (55) - (57) change into (58) - (60).

  4. Once β = 0 in γ2 = α2 + β2 then γ = α. Thus, solutions (43), (46), (49), (52), (55) and (58) are converted to solutions (44), (47), (50), (53), (56) and (59), respectively.

  5. If γ = 0 in β2 = α2 + γ2 so β = α and consequently solutions (43), (46), (49), (52), (55) and (58) become solutions (45), (48), (51), (54), (57) and (60), respectively.

Now, we exhibit the graphical representations for some solutions of the generalised KdV equation. Figures 5-8 illustrate three types of waves such as the bell-shaped solitary waves u3, the kink-shaped wave u6, u18 and the singular wave u11. The solutions u3, u6 and u11 are drawn with n = 2 which is not valid for u18 so n is selected to be 4 for this case. The dependence of the constant c on the parameter n leads to c = 1/4 for u3 and u11, c = 1 for u6, and c = 1/16 for u18. It is assumed that r = 2 in u3 and a = 1 in all cases.

Graphical representation of solution (45)
Figure 5

Graphical representation of solution (45)

Graphical representation of solution (48)
Figure 6

Graphical representation of solution (48)

Graphical representation of solution (53)
Figure 7

Graphical representation of solution (53)

Graphical representation of solution (60)
Figure 8

Graphical representation of solution (60)

5 Discussion and conclusion

In this paper, we have studied the solitary wave solutions of two KdV-type equations with power nonlinearties. The projective Riccati equation method is applied to solve the two examples of KdV-type equations. Various types of solitary wave solutions of bell-shaped, kink-shaped and singular wave profiles have been obtained. The results have shown that the projective Riccati equation approach can derive many kinds of exact solutions of NEEs.

Acknowledgement

The author is grateful to Ministry of Higher Education, Oman for the continuous support and encouragement.

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About the article

Received: 2018-02-19

Accepted: 2018-04-16

Published Online: 2018-05-30


Citation Information: Open Physics, Volume 16, Issue 1, Pages 311–318, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0043.

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© 2018 K. S. Al-Ghafri, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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