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Revista mexicana de física

versión impresa ISSN 0035-001X

Rev. mex. fis. vol.63 no.4 México jul./ago. 2017

 

Research

Dynamics of solitons of the nonlinear dispersion Drinfel’d-Sokolov system by ansatz method and He’s varitional principle

F. Tchiera 

L. Ragoubb 

D. Baleanuc  d 

M. Ince 

aDepartment of Mathematics, King Saud University, P. O. Box 22452, Riyadh 11495, Saudi Arabia. e-mail: ftchier@ksu.edu.sa

bMathematics Department, College of Computers and Information Systems, Al Yamamah University, P. O. Box 45 180 Riyadh 11512, Saudi Arabia. e-mail: radhkla@hotmail.com

cDepartment of Mathematics, Cankaya University, Ankara, Turkey. e-mail: dumitru@cankaya.edu.tr

dInstitute of Space Sciences, Magurele, Romania.

eFirat University, Science Faculty, Department of Mathematics, 23119 Elazǧ/Türkiye. e-mail: minc@firat.edu.tr

ABSTRACT

In this article, the exact-special solutions of the nonlinear dispersion Drinfel’d-Sokolov (shortly D(m, n)) system are analyzed. We use the ansatz approach and the He’s variational principle for the mentioned equation. The general formulae for the compactons, solitary patterns, solitons and periodic solutions are acquired. These types of solutions are useful and attractive for clarifying some types of nonlinear physical phenomena. These two methods will be used to carry out the integration.

Keywords: Nonlinear dispersion D(m, n) system; ansatz method; he’s variational principle; compacton

PACS: 02.30.Jr; 02.30.Ik

1. Introduction

Nonlinear partial differential equations (NPDE) have been analyzed by different type of mathematical approach, among which include the Darboux transformation, the inverse scattering method, the Hirota method, the Backlund transformation, the tanh method, the sine-cosine method, the exp-function method, the variational iteration method, the homogenous balance method and among others 1-41.

In this article, the D(m, n) system 1 are considered:

ut+(νm)x=0,νt+a(νn)xxx+buxν+cuνx=0. (1)

For m = 2 and n = 1, the system (1) is called “The normal Drinfel’d-Sokolov system”

ut+(ν2)x=0,νt +aνxxx+buxν+cuνx=0, (2)

where a, b, c are unchanged. The system (2) is considered as an example of a system of nonlinear equations possessing Lax pairs of a special form 13. Wang obtained its Hamiltonian, recursion, symplectric and cosymplectric operators and roots of its symmetries and scaling symmetry of the system (2) 14.

For n = 1 the system (1) changes to “The generalized Drinfel’d-Sokolov system” as:

ut+(νm)x=0,νt +aνxxx+buxν+cuνx=0. (3)

Wazwaz obtained some exact traveling wave solutions of the D(m, n) system with compact and noncompact structures by applying the tanh method and the sine-cosine method 15.

We firstly apply the ansatz method 16,17,18,19,20 to obtain the exact special solutions to the D(m, n) system. Then, we use the He’s variational approach 21 to obtain unknown traveling wave solution to the subsidiaries of the D(m, n) system. We give a comparison between the obtained solutions and those exist in the literature.

2. Ansatz method

We start by considering the solution of the equation

dwdz2=a0-b0w2, (4)

where a0 ≠ 0 and b0 ≠ 0 are constants. When b0 > 0, Eq. (4) admits two solutions as:

w1=±a0b0sin[b0z+A],w2=±a0b0cos[b0z+A], (5)

where A is an arbitrary unchanged. If b0 < 0, recognizing that cosh2 z + sinh2 z = 1 we know that Eq. (4) has two solutions of the form as:

w1=±a0b0sinh[b0z+A],w2 =±ia0b0cosh[b0z+A],i2=-1. (6)

Secondly, we make a consideration of the solutions of the equation of the form

dwdz2=w2(c0+d0w2) (7)

where c0 ≠ 0 and d0 ≠ 0 are constants. If c0 < 0, Eq. (7) confesses two solutions

w5 =±c0d0sec[-c0(z)],w6=±a0b0csc[-c0z]. (8)

If c0 > 0 in Eq. (7), then the equation offers two solutions of the form

w7 =±c0d0sech[-c0(z)],w6 =±ia0b0csch[-c0z]. (9)

2.1. Nonlinear Dispersion D(m, n) System

We assume that the traveling wave solution has the form u(x, t) = u(ξ) with wave variable ξ = k(x - λt), (k, λ ≠ 0). Then, we get the following ordinary differential equation:

-kλu'+kνm'=0, (10)

-kλν'+ak3νn+bku'ν+ckuν'=0. (11)

We get

u=1λνm+c1, (12)

by (11), where c1 is arbitrary constant. Substituting (12) into (11) we obtain

(13)

By integrating Eq. (13), we get

(cc1-λ)ν+ak2(νn)+bm+cλ(m+1)νm+1=c2. (14)

where c2 is integration constant.

Case 1. When c = -bm = λ/c1, the nonlinear ODE (14) becomes

ak2(νn)-c2=0. (15)

Therefore, we get the rational solution of Eq. (15),

ν(x,t)={c22ak2(x-λt)2+c3(x-λt)+c4}1n, (16)

and

u(x,t)=1λ{c22ak2(x-λt)2+c3(x-λt)+c4}mn+c1, (17)

where c3 and c4 are arbitrary constants.

Case 2. If we take m = n - 1 and c = λ/c1 in Eq. (14), the following traveling wave solutions are obtained

(18)

and

(19)

where c5 and c6 arbitrary constants. In view of (18) and (19), we clearly see that these solutions exist provided that (bm + c)/((m + 1)) > 0

(20)

and

(21)

where c7 and c8 arbitrary constants. In view of (20) and (21), we clearly see that these solutions exist provided that (bm + c)/((m + 1)) < 0.

Case 3. c + bm ≠ 0, cc1 - λ ≠ 0 and specially c2 = 0.

Let

dνndξ=z,d2νndξ2=zdzdνn. (22)

Substituting (22) into (14) leads to the following equation

(23)

Letting ν = w2/m, we have

ν=w2mdv=2mw2m-1dw, (24)

which changes Eq. (23) to

(25)

If we take n = 2m + 1 in Eq. (25), we get the algebraic traveling wave solution of the form:

ν(ξ)=1B(A-B2ξ2-2B2ξC-B2C2)1m, (26)

and

u(ξ)=1λB(A-B2ξ2-2B2ξC-B2C2)+c1, (27)

where

A=-(cc1-λ)m24ak2(m+1)(2m+1)

and

B=m2(bm+c)2ak2λ(m+1)(3m+2)(2m+1).

Case 4. If m = n - 1, we know that Eq. (25) becomes

(28)

where n ≠ 1 and a, k, n, λ ≠ 0.

If we take anλ(c + bn - b) > 0, then we acquire from Eqs. (5) and (28)

(29)

and

(30)

Theorem 1. The D(m, n) system has solutions in Eq. (28) described as follows:

1. When anλ(c + bn - b) > 0,

ν=*20c2n2λ(λ-cc1)(n+1)(bn-n+c)cos2n-12knbn-b+cλan(ξ+A)1n-1,b0(ξ)π2,0,Otherwise (31)

is a solitary wave solution with compact support.

2. When anλ(c + bn - b) > 0,

ν=*20c2n2λ(λ-cc1)(n+1)(bn-n+c)cos2n-12knbn-b+cλanξ+A1n-1,0b0(ξ)π,0,Otherwise (32)

is a compacton solution for Eq. (1) and

b0=n-12knbn-b+cλan.

3. Equation (25) can be written as following

(33)

If n = 1in Eq. (33) that yields as

(34)

When a(cc1 - λ) < 0,

(35)

which is a singular soliton solution for the D(m, n) equation for

0<ξ<πm2k(cc1-λ)a

4. When a(cc1 - λ) > 0 and m < 0,

(36)

is a traveling wave solution for the D(m, n) equation for

ξ<πm4k(cc1-λ)a.

Remark 1. If (bn - b - c)/(λan) < 0, it follows from (6) and (28) that

(37)

and

(38)

Theorem 2. The D(m, n) equation with when m = n equation has the following solutions:

1. When (c + bn - b) < 0, anλ > 0, (λ - cc1)(n + 1) < 0 and n ≠ 1

(39)

is a solitary solution of Eq. (1).

2. When (c + bn - b) < 0, anλ > 0, (λ - cc1)(n + 1) > 0 and n ≠ 1

(40)

is a solitary solution of Eq. (1). If we take n < 1, then Eq. (40) is a bounded solution.

3. m = n < 1solutions (39) and (40) turns to solitary wave solutions

(41)

and

(42)

Case 5. a(cc1 - λ) > 0 and m > 0,

Thus, by using (8) and (34), the periodic solutions of Eq. (1) are obtained as:

(43)

and

(44)

Case 6. a(cc1 - λ) > 0 and m > 0,

Therefore, by considering (9) and (34), solitary pattern and bell-shaped solitary wave solutions of (1) are obtained as:

(45)

and

(46)

Case 7. a(cc1 - λ) > 0 and m ≠ 0,

Using the solutions of (43) and (44), gives the following compacton solutions as:

(47)

for

0πm2k(cc1-λ)a|π2

and u = 0, otherwise.

(48)

for

0|πm2k(cc1-λ)a|π

and u = 0, otherwise.

Case 8. a(cc1 - λ) > 0; m < 0 and m ≠ 1.

Using cosh(x) = cos(ix) and sinh(x) = -sin(ix) we have the following solitary pattern solutions of Eq. (1):

(49)

and

(50)

Remark 2. When m = n < 1, the obtained solution (42) agrees with the outcomes (2.13a), (2.13b) in 36 and (25) in 37. The solution (42) changes to the compacton solution (2.12a) and the periodic solution (2.12b) in 36,37.

If m > 0, the obtained solution (46) is consented with the outcomes (3.18a) and (3.18b) described in 36 and (26) in 37. The solution (46) changes to the solitary pattern solution (3.17a) and solitary wave solution (3.17b) in 36.

Remark 3. If a(cc1 - λ) < 0, the obtained solution (35) agrees with the outcomes (3.9a) and (3.9b) in 36 and (32) in 37. The solution (35) changes to the singular solitary wave solution.

Remark 4. If we take (c + bn - b) < 0, anλ > 0, (λ - cc1)(n + 1) < 0 then the obtained solution (39) similar to the solitary pattern solutions (3.7a) and (3.7b) in 36 and (46) in 37.

3. Variational principle

In this Section, He’s variational principle will be applied to the system (1). This technique was first proposed by He21 and it is popularly known as He’s semi-inverse variational principle. Some years back, it was applied mainly to extract soliton solutions of nonlinear PDEs and systems by many authors 16,17,18,19,20,21,22,23,24. Biswas and co-workers 17,18,19,20 obtained optical solitons and soliton solutions with higher order dispersion by applying the He’s variational principle. Xu and Zhang’s25 used a variational principle to construct catalytic reactions in short monoliths by He’s semi-inverse approach. He’s variational method was used to the effective nonlinear oscillators with high nonlinearity by Liu 26. Zheng et al. 27 established a class of generalized variational principles for the initial-boundary value problem of micromorphic magneto electrodynamics by He’s semi-inverse technique. In order to seek traveling wave solutions of the system (1). We consider

(cc1-λ)ν+ak2(νn)+bm+cλ(m+1)νm+1=c2. (51)

Let νn = V

(52)

the 1-soliton solution ansatz, given by

V(ξ)={psech2(qξ)}1n, (53)

is substituted into (51). Here, in (53), the parameters p and q represent the amplitude and inverse width of the soliton, respectively.

(54)

where p1 = cc1 - λ, p2 = ak2 and

p3=bm+cλ(m+1).

From the above equation it is obtained as

(55)

where F is Gauss’ hypergeometric function defined as

F[α,β,γ,z]=Γ(γ)Γ(α)Γ(β)n=0Γ(α+n)Γ(β+n)Γ(γ+n)znn! (56)

and Re[(2 + m)q] > 0, Re[(1 + n)q] > 0, Re[q] > 0.

Making J stationary with respect to p and q results in

(57)

(58)

Solving Eqs. (57) and (58) for m = n = 2 simultaneously, we get

p=35ak2λ9(2b+c),q=k3apλ-cc1, (59)

Therefore, by substituting p and q in (52) we have the following a new solitary wave solution for the system (1) as:

(60)

So, the solitary wave solution (60) will exist for ap(λ - cc1) > 0.

4. Results and Discussions

In this article, we investigated the nonlinear dispersion D(m, n) system and obtained some traveling wave solutions by applying the ansatz technique and the He’s variational principle. Several forms of solutions including topological, non-topological, compacton, solitary pattern, singular soliton, algebraic and periodic wave solutions were acquired. The approaches can be used to a lot of other nonlinear differential equations and coupled systems. Some new obtained exact solutions were previously unknown by other methods. We proved the existence of these solutions for a generalized form of the D(m, n) system under specific conditions. In general, the outcome expose that the ansatz approach and the He’s variational principle are important mathematical techniques for solving nonlinear partial differential equations in terms of correctness and ability to avoid errors.

Acknowledgments

This research project was supported by a grant from the “Research Center of the Center for Female Scientific and Medical Colleges”, Deanship of Scientific Research, King Saud University.

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Received: January 13, 2017; Accepted: May 25, 2017

Conflict of Interests The authors declare that they have no conflict of interests.

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