On exact special solutions of integrable nonlinear dispersive equation

Chaos, Solitons and Fractals 39 (2009) 1920–1927 www.elsevier.com/locate/chaos

On exact special solutions of integrable nonlinear dispersive equation Mustafa Inc *, Mehmet Bektasß Fırat University, Department of Mathematics, 23119 Elazıg˘, Turkey Accepted 20 June 2007

Abstract In this paper, we obtained integrable evolution equation from binormal motions of curves in the 3-dimensional Euclidean space sb ¼ ðsm Þsss  ðsmþ2 Þs þ ðsm Þs  Afterwards, this equation is solved by sine–cosine method and compacton solutions are obtained. As a result, abundant new compactons solitons with the absence of infinite wings, solitary pattern solutions having infinite slopes or cups, solitary wave and periodic wave solutions are obtained. Ó 2007 Elsevier Ltd. All rights reserved.

1. Introduction The investigation of the travelling wave solutions plays an important role in nonlinear science. These solutions may well describe various phenomena in nature, such as vibrations, solitons and propagation with a finite speed. The wave phenomena are observed in fluid dynamics, plasma and elastic media. Various methods for obtaining explicit travelling solitary wave solutions to nonlinear evolution equations have been proposed. In recent years, direct searching for exact solutions of nonlinear partial differential equations (PDEs) has become more and more attractive partly due to the availability of computer symbolic systems like Maple or Mathematica, which allow us to perform some complicated and tedious algebraic calculation on a computer, as well as help us to find new exact solutions of PDEs. Many methods were developed for finding the exact solutions of nonlinear evolution equations [1–8]. To investigate the role played by the nonlinear dispersion in pattern formation Rosenau and Hayman [9,10] have proposed K(n,m) models of form /t  ð/n Þs þ ð/m Þsss ¼ 0 where m and n are integers, / is a smooth function of time t and the space variable s. In particular, the K(2, 1) and K(3 ,1) models are the well-known KdV and mKdV equations. A remarkable property to the K(m, m) models is that it has solitons with compact support [9]. *

Corresponding author. Tel.: +90 424 237 00 00; fax: +90 424 233 00 62. E-mail address: minc@firat.edu.tr (M. Inc).

0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.123

M. Inc, M. Bektasß / Chaos, Solitons and Fractals 39 (2009) 1920–1927

1921

It is well-known that certain motions of space curves are closely related to integrable equations. In [11], Hasimoto discovered the remarkable fact that the binormal motion of a nonstretching vortex filament with speed equal to its curvature produces the cubic nonlinear Schro¨dinger equation. By considering other binormal motions, Lamb [12] obtained the sine-Gordon and mKdV equations. Soon the Heisenberg spin chain model was found to be related to motion of space curves as well [13]. The NKLS hierachy is also related to motion of space curves [14]. More recently, Schief and Rogers [15] obtained and extended Harry-Dym equation and classical Sine-Gordon equation from binormal motions of curves of constant curvature or torsion. The line of research has been extended to motions of curves in three-dimensional space forms, surfaces or even the plane. These works on motions of plane curves or surfaces can be found in [16–21]. Compacton is a soliton solution, which has finite wavelength or free of exponential wings. Unlike soliton that narrows as the amplitude increases, the compactons width is independent of its amplitude. Compacton solutions have been used in many fields of scientific applications such as the super deformed nuclei, phonon, photon, the fission of liquid drops (nuclear physics), and pre-formation of cluster in hydrodynamic models. Many mathematical methods have been involved in the compactons concept for various approaches such as the Ba¨cklund transformation, the Painleve´ analysis, the Inverse scattering method and the Darboux transformation. In addition, the compacton solutions have been worked by many numerical methods, for example, the finite difference method, the pseudo-spectral method and Adomian decomposition method. This paper is organized as follows: In the second part we obtain integrable nonlinear dispersive equation from Serret–Frenet equations in the three-dimensional Euclidean space R3 . We present the sine–cosine method to obtain exact special solutions for Eq. (2.14) in the third part. The method is used to seek compact and solitary pattern solutions for this equation in general formulas in the same part. In the last part is given some conclusions for obtained solutions.

2. Preliminaries We use the same notations and terminologies as in [22], unless otherwise stated. In the present chapter, we are concerned with curves and surfaces in Euclidean space R3 If a curve C:r = r(s) is parametrized in terms of arc length then the orthonormal triad (t, n, b) consisting of the unit tangent vector t = rs, the principal normal n and the binormal b varies along C according to the Serret–Frenet equations, 10 1 0 1 0 t t 0 j 0 CB C B C B ð2:1Þ @ n A ¼ @ j 0 s A@ n A b b s 0 s 0 where the quantities j and s denote the curvature and torsion of curve, respectively. An offset curve C* along the principal n normals defined by r ¼ r þ an

ð2:2Þ

where a constitues a prescribed function of s. If one demands that the parent curve C and its offset curve C* occur on an equal footing then the principal normals to C and C* must coincide, i.e., ð2:3Þ n ¼ n: This imposes constraints on the parent curve and the ‘‘distance’’ function a. Thus, differentiation of (2.2) yields rs ¼ ð1  ajÞt þ as n þ asb

ð2:4Þ

which, by virtue of rs  n ¼ 0 has the important implication that the offset curve is at a constant distance a from the parent curve, i.e., as = 0. The unit tangent vector t* is therefore given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1  ajÞt þ asb ð2:5Þ ; D ¼ ð1  ajÞ2 þ a2 s2 : t ¼ D Further differentiation produces   as ð1  ajÞ ð1  ajÞj  as2 þ b nþ ts ¼ D D s D s

ð2:6Þ

The t- and b-components of the above are required to vanish since ts kn : It is readily verified that this requirement leads to the curvature–torsion relation aj þ bs ¼ 1

ð2:7Þ

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where b constitutes a constant of integration. Curves for which there exist constants a and b such that (2.7) holds are known as Bertrand curves. If one chooses a one-parameter family of geodesics and their orthogonal trajectories as the coordinate lines on a surface R then in terms of the associated geodesic coordinates s and b, the first fundamental form of surface reads dr2 ¼ ds2 þ g2 db2

ð2:8Þ

Here, the lines b = constant are the arc length parametrized geodesics and the lines s = constant from the orthogonal parallels. Since rs Æ rb = 0 and the principal normal n of the geodesics is orthogonal to the surfaces the tangent vectors to the coordinate lines are given by rs ¼ t;

rb ¼ gb;

ð2:9Þ

where b denotes the usual binormal of the geodesics. One may therefore think of the surfaces R as being generated by the motion of an inextensible curve, which moves in binormal direction at speed g, wherein the coordinate b is identified with time. In particular, a Razzaboni surface is generated by the binormal motion of a Bertrand curve which does not change the constants a and b . Here, it is emphasized that binormal motions are only possible for inextensible curve, i.e., binormal motions automatically preserve arc length. The variation of the orthonormal triad (t,n,b) in s-direction is given by the Serret–Frenet equations (2.1). The b-dependence must be of the general form 10 1 0 1 0 t t 0 u w CB C B C B ð2:10Þ @ n A ¼ @ u 0 v A@ n A b b b w v 0 The compatibility condition rsb = rbs applied to (2.1) yields un þ wb ¼ sgn þ gs b leading to the b-evolution 0 1 0 t 0 sg B C B 0 @ n A ¼ @ sg b b gs v

ð2:11Þ 10 1 t gs CB C v A@ n A b 0

ð2:12Þ

The latter is compatible with the Serret–Frenet equations (2.1) if and only if j, s, g and v constitute a solution of the underdetermined system jb ¼ 2sgs  ss g;

sb ¼ vs þ jgs ;

gss ¼ s2 g þ jv:

ð2:13Þ

m

If we choose j = 1 and g = s , we obtained integrable evolution equation sb ¼ ðsm Þsss  ðsmþ2 Þs þ ðsm Þs :

ð2:14Þ

3. Compacton and solitary pattern solutions We first make the travelling wave transformation as follows: sðs; bÞ ¼ sðfÞ;

f ¼ s  cb

ð3:1Þ

where c is a constant. Then (2.14) reduces to be 000 0 0 cs0  ðsm Þ þ ðsmþ2 Þ  ðsm Þ ¼ 0:

ð3:2Þ

Integrating (3.2) one and setting the integration constant to zero, we have cs0  ðsm Þ00 þ sm ðs2  1Þ ¼ 0:

ð3:3Þ

To search for compacton solutions we assume that the general compacton solution is of the form [23–25]: sðfÞ ¼ k cosb ðlfÞ;

ð3:4Þ

sðfÞ ¼ k sinb ðlfÞ:

ð3:5Þ

or

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The parameters k, l and b will be determined, and l and c are the wave number and wave speed, respectively. In view of (3.4), we set 8 m s ðfÞ ¼ km cosmb ðlfÞ; > > < smþ2 ðfÞ ¼ kmþ2 cosðmþ2Þb ðlfÞ; > > : m 00 ðs Þ ¼ m2 l2 b2 km cosmb ðlfÞ þ ml2 km bðmb  1Þ cosmb2 ðlfÞ;

ð3:6Þ

and for (3.5), we get 8 m s ðfÞ ¼ km sinmb ðlfÞ; > > < smþ2 ðfÞ ¼ kmþ2 sinðmþ2Þb ðlfÞ; > > : m 00 ðs Þ ¼ m2 l2 b2 km sinmb ðlfÞ þ ml2 km bðmb  1Þ sinmb2 ðlfÞ:

ð3:7Þ

Substituting (3.6) into (3.3) gives rise to ck cosb ðlfÞ þ m2 l2 b2 km cosmb ðlfÞ  ml2 km bðmb  1Þ cosmb2 ðlfÞ þ kmþ2 cosðmþ2Þb ðlfÞ  km cosmb ðlfÞ ¼ 0

Fig. 1. The compacton solutions Eqs. (3.11) and (3.12) at m = 5 and c = 0.5, respectively.

ð3:8Þ

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Thus, we obtain the following system of algebraic equations: 8 mb  1–0; > > > < mb  2 ¼ b; > km þ m2 l2 b2 km ¼ 0; > > : ck þ ml2 km bðmb  1Þ ¼ 0; So that this system gives 8 2 b ¼ m1 ; m>1 > > < m1 b ¼ 2m ; m–0; m > 1;   > > : k ¼ 2mc ; c–0; m–0; mþ1

ð3:9Þ

ð3:10Þ m > 1;

that can also be obtained by using the ansatz (3.7). 3.1. Compacton solutions By using Eq. (3.10), we obtain a family of compacton solutions 8n 1  om1 < 2mc p cos2 m1 ðs  cbÞ ; js  cbj < 2l mþ1 2m sðs; bÞ ¼ : 0; otherwise

Fig. 2. The solitary pattern solutions Eqs. (3.13) and (3.14) at m = 3 and c = 1, respectively.

ð3:11Þ

M. Inc, M. Bektasß / Chaos, Solitons and Fractals 39 (2009) 1920–1927

and sðs; bÞ ¼

8n < 2mc :

mþ1

0;

sin2

m1 2m

ðs  cbÞ

1 om1

;

js  cbj < p

1925

ð3:12Þ

otherwise:

3.2. Solitary pattern solutions By using Eq. (3.10), we obtain a family of solitary pattern solutions 1 

m1 2mc 1m sðs; bÞ ¼ ; cosh2 ðs  cbÞ mþ1 2m

ð3:13Þ

and sðs; bÞ ¼

1 

m1 2mc m1 : sinh2 ðs  cbÞ  mþ1 2m

ð3:14Þ

3.3. Periodic solutions It is normal to examine the effect of the negative exponents on the previous compacton and solitary pattern solutions. If we set m = n, n > 0, Eq. (2.14) becomes sb  ðsn Þsss þ ðs2n Þs  ðsn Þs ¼ 0;

n > 0:

Fig. 3. The periodic wave solutions Eqs. (3.16) and (3.17) at n = 3, c = 2, respectively.

ð3:15Þ

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Hence, we obtain the following periodic solutions: 1 

nþ1 2nc 1n ; sec2 ðs  cbÞ sðs; bÞ ¼ 1n 2n

ð3:16Þ

and  sðs; bÞ ¼

1

nþ1 2nc 1n : csc2 ðs  cbÞ 1n 2n

ð3:17Þ

3.4. Soliton solutions From the last two solutions, we get the soliton solutions of the form 1 

nþ1 2nc 1n sðs; bÞ ¼ ; sech2 ðs  cbÞ 1n 2n and

Fig. 4. The soliton solutions Eqs. (3.18) and (3.19) at n = 3 and c = 2, respectively.

ð3:18Þ

M. Inc, M. Bektasß / Chaos, Solitons and Fractals 39 (2009) 1920–1927

sðs; bÞ ¼

1 

nþ1 2nc 1n  : csch2 ðs  cbÞ 1n 2n

1927

ð3:19Þ

4. Conclusions In this paper, we obtained integrable nonlinear dispersive equation from Serret–Frenet equations in the three-dimensional Euclidean space R3 and then, we have used the sine–cosine method to derive exact special solutions with distinct physical structures. Some of these results are in agreement with the results reported by others in the literature, and new results are formally developed. The properties of obtained solutions are shown in Figs. 1–4. The present method is direct and efficient to obtain exact special solutions of Eq. (2.14). This method is very easily applied to both this type of nonlinear dispersive and the modified nonlinear dispersive equation in higher-dimensional spaces.

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