Solitary waves in a quartic nonlinear elastic rod

Chaos, Solitons and Fractals 11 (2000) 1265±1267

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Solitary waves in a quartic nonlinear elastic rod Wen-shan Duan, Jing-bao Zhao Department of Physics, Northwest Normal University, Lanzhou 730070, People's Republic of China Accepted 9 February 1999 Communicated by Prof. M. Roseau

Abstract The bright and dark solitons described by the nonlinear Schr odinger equation (NLSE) are given for a quartic nonlinear elastic rod. It has also been found that the KdV soliton does not exist in this system. Ó 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction In recent years, the interest in the study of nonlinear phenomena in many branches of physics has considerably increased. In particular, great attention is paid to the dynamics of nonlinear localized waves, i.e., solitons. In wide-ranging dispersive media, the existence of solitons has been extensively studied by many authors, for example, in nonlinear optics [1,2], nonlinear lattices [3,4], nonlinear surface water waves [5,6], nonlinear waves in ¯uid-®lled elastic tube [8±10] and so on. These solitons may be classi®ed by two physically interesting waves which can be described by the KdV equation and the nonlinear Schr odinger equation (NLSE), respectively. In this paper we ®rstly and theoretically studied the NLS solitons in a quartic nonlinear elastic rod.

2. Equation of motion Taking the cylindrical coordinates (r; h; x), we consider one-dimensional wave in a elastic rod which is in®nitely long, homogeneous and having an arbitrary cross-sectional area. We assume that 1. rr ˆ rh ˆ 0; 2. sr ˆ ÿmx and ur ˆ rr ˆ ÿmr…ou=ox†; 3. rx ˆ Ex ‡ an Enx ; where E is the Young's modulus of the rod, r the stress, x and r are the elongations in the directions of x and r, respectively, an and n the constants. For the soft nonlinear materials an < 0, for example, majority of the metals. For the hard nonlinear materials such as rubbers and polymers, an > 0. The kinetic energy per unit length of the rod is  2  2 2 1 ou 1 2 ou ‡ qm Jm : T ˆ qA 2 ot 2 ot ox The potential energy per unit length is 0960-0779/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 9 9 ) 0 0 0 1 4 - 4

…1†

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W. Duan, J. Zhao / Chaos, Solitons and Fractals 11 (2000) 1265±1267

 2  n‡1 1 ou 1 ou EAan W ˆ EA ‡ : 2 ox n‡1 ox The equation of motion for this system can be got from the Hamilton principle [7] "  nÿ1 # 2 o2 u ou o u m 2 J m o4 u 2 ÿ c0 1 ‡ nan ÿ ˆ 0; 2 ot ox ox2 A ot2 ox2

…2†

…3†

where c20 ˆ E=q. We only consider the case n ˆ 3. According to the general method of reductive perturbation theory, we introduce the stretched variables n and s n ˆ … x ÿ c0 t†;

…4†

s ˆ 3 x:

…5†

The smallness parameter  represents the size of the amplitude of perturbation. The u is expanded into the following power series u ˆ u…1† ‡ 3 u…2† ‡




…6†

Substitution of Eqs. (4)±(6) into (3), we got nothing at O…3 † and the following equation at O…5 † oY mJq o3 Y ‡ ˆ 0; os A on3

…7†

where Y ˆ ou…1† =on. Compared with the KdV equation, we ®nd that the nonlinear term disappeared in this equation. Therefore there are no KdV solitons. 3. The NLSE In order to introduce the NLSE for a elastic rod, we introduce the following transformations according to the general perturbation method n ˆ …x ÿ Vt†;

…8†

s ˆ 2 x;

…9†

uˆ

1 X

n

n X

u…n;l† …n; s† eil…kxÿxt† :

…10†

lˆÿn

nˆ1

Substitution of Eqs. (8)±(10) into (3), we got at O…† for j l jˆ 1 x2 ˆ

c20 k 2 1‡

m2 Jq A

k2

:

…11†

With the x determined by Eq. (11), we got that u…1;l† ˆ 0 for j l j> 1. At O…2 †, we got for l ˆ 1 V ˆ

k  x

c20 1‡

m2 Jq A

k2

2

…12†

and for l ˆ 2 u…2;2† ˆ 0:

…13†

W. Duan, J. Zhao / Chaos, Solitons and Fractals 11 (2000) 1265±1267

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We got for l ˆ 0 at O…3 † …1;0†

unn

ˆ0

…14†

and for l ˆ 1 at O…3 † …1;1†

…15†

;

…16†

iC1 us…1;1† ‡ C2 unn ‡ C3 j u…1;1† j2 u…1;1† ˆ 0; where C1 ˆ

ÿ2kc20 1‡

C2 ˆ 

m2 Jq A

k2

3k 2 c20 1‡

m2 Jq A

m2 Jq A

k2

2 ;

C3 ˆ ÿ12a3 c20 k 4 :

…17†

…18†

It can be found that C2 > 0, but C3 < 0 if a3 > 0 and C3 > 0 if a3 < 0. The envelop solitons described by NLSE have been studied extensively during recent years [11]. If a3 < 0 (for soft nonlinear materials), there are bright solitons in this system since C3 > 0. If a3 > 0 (for hard nonlinear materials), there are dark solitons in this system since C3 < 0. Acknowledgements This work is supported by the Foundations of Natural Science of Gansu province of the People's Republic of China. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, Boston, MA, 1989. A. Hasegawa, Optical Solitons in Fibers, Springer, Berlin, 1989. K. Yoshimura, S. Watanabe, J. Phys. Soc. Jpn. 60 (1991) 82±87. V.V. Konotop, Phys. Rev. E 53 (1996) 2843±2858. H. Hashimoto, H. Ono, J. Phys. Soc. Jpn. 33 (1972) 605±811. C.C. Mei, Dynamics of Water Waves (in Chinese), Academic Press, Beijing, 1984. Z. Shanyuan, Z. Wei, Acta Mechanica Sinica (in Chinese) 20.1 (1988) 58±67. W.S. Duan, B.R. Wang, R.J. Wei, J. Phys. Soc. Jpn. 65 (1996) 945±947. W.S. Duan, B.R. Wang, R.J. Wei, Phys. Lett. A 224 (1997) 154±158. W.S. Duan, B.R. Wang, R.J. Wei, Phys. Rev. E 55 (1997) 1773±1778. G.L. Lamb, Elements of Soliton Theory, Wiley, New York, 1982.