An exponential expansion method and its application to the strain wave equation in microstructured solids

Ain Shams Engineering Journal (2015) 6, 683–690

Ain Shams University

Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com

ENGINEERING PHYSICS AND MATHEMATICS

An exponential expansion method and its application to the strain wave equation in microstructured solids M.G. Hafez a b

a,*

, M.A. Akbar

b

Department of Mathematics, Chittagong University of Engineering and Technology, Chittagong, Bangladesh Department of Applied Mathematics, University of Rajshahi, Bangladesh

Received 20 August 2014; revised 24 October 2014; accepted 19 November 2014 Available online 7 January 2015

KEYWORDS An exponential expansion method; Strain wave equation; Explicit solution; Solitary wave solution; Periodic solution

Abstract The modeling of wave propagation in microstructured materials should be able to account for various scales of microstructure. Based on the proposed new exponential expansion method, we obtained the multiple explicit and exact traveling wave solutions of the strain wave equation for describing different types of wave propagation in microstructured solids. The solutions obtained in this paper include the solitary wave solutions of topological kink, singular kink, nontopological bell type solutions, solitons, compacton, cuspon, periodic solutions, and solitary wave solutions of rational functions. It is shown that the new exponential method, with the help of symbolic computation, provides an effective and straightforward mathematical tool for solving nonlinear evolution equations arising in mathematical physics and engineering. Ó 2014 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Nonlinear evolution equations (NLEEs) are very important model equations in mathematical physics and engineering for describing diverse types of physical mechanisms of natural phenomena in the field of applied sciences and engineering. For this reason, the search of exact traveling wave solutions to NLEEs plays very important role in the study of these * Corresponding author. Tel.: +880 1712340602. E-mail address: [email protected] (M.G. Hafez). Peer review under responsibility of Ain Shams University.

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physical phenomena. The wave propagation phenomena are observed in microstructured solids, plasma physics, chemical physics, elastic media, optical fibers, fluid dynamics, quantum mechanics, etc. With the rapid development of nonlinear science based on computer algebraic system, many effective methods have been presented, such as, the tanh method [1], the extended tanh method [2,3], the modified extended tanhfunction method [4,5], the Exp-function method [6–8], the improved F-expansion method [9], the exp(U(n))-expansion method [10–14], the sine–cosine method [15], the modified simple equation method [16–22], the (G0 /G)-expansion method [23,24], the novel (G0 /G)-expansion method [25], new approach of the generalized (G0 /G)-expansion method [26,27], the Jacobi elliptic function method [28,29], the homogeneous balance method [30–32], the Hirota’s bilinear method [33], the homotopy perturbation technique [34] and others.

http://dx.doi.org/10.1016/j.asej.2014.11.011 2090-4479 Ó 2014 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

684

M.G. Hafez, M.A. Akbar

The mathematical modeling of wave propagation in microstructured solids is also described by nonlinear PDEs and should be able to account for several scales of microstructure. If the scale reliance involves dispersive effects and materials behave nonlinearity, then dispersive and nonlinear effects may be balanced and thus the solitary wave causes. The existence and appearance of solitary waves in complicated physical problems apart from the model equations of mathematical physics must be analyzed with sufficient accuracy. There is an amount of paper where the governing equations for waves in microstructured solids have been derived and the solitary waves were analyzed [27,35–38]. For instance, the microstrain wave function u(x, t) in micro-structured solids is characterized by nonlinear partial differential equation. The governing nonlinear PDE equation (see Ref. [27,37]) for the microstrain wave function u(x, t) in micro-structured solids is given by ut t  ux x  e a1 ðu2 Þx x  c a2 ux x t þ d a3 ux x x x  ðd a4  c2 a7 Þ ux x t t þ c d ða5 ux x x x t þ a6 ux x t t t Þ ¼ 0:

ð1Þ

where e accounts for elastic strains, d characterizes the ratio between the microstructure size and the wave length, c characterizes the influence of dissipation and a1 ; a2 ; a3 ; a4 ; a5 ; a6 are constants. The balance between nonlinearity and dispersion takes place when d ¼ OðeÞ. If we set c ¼ 0, we have the non-dissipative case, and governed by the double dispersive equation [27,37] as follows: 2

ut t  ux x  e fa1 ðu Þx x  a3 ux x x x þ a4 ux x t t g ¼ 0:

ð2Þ

The aim of this paper was to apply the proposed exponential expansion method to construct the new exact traveling wave solutions to the strain wave Eq. (2) for describing various types of solitary wave propagation in microstructured solids. The remainder of the paper is organized as follows: In Section 2, we give the brief description of the proposed new exponential expansion method. In Section 3, we apply this method for finding the explicit and solitary wave solutions to the strain wave equation in microstructured solids. The physical explanations of the obtained solutions and the advantages of the new exponential expansion method are presented in Sections 4 and 5 respectively. Conclusions are given in the last section. 2. Description of the proposed exponential expansion method This section presents the brief descriptions of the new exponential expansion method. Let us consider the NLEE as follows: Fðu; ut ; ux ; ux x ; ut t ; ut x ; . . .Þ ¼ 0;

ð3Þ

where F is a function of u; ut ; ux ; ux x ; ut t ; ut x ; . . . and the subscripts denote the partial derivatives of uðx; tÞ with respect to x and t. Suppose uðx; tÞ ¼ uðnÞ; n ¼ x  V t; where the constant V is the velocity of the traveling wave, then the Eq. (3) reduces to a nonlinear ordinary differential equation (ODE) for u ¼ uðnÞ: Qðu; u0 ; u00 ; u000 ;   Þ ¼ 0;

ð4Þ

where Q is a function of u; u0 ; u00 ; u000 ;    and its derivatives point out the ordinary derivatives with respect to n.

Let us consider the traveling wave solution of Eq. (4) is of the form: uðnÞ ¼

N X Ai ðexp ðUðnÞÞÞi ; AN –0

ð5Þ

i¼0

where the coefficients Ai ð0 6 i 6 NÞ are constants to be evaluated and U = U(n) satisfies the first order nonlinear ordinary differential equation: U0 ðnÞ ¼ expðUðnÞÞ þ l expðUðnÞÞ þ k;

ð6Þ

where k and l are arbitrary constants. The value of the positive integer N can be determined by balancing the highest order derivatives with the nonlinear terms of the highest order appearing in Eq. (4). By substituting (5) into (4) and using (6) when required, we obtain a system of algebraic equations for Ai ð0  i  NÞ; k; l; and V. With the help of symbolic computation, such as, Maple, we can evaluate the obtaining system and find out the values Ai ð0  i  NÞ; k; l; and V. It is notable that Eq. (6) has the following five types of general solutions [10–14]: ! pffiffiffiffi  pffiffiffiffi   H tanh 0:5 Hðn þ n0 Þ  k UðnÞ ¼ ln ; l–0; H ¼ k2  4l > 0; 2l

ð7aÞ ! pffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffi  H tan 0:5 Hðn þ n0 Þ  k UðnÞ ¼ ln ; l–0; H ¼ k2  4l < 0; 2l



 k ; l ¼ 0; k–0; H UðnÞ ¼ ln expðkðn þ n0 ÞÞ  1 ¼ k2  4l > 0;   2ðkðn þ n0 Þ þ 2Þ ; l–0; k–0; H ¼ k2  4l ¼ 0; UðnÞ ¼ ln  k2 ðn þ n0 Þ UðnÞ ¼ ln ðn þ n0 Þ; k ¼ 0; l ¼ 0:

ð7bÞ

ð7cÞ ð7dÞ

ð7eÞ

where n0 is the integration constant. Thus the multiple explicit solutions to the NLEE (3) are obtained by means of the Eqs. (5) and (7). Again, suppose Eq. (4) has solution of the form (5) and U ¼ UðnÞ satisfies another first order nonlinear ordinary differential equation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U0 ðnÞ ¼  k þ lðexpðUðnÞÞÞ2 ; k; l 2 R: ð8Þ By substituting (5) into (4) and using (8) repeatedly, we obtain a system of algebraic equations for Ai ð0  i  NÞ; k; l; V. With the help of symbolic computation, such as, Maple we can evaluate the resulting system and find out the values Ai ð0  i  NÞ; k; l; V. It is notable that Eq. (8) has the following general solutions: sffiffiffi ! hpffiffiffi i k csc h k ðn þ n0 Þ ; k > 0; l > 0; ð9aÞ UðnÞ ¼  ln  l sffiffiffiffiffiffiffi ! hpffiffiffiffiffiffiffi i k sec k ðn þ n0 Þ ; k < 0; l > 0; UðnÞ ¼  ln l sffiffiffiffiffiffiffi ! hpffiffiffi i k UðnÞ ¼  ln sec h k ðn þ n0 Þ ; k > 0; l < 0; l

ð9bÞ

ð9cÞ

An exponential expansion method and its application to the strain wave sffiffiffiffiffiffiffi ! hpffiffiffiffiffiffiffi i k csc k ðn þ n0 Þ ; k < 0; l > 0; UðnÞ ¼  ln l  UðnÞ ¼  ln

 1 ; k ¼ 0; l > 0; pffiffiffi  l ðn þ n0 Þ

ð9dÞ

ð9eÞ



 i ; k ¼ 0; l < 0; UðnÞ ¼  ln pffiffiffi  l ðn þ n0 Þ

ð9fÞ

where n0 is the constant of integration. Thus the new multiple exact solutions to the NLEE (3) can be found by combining (5) and (9). Finally, suppose Eq. (4) has solution of the form (5) where U = U(n) satisfies a different first order nonlinear ordinary differential equation: U0 ðnÞ ¼ k expðUðnÞÞ  l expðUðnÞÞ; k; l 2 R:

ð10Þ

By means of (5) and (4) and using (10) recurrently, we obtain a system of algebraic equations for Ai ð0  i  NÞ; k; l; V. With the help of symbolic computation, such as, Maple we can calculate the resulting system and find out the values Ai ð0  i  NÞ; k; l; V. It is notable that Eq. (10) has the following general solutions as follows: sffiffiffi ! hpffiffiffiffiffiffi i k tan ð11aÞ UðnÞ ¼  ln kl ðn þ n0 Þ ; kl > 0; l sffiffiffi ! hpffiffiffiffiffiffi i k cot UðnÞ ¼  ln  kl ðn þ n0 Þ ; kl > 0; l sffiffiffiffiffiffiffi ! hpffiffiffiffiffiffiffiffiffi i k tanh UðnÞ ¼  ln kl ðn þ n0 Þ ; kl < 0; l sffiffiffiffiffiffiffi ! hpffiffiffiffiffiffiffiffiffi i k coth UðnÞ ¼  ln kl ðn þ n0 Þ ; kl < 0; l  UðnÞ ¼  ln 

ð11bÞ

ð11dÞ

ð11eÞ

ð14Þ

where U(n) satisfies the nonlinear ODEs (6), (8) and (10). By substituting (14) into (13) and using (6) repeatedly, we obtain a system of algebraic equations and by equating the coefficients ofðexpðUðnÞÞÞi , ði ¼ 0; 1; 2;   Þ to zero as follows: 9 > > > > > 2eV2 a4 A1  10eV2 a4 A2 k  2ea1 A1 A2 þ 2ea3 A1 þ 10ea3 A2 k ¼ 0; > > > > > 2ea1 A0 A2  3eV2 a4 A1 k  4eV2 a4 A2 k2 þ 3ea3 A1 k  A2 þ 8ea3 A2 l > > > > = 2 þV2 A2 þ 4ea3 A2 k  ea1 A21  8eV2 a4 A2 l ¼ 0; ; 2 > 2ea1 A0 A1  2eV2 a4 A1 l þ 2ea3 A1 l þ 6ea3 A2 lk  A1 þ ea3 A1 k > > > > 2 > þV2 A1  eV2 a4 A1 k  6eV2 a4 A2 lk ¼ 0; > > > > > ea1 A20  eV2 a4 A1 kl  2eV2 a4 A2 l2 þ ea3 A1 lk  A0 þ K > > > ; þ2ea3 A2 l2 þ V2 A0 ¼ 0 6eV2 a4 A2  ea1 A22 þ 6ea3 A2 ¼ 0;

ð15Þ

Solving the system of algebraic Eq. (15) with aid of symbolic computation, such as Maple, we have 9 K ¼ 4ea1 1 ð2V2 þ e2 a23 k4  V4  32e2 a3 l2 V2 a4 > > > > > > > 8e2 V4 a24 lk2  2e2 V2 a4 k4 a3 = 2 ð16Þ þ16e2 a23 l2 þ 16e2 V2 a4 la3 k þ 16e2 V4 a24 l2 > > > 2 4 2 4 2 2 2 > þe V a4 k  8e a3 k l  1Þ; V ¼ V > > > ; A ¼ m1 ; A ¼ km ; A ¼ m ; 1

2ea1

2

2

2

where m1 ¼ ð8eV2 a4 l þ eV2 a4 k2  V2  ea3 k2 þ 1  8ea3 lÞ, m2 ¼  a61 ða3 þ V2 a4 Þ; k and l are arbitrary constants. If we put the integrating constant equal to zero, i.e. K = 0 in Eqs. (13) and (15) and solving the algebraic equations given in (15), then we obtain the following set of solutions: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 4ea3 l  ea3 k2  1 m3 ¼ ;A ¼ km ; A ¼ m ; A 0 1 4 2 4 a1 4ea4 l  ea4 k2  1

V¼

Set 1 :

ð16aÞ

where n0 is the integrating constant and kl > 0 or kl < 0 are dependent on sign of k. Finally, we obtain the new multiple explicit solutions of NLEE (3) by combining the Eqs. (5) and (11).

2

3 lþa3 k 2la4 k where m3 ¼  2a1þ4ea 2 4 lea4 k l are arbitrary constants.

( Set 2 :

3. Applications of the method This section presents the application of the proposed exponential expansion method to find the more general and exact traveling wave solutions to the strain wave Eq. (2). We now utilize the wave variable uðx; tÞ ¼ uðnÞ; n ¼ x  V t; where V is the speed of traveling, then the Eq. (2) reduces to a nonlinear ODE as 00

ðV2  1Þ u00  e a1 ðu2 Þ þ e ða3  V2 a4 Þ uð4Þ ¼ 0;

ð12Þ (4)

where u00 denotes the second derivative, and u denotes the fourth derivative with respect to n. Integrating (12) twice with respect to n, we get ðV2  1Þ u  ea1 u2 þ eða3  V2 a4 Þ u00 þ K ¼ 0;

uðnÞ ¼ A0 þ A1 ðexpðUðnÞÞÞ þ A2 ðexpðUðnÞÞÞ2 ;

(



1 ; k ¼ 0; l > 0; lðn þ n0 Þ

where K is the integral constant. It can be easily shown that, the pole of Eq. (13) is N = 2. According to the introduced method, the solution of the Eq. (13) can be written as

0

ð11cÞ

685

ð13Þ

2

a4

; m4 ¼

6ða3 þa4 Þ  ð1þ4ea 2 4 lea4 k Þa1

; k and

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 4ea3 l  ea3 k2 þ 1 m5 V¼ ; A0 ¼ ;A1 ¼ km6 ; A2 ¼ m6 a1 4ea4 l  ea4 k2 þ 1 ð16bÞ

6lða3 a4 Þ 6ða3 þa4 Þ where m5 ¼ 1þ4ea ; k and l are arbi2 ; m6 ¼ ð1þ4ea4 lea4 k2 Þa1 4 lea4 k trary constants. Combining Eqs. in (7), (14) and (16), the strain wave Eq. (2) has the following explicit solutions: ! m1 2kl pffiffiffiffi u1 ðx; tÞ ¼  m2 pffiffiffiffi 2ea1 H tanhð0:5 Hðx  Vt þ n0 ÞÞ þ k !2 2l pffiffiffiffi ; þ m2 pffiffiffiffi H tanhð0:5 Hðx  Vt þ n0 ÞÞ þ k

l–0; H > 0;

ð17Þ

686

M.G. Hafez, M.A. Akbar !

m1 2lk pffiffiffiffiffiffiffiffi þ m2 pffiffiffiffiffiffiffiffi 2ea1 H tanð0:5 Hðx  Vt þ n0 ÞÞ  k !2 2l pffiffiffiffiffiffiffiffi ; þ m2 pffiffiffiffiffiffiffiffi H tanð0:5 Hðx  Vt þ n0 ÞÞ  k

u2 ðx; tÞ ¼

ð18Þ

l–0; H < 0;   m1 k2 þ m2 2ea1 expðkðx  Vt þ n0 ÞÞ  1  2 k ; l ¼ 0; H > 0; þ m2 expðkðx  Vt þ n0 ÞÞ  1

ð19Þ

 3  m1 k ðx  Vt þ n0 Þ u4 ðx; tÞ ¼  m2 2ea1 2kðx  Vt þ n0 Þ þ 4  2 2 k ðx  Vt þ n0 Þ ; l–0; k–0; H ¼ 0 ð20Þ þ m2 2kðx  Vt þ n0 ÞÞ þ 4 u5 ðx; tÞ ¼

m1 m2 k 1 þ þ m2 ðx  Vt þ n0 Þ 2ea1 x  Vt þ n0

l ¼ 0; k ¼ 0:

;

In the same way combining Eqs. in (7), (14), (16a) and (16b), we obtain further ten solutions to the strain wave Eq. (2) which are omitted in this article for simplicity. Again, by substituting (14) into (13) and using (8) recurrently, we obtain a system of algebraic equations and by equating the coefficients of ðexpðUðnÞÞÞi ; ði ¼ 0; 1; 2; 3;   Þ to zero as follows: 9 A0 þ V2 A0  ea1 A20 þ K ¼ 0; > > > > > ea3 A1 k  A1  eV2 a4 A1 k2  2ea1 A0 A1 þ V2 A1 ¼ 0; > = 2 2 2 4ea3 A2 k  2ea1 A0 A2  A2  ea1 A1 þ V A2  4eV a4 A2 k ¼ 0; ; > > > > 2ea3 A1 l  2eV2 a4 A1 l  2ea1 A1 A2 ¼ 0; > > ; 2 2 6ea3 A2 l  6eV a4 A2 l  ea1 A2 ¼ 0: ð22Þ Solving the system of algebraic Eqs. (22), we obtain 9   K ¼ 4ea1 1 1 þ 2V2  V4 þ 16e2 a23 k2  32e2 a3 k2 V2 a4 ; V ¼ V = 2

2a k 4

;A1 ¼ 0; A2 ¼ 6lða3 aV 1

2a Þ 4

:

; ð23Þ

where V,k and l are arbitrary constants. Again, if we put the integrating constant equal to zero, i.e., K = 0 in Eqs. (22) and solving the algebraic Eq. (22), then we obtain the following set of solutions: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( 4ea3 k þ 1 4kða3  a4 Þ ; A0 ¼ Set 3 : V¼ ; 4ea4 k þ 1 ð1 þ 4ea4 kÞa1 6lða3 þ a4 Þ ð23aÞ A1 ¼ 0; A2 ¼ ð1 þ 4ea4 kÞa1 (

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 4ea3 k  1 6lða3 þ a4 Þ ;A0 ¼ 0; A1 ¼ 0;A2 ¼  4ea4 k  1 ð1 þ 4ea4 kÞa1

V¼

ð23bÞ

where k and l are arbitrary constants.

4ea3 k  1 þ V2  4eV2 a4 k 2ea1 i 6kða3  V2 a4 Þ 2 hpffiffiffiffiffiffiffi  sec k ðx  Vt þ n0 Þ ;k < 0; l > 0; ð26Þ a1

u8 ðx;tÞ ¼

4ea3 k  1 þ V2  4eV2 a4 k 2ea1 hpffiffiffi i 6kða3  V2 a4 Þ  sec h2 k ðx  Vt þ n0 Þ ;k > 0; l < 0; ð27Þ a1

u10 ðx; tÞ ¼ 

1  V2 6ða3  V2 a4 Þ þ a1 2ea1



1 x  Vt þ n0

2 ;

k ¼ 0; l > 0;

u11 ðx; tÞ ¼ 

ð28Þ

1  V2 6ða3  V2 a4 Þ  a1 2ea1



1 x  Vt þ n0

2 ;

k ¼ 0; l < 0:

ð29Þ

In the same way combining Eqs. in (9), (14), (23a) and (23b) we obtain further twelve solutions to the strain wave Eq. (2) which are omitted in this article for simplicity. Finally, substituting Eq. (14) into (13) and using nonlinear ODE (10) at every step, we obtain a system of algebraic and equating the coefficients of ðexpðUðnÞÞÞi ði ¼ 0; 1; 2;   Þ, as: 9 2ea3 A1 l k  2eV2 a4 A1 lk ¼ 0; > > > > K  A0 þ V2 A0 þ 6ea3 A2 l k  ea1 A20  6eV2 a4 A2 lk ¼ 0; > > > > 2ea3 A1 l2  2ea1 A0 A1  2eV2 a4 A1 l2 þ V2 A1  A1 ¼ 0; = ; ea1 A21  A2  6eV2 a4 A2 l2 þ 2ea3 A2 k2  2eV2 a4 A2 k2 > > > > > þV2 A2  2ea1 A0 A2 þ 6ea3 A2 l2 ¼ 0; 2ea1 A1 A2 ¼ 0; > > > ; 2 2 2ea3 A2 lk  2eV a4 A2 lk  ea1 A2 ¼ 0: ð30Þ Solving the algebraic Eqs. in (30), we obtain 9 > > > > > þ4e2 V4 a24 k4 þ 4e2 a23 k4 þ 36e2 a23 l4  24e2 V4 a24 k2 l2  72e2 V2 a4 l4 a3 = 8e2 a3 k4 V2 a4  1Þ; V ¼ V; A0 ¼  2ea1 1 ð1 þ 6eV2 a4 l2  2ea3 k2 > > > > > 2klða3 þV2 a4 Þ 2 ; 2 2 2 þ2eV a k  V  6ea l Þ;A ¼ 0;A ¼  K ¼ 4ea1 1 ð2V2  V4 þ 48e2 V2 a4 l2 k2 a3 þ 36e2 V4 a24 l4  24e2 a23 k2 l2

4

3

1

2

a1

ð31Þ

where k and l are arbitrary constants. Set 4 :

4ea3 k  1 þ V2  4eV2 a4 k 2ea1 i 6kða3  V2 a4 Þ 2 hpffiffiffiffiffiffiffi k ðx  Vt þ n0 Þ ; k < 0; l > 0; ð25Þ  csc a1

u9 ðx;tÞ ¼

2 ð21Þ

4eV A0 ¼ 4ea3 k1þV 2ea1

4ea3 k  1 þ V2  4eV2 a4 k 2ea1 hpffiffiffi i 6kða3  V2 a4 Þ þ csc h2 k ðx  Vt þ n0 Þ ;k > 0; l > 0; ð24Þ a1

u6 ðx;tÞ ¼

u7 ðx; tÞ ¼

u3 ðx; tÞ ¼



By means of Eqs. (9), (14) and (23), the stain wave Eq. (2) has the following new explicit solutions:

where V, k and l are arbitrary constants. Again, if we put the integrating constant equal to zero, that is, K = 0 in Eq. (30) and solving the algebraic Eq. (30), then we obtain the following set of solutions:

An exponential expansion method and its application to the strain wave

Set 5 :

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 < V ¼  6ea3 l22 2ea3 k22 1; A0 ¼  ð6ea 6ea l 2ea k 1 4

:

4

2k2 ða3 a4 Þ 2 2ea k2 1Þa 4 1

4l

3 þa4 Þ A1 ¼ 0; A2 ¼  ð6ea 2klða l2 2ea k2 1Þa 4

4

1

687

9 = ; ð31aÞ

where k and l are arbitrary constants. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 9 6l2 ða3 a4 Þ < V ¼  6ea3 l22 2ea3 k22 þ1; A0 ¼ = 6ea4 l 2ea4 k þ1 ð6ea4 l2 2ea4 k2 þ1Þa1 Set 6 : : ; 3 þa4 Þ A1 ¼ 0; A2 ¼ ð6ea 2klða l2 2ea k2 þ1Þa 4

4

1

ð31bÞ where k and l are arbitrary constants. Using the Eqs. (11), (14) and (31), the fourth order strain wave Eq. (2) has the following unsullied explicit solutions: hpffiffiffiffiffiffi i 2k2 ða3 þ V2 a4 Þ u12 ðx; tÞ ¼ C þ tan2 kl ðx  Vt þ n0 Þ ; a1 kl > 0; ð32Þ u13 ¼ C þ

Figure 1 solution.

Surface plot of topological kink type solitary wave

Figure 2 solution.

Surface plot of singular kink type solitary wave

i 2k2 ða3 þ V2 a4 Þ 2 hpffiffiffiffiffiffi cot kl ðx  Vt þ n0 Þ ; kl > 0; a1 ð33Þ hpffiffiffiffiffiffiffiffiffi i 2k2 ða3 þ V2 a4 Þ tanh2 kl ðx  Vt þ n0 Þ ; a1 kl < 0; ð34Þ

u14 ðx; tÞ ¼ C 

hpffiffiffiffiffiffiffiffiffi i 2k2 ða3 þ V2 a4 Þ coth2 kl ðx  Vt þ n0 Þ ; a1 kl < 0;

u15 ðx; tÞ ¼ C 

where, C ¼  2ea1 1 ð1 þ 6eV2 a4 l2  2ea3 k2 þ 2eV2 a4 k2  V2  2 6ea3 l Þ. In the same way combining Eqs. in (11), (14), (31a) and (31b) we obtain further eight solutions to the strain wave Eq. (2) which are omitted in this article for simplicity. 4. Physical explanation This section presents the physical significances and graphical representations of the above determined solutions to the strain wave equation. The solutions u(x, t) to the strain wave Eq. (2) play an important role for describing different types of wave propagation in micro-structured solids. The Eq. (2) is given not only more new multiple explicit solutions but also many types of exact traveling wave solutions. The exact traveling wave solutions are obtained from the explicit solutions by choosing the particular value of the physical parameters. So, we can select the appropriate values of the parameters to obtain exact solutions, we need in varied instances. There are various types of traveling wave solutions that are particular interest in solitary wave theory. In this research work some important traveling wave solutions are described and presented graphically. The solitary wave solutions of kink type corresponding to u1 ðx; tÞ for the fixed values of the parameters a1 ¼ 1, a3 ¼ 1, a4 ¼ 2, e ¼ 0:5,V ¼ 1,k ¼ 3,l ¼ 1, n0 ¼ 1 within 5  x; t  5 have presented in Fig. 1. Kink solitons are rise or descent from one asymptotic state at n ! 1 to another asymptotic state at n ! 1. These solitons are referred to as

topological solitons. Fig. 2 shows the solitary wave solution of singular kink type corresponding to u4(x, t) with fixed parameters a1 = 3, a3 = 1, a4 = 2, e = 1, V = 1, k ¼ 2, l = 1, n0 = 0 within 10  x; t  10. Fig. 3 presents the solitary wave solution of singular soliton type corresponding to u15(x, t) with fixed parameters a1 = 1, a3 = 2, a4 = 1, e = 0.5, V = 1, k ¼ 0:5, l = 0.5, n0 = 0 within 5  x; t  5. Fig. 4 shows an exact cuspon type solitary wave profile corresponding to u6(x, t) with fixed parameters a1 = 1, a3 = 1, a4 = 2, e = 0.5, V = 1, k ¼ 0:5, l = 1, n0 = 1 within 5  x; t  5: Fig. 5 shows an exact compacton profile corresponding to the solution u14(x, t) with fixed parameters a1 = 1, a3 = 3, a4 = 1, e = 0.5, V = 1, k ¼ 0:1, l = - 0.5, n0 = 0 and 3  x; t  3. Compacton is a new category of solitons with compact spatial hold up such that each compacton is a soliton restricted to a finite core. It has a remarkable soliton property that after colliding with other compactons, they come back with the same coherent shape. These particles like waves exhibit elastic collision that is similar to the soliton collision. That is, a compacton is a solitary wave with a compact

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Figure 3

Figure 4

M.G. Hafez, M.A. Akbar

Surface plot of singular soliton solitary wave solution.

Figure 5

Surface plot of compacton type solitary wave solution.

Figure 6 solution.

Surface plot of non-topological bell type solitary wave

Surface plot of cuspon type solitary wave solution.

support where the nonlinear dispersion confines it to a finite core, therefore the exponential wings disappear. The bell type solitary wave solution corresponding to u8(x, t) for the fixed values of the parameters a1 = 1, a3 = 1, a4 = 2, e = 0.5, V = 1, k ¼ 0:5, l = 1, n0 = 1 and 5  x; t  5 is shown in Fig. 6. It has infinite wings or infinite tails. This soliton referred to as non-topological solitons. This solution does not depend on the amplitude and high frequency soliton. Fig. 7 represents the periodic traveling wave solution corresponding to u12 ðx; tÞ with fixed parameters a1 ¼ 1, a3 ¼ 3, a4 ¼ 1, e ¼ 1, V ¼ 1, k ¼ 0:5, l ¼ 0:5, n0 ¼ 0 and 3  x; t  3. Finally, Fig. 8 shows exact singular periodic traveling wave profile corresponding to solution u13 ðx; tÞ a with fixed parameters a1 ¼ 1, a3 ¼ 2, a4 ¼ 1, e ¼ 0:5, V ¼ 1, k ¼ 1, l ¼ 1, n0 ¼ 0 and 3  x; t  3: 5. Advantages and comparison The main advantage of the introduced method is that it offers more general and huge amount of new exact traveling wave solutions with some free parameters. The exact solutions have its extensive importance to interpret the inner structures of the natural phenomena. The explicit solutions represented various types of solitary wave solutions according to the variation of the physical parameters. In different literatures [10–14,39], the exp(U(n))-expansion method has been employed to look into exact solutions, wherein the nonlinear ordinary differential equation U0 ðnÞ ¼ expðUðnÞÞ þ l expðUðnÞÞ þ k ;k; l 2 R

provides only a few traveling wave solutions to the NLEEs which does not give a complete picture of physical phenomena. But, in this article, we extended exp(U(n))-expansion method for finding the new solutions of these physical phenomena. With the assistance computer algebra system, such as Maple, Mathematica, the proposed new exponential expansion method according to the nonlinear ordinary differential equaqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tions U0 ðnÞ ¼  k þ lðexpðUðnÞÞÞ2 ; k; l 2 R and U0 ðnÞ ¼ k expðUðnÞÞ  l expðUðnÞÞ; k; l 2 R provides some new solutions of the NLEEs. This method not only re-derives all known solutions in a systemic way but also obtains some entirely new and more explicit solutions to the NLEEs. The algebraic manipulation of this method with the help of Maple is much easier than the other existing method. Sometimes the method gives solutions in disguised versions of known solutions that may be found by other methods. Many researchers applied different methods for finding the traveling wave solutions to the strain wave equation, such as Alam et al. [27] used the new approach of the generalized (G0 /G)-expansion method for investigating the general traveling wave solutions. From the article [27], we observe that the obtained solutions to the NLEEs can be reduced to our

An exponential expansion method and its application to the strain wave

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engineering. The obtained solutions can be utilized to further analyze by physicists or engineers on varied instances. References

Figure 7

Figure 8 solution.

Surface plot of periodic wave solution.

Surface plot of singular periodic traveling wave

obtained solutions u10 ðx; tÞ; u11 ðx; tÞ; u13 ðx; tÞ and u15 ðx; tÞ. The other obtained traveling waves solutions in this paper are new and have not been found in the previous literature [27,37,40]. Therefore, it can be decided that the proposed method is powerful mathematical tool for solving nonlinear evolutions equations and all kinds of NLEEs can be solved through this method. We are working on another article in which the other NLEEs are considered for showing its effectiveness. 6. Conclusions In this article, we have developed a new exponential expansion method and successfully implemented to obtain more explicit and exact traveling wave solutions to the nonlinear evolution equation via the strain wave equation. This method not only produces the same solutions but can also pick up what we believe to be new solutions missed by other researchers. The results indicate the strain wave equation admit multiple exact traveling wave solutions with few parameters. The type of exact solitary wave solution is different along with different value of arbitrary parameters. So we can choose appropriate parameter value to obtain exact solitary wave solutions we need in varied instances. The paper has shown that the new exponential expansion method is sufficient to search for more new exact solutions of NLEEs in mathematical physics and

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Md. Golam Hafez has received his M. Phil. in Mathematical Physics from University of Chittagong, Bangladesh in 2011. He has received his M.Sc. in Applied Mathematics from University of Rajshahi, Bangladesh in 2006. Currently, he is an Assistant Professor at the Dept. of Mathematics, Chittagong University of Engineering and Technology, Chittagong, Bangladesh. His fields of interest are Nonlinear Differential equations, Plasma Physics, and Quantum Field Theory.

M. Ali Akbar has received his M. Sc. in Mathematics from University of Rajshahi, Bangladesh in 1996. He has received his Ph.D. in Mathematics from the same university in 2005. At present he is an Associate Professor at the Dept. of Applied Mathematics, University of Rajshahi, Bangladesh. His fields of interest are nonlinear ordinary and partial differential equations, population dynamics, computational mathematics and analysis.