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Topological soliton solutions for some nonlinear evolution equations
Ain Shams Engineering Journal (2014) 5, 257–261
Ain Shams University
Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com
ENGINEERING PHYSICS AND MATHEMATICS
Topological soliton solutions for some nonlinear evolution equations Ahmet Bekir a b
, O¨zkan Gu¨ner
a,*
b
Eskisehir Osmangazi University, Art-Science Faculty, Department of Mathematics-Computer, Eskisehir, Turkey Dumlupınar University, School of Applied Sciences, Department of Management Information Systems, Ku¨tahya, Turkey
Received 12 April 2013; revised 15 July 2013; accepted 10 August 2013 Available online 23 September 2013
KEYWORDS Exact solutions; Topological soliton; (2 + 1)-Dimensional breaking soliton (Calogero) equation; The sixth-order Ramani equation
Abstract In this paper, the topological soliton solutions of nonlinear evolution equations are obtained by the solitary wave ansatz method. Under some parameter conditions, exact solitary wave solutions are obtained. Note that it is always useful and desirable to construct exact solutions especially soliton-type (dark, bright, kink, anti-kink, etc.) envelope for the understanding of most nonlinear physical phenomena. Ó 2013 Production and hosting by Elsevier B.V. on behalf of Ain Shams University.
1. Introduction Nonlinear evolution equations are special classes of the category of partial differential equations (PDEs), which have been studied intensively in past few decades [1]. It is well known that seeking explicit solutions for nonlinear evolution equations, by using different numerous methods, plays a major role in mathematical physics and becomes one of the most exciting and extremely active areas of research investigation for mathematicians and physicists. In fact, when we want to understand the physical mechanism of phenomena in nature, described by nonlinear PDEs, exact solution for the nonlinear PDEs has to * Corresponding author. Tel.: +90 222 2393750; fax: +90 222 2393578. E-mail addresses: [email protected] (A. Bekir), [email protected] (O¨. Gu¨ner). Peer review under responsibility of Ain Shams University.
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be explored. In particular, the soliton solutions play an important role in the study of the models arising from various natural phenomena and scientific fields; for instance, the wave phenomena observed in fluid mechanics, elastic media, optical fibers, nuclear physics, high-energy physics, plasma physics, biology, solid-state physics, chemical kinematics, chemical physics and geochemistry, etc. Most famous models of such equations admitting solitons are, for example, the nonlinear Schro¨dinger equations, the Korteweg–de Vries equations, the Kadomtsev–Petviashvili equations, the Boussinesq equations, and the Zakharov–Kuznetsov equations. A large number of such equations have been studied in these contexts, and numerous analytic and computational effective techniques have been proposed to investigate these types of equations. Some of these methods that have been recently developed 0 are the GG -expansion method [2,3], exponential function method [4,5], Fan’s F-expansion method [6,7], the tanh–sech method [8–10], extended tanh method [11–13], sine–cosine method [14–16], homogeneous balance method [17,18], first integral method [19–21], simplest equation method [22,23], ansatz method [24–27] and many others. There are various
2090-4479 Ó 2013 Production and hosting by Elsevier B.V. on behalf of Ain Shams University. http://dx.doi.org/10.1016/j.asej.2013.08.002
A. Bekir, O¨. Gu¨ner
258 advantages and disadvantages of these modern methods of integrability that includes these methods. Although a closed form soliton solution can be obtained by these techniques, the disadvantage of these methods is that these techniques cannot compute the conserved quantities of nonlinear evolution equations nor it can lay down an expression of the soliton radiation. Nevertheless, the fact that soliton solutions can be obtained is itself a big blessing [28–30]. In this paper, one such method of integration will be used to carry out the integration for some nonlinear evolution equations (NEEs) to obtain the topological soliton solutions.
Now equating the highest exponents of tanh2p+1 s and tanhp+2 s terms in Eq. (4) gives 2p þ 1 ¼ p þ 2 ð5Þ which yields p¼1
ð6Þ
It should be noted that the same value of p is yielded when the exponents pair 2p + 3 and p + 4; 2p 3 and p 2; and 2p 1 and p are equated with each other, respectively. pðp þ 1Þðp þ 2Þðp þ 3Þg31 g2 k þ 6k2 p2 ðp þ 1Þg21 g2 ¼ 0
ð7Þ
4pðp þ 1Þðp2 þ 2p þ 2Þg31 g2 k 6k2 p2 ð3p þ 1Þg21 g2 pðp þ 1Þkvg1 ¼ 0 ð8Þ
2. Ansatz method In this section, the ansatz method will be used to obtain the Calogero and Ramani equations of solutions. The search is going to be for a topological 1-soliton solution, which is also known as a kink solution or a shock wave solution. This will be demonstrated in the following two subsections. For both equations, arbitrary constant coefficients will be considered. 2.1. (2 + 1)-Dimensional breaking soliton (Calogero) equation
2p2 ð3p2 þ 5Þg31 g2 k þ 6k2 p2 ð3p 1Þg21 g2 þ 2p2 kvg1 ¼ 0
ð9Þ
If we put p = 1 in (7)–(9), the system reduces to 24g31 g2 k þ 12k2 g21 g2 ¼ 0
ð10Þ
40g31 g2 k 24k2 g21 g2 2kvg1 ¼ 0
ð11Þ
16g31 g2 k þ 12k2 g21 g2 þ 2kvg1 ¼ 0
ð12Þ
Solving the above equations yields Consider the following nonlinear (2 + 1)-dimensional breaking soliton (Calogero) equation is given uxxxy 2uy uxx 4ux uxy þ uxt ¼ 0
ð1Þ
which was first introduced by Calogero et al. [31]. Mei and Zhang obtained some soliton-like solutions and periodic solutions by using the projective Riccati equation expansion method [32]. Geng and Cao are obtained some N-soliton solutions and algebro-geometric solutions [33]. The topological soliton solution: The topological solitons that are also known as optical solitons are also supported by the NEEs [34–39]. In this case, as it will be seen that for the case of generalized NEEs, the topological solitons will exist under certain restrictions. The subject of topological solitons will be studied for the case of power law nonlinearity only. Thus, the equation of study in this section will be (1). Let us begin the analysis by assuming an ansatz solution of the from [40–45]: uðx; y; tÞ ¼ ktanhp s
ð2Þ
k ¼ 2g1
ð13Þ
v ¼ 4g21 g2
ð14Þ
Hence, finally, the topological 1-soliton solution to (1) is respectively given by ð15Þ uðx; y; tÞ ¼ 2g1 tanh g1 x þ g2 y 4g21 g2 t and where the parameter k is given by (13), the velocity of the solitons is given in (14). 2.2. The sixth-order Ramani equation We consider the sixth-order Ramani equation or the KdV6 equation is given [46]
:uxxxxxx þ 15ux uxxxx þ 15uxx uxxx þ 45u2x uxx 5ðuxxxt þ 3ux uxt þ 3ut uxx Þ 5utt ¼ 0
ð16Þ
where the parameters k, gi (i = 1, 2) are the free parameters and is the velocity of the soliton. The value of the unknown exponent p will be determined during the course of derivation of the soliton solution of (1) From Eqs. (2) and (3), we find ux, uy, uxt, uxx, uxy, uyuxx, uxuxy, uxxxy and substituting these equations into Eq. (1), we obtain
In [47], it was indicated that there are four distinct cases for Eq. (16) to pass the Painleve´ test and was examined for complete integrability by using the Ba¨cklund transformation and Lax pairs. Now, topological soliton solution of this equation will be obtained. The Hirota’s bilinear method is used to determine the two distinct structures of solutions by Wazwaz and Triki in [48]. The topological soliton solution: In this section, we are interested in finding the topological soliton solution, as defined in, [49–54] for the considered sixth-order Ramani Eq. (16).
pg31 g2 kfðp 1Þðp 2Þðp 3Þtanhp4 s 4ðp 1Þðp2 2p
uðx; tÞ ¼ ktanhp s
ð17Þ
and s ¼ gðx vtÞ
ð18Þ
and s ¼ g1 x þ g2 y vt
p2
þ 2Þtanh
ð3Þ
2
p
s þ 2pð3p þ 5Þtanh sg
þ 2p þ 2Þtanh
pþ2
pg31 g2 kf4ðp pþ4
s ðp þ 1Þðp þ 2Þðp þ 3Þtanh
2
þ 1Þðp sg
6k2 p2 g21 g2 fðp 1Þtanh2p3 s ð3p 1Þtanh2p1 s þ ð3p þ 1Þtanh2pþ1 s ðp þ 1Þtanh2pþ3 sg pkvg1 fðp þ 1Þtanhpþ2 s 2ptanhp s þ ðp 1Þtanhp2 sg ¼0
ð4Þ
Here, k, g will be determined free parameters and is the soliton velocity. The unknown index p will be determined in terms of derivation of the exact solution. From Eqs. (17) and (18), it is possible to obtain uxxxxxx, uxuxxxx, utuxx, uxxuxxx, uxxxt, uxuxt, u2x uxx ; utt these terms and substituting these equations into Eq. (16), we obtain
Topological soliton solutions for some nonlinear evolution equations pkg6 ðp 1Þðp 2Þðp 3Þðp 4Þðp 5Þtanhp6 s
in Eq. (19), one gets:
6
2
pkg ðp 1Þðp 2Þðp 3Þfðp 4Þðp 5Þ þ ½2p 2
p4
þ 2ðp 2Þ þ ðp 3Þðp 4Þgtanh
259
3p þ 4 ¼ p þ 6
6
s þ pkg ðp 1Þ
ð20Þ
which yields the following analytical condition:
2
2
fðp 2Þðp 3Þ½2p þ 2ðp 2Þ þ ðp 3Þðp 4Þ
p¼1
þ ðp 1Þðp 2Þ½2p2 þ 2ðp 2Þ2 gtanhp2 s þ pkg6 ðp
It should be noted that the same value of p is yielded when the exponents pair 2p + 1 and p + 2; 3p and 2p + 1; 3p + 2 and p + 4; 2p + 5 and p + 6; 3p 4 and p 2; 2p 5 and p 4; 3p 2 and p and 2p 1 and p are equated with each other, respectively.
1Þf4p4 þ pðp 1Þ2 ðp 2Þ þ pðp þ 1Þ2 ðp þ 2Þgtanhp2 s pkg6 fðp 1Þ2 ðp 2Þ½2p2 þ 2ðp 2Þ2 þ 4p4 ðp 1Þ þ pðp 1Þ3 ðp 2Þgtanhp s pkg6 fpðp 1Þðp þ 1Þ2 ðp þ 2Þ þ ðp þ 1Þ2 ðp þ 2Þ½2p2
pkg6 ðp þ 1Þðp þ 2Þðp þ 3Þðp þ 4Þðp þ 5Þ
þ 2ðp þ 2Þ2 gtanhp s pkg6 f4p4 ðp þ 1Þ þ pðp 2
3
15p2 k2 g5 ðp þ 1Þ2 ðp þ 2Þ 15p2 k2 g5 ðp þ 1Þ2 ðp þ 2Þ
p
þ 1Þðp 1Þ ðp 2Þ þ pðp þ 1Þ ðp þ 2Þgtanh s 6
ð21Þ
þ 45p3 ðp þ 1Þk3 g4 ¼ 0
2
2
þ pkg ðp þ 1Þfðp þ 2Þðp þ 3Þ½2p þ 2ðp þ 2Þ þ ðp pkg6 ðp þ 1Þðp þ 2Þðp þ 3Þfðp þ 4Þðp þ 5Þ þ ½2p2
þ 3Þðp þ 4Þgtanhpþ2 s þ pkg6 ðp þ 1Þfðp þ 1Þðp þ 2Þ ½2p2 þ 2ðp þ 2Þ2 þ 4p4 þ pðp 1Þ2 ðp 2Þ
þ 2ðp þ 2Þ2 þ ðp þ 3Þðp þ 4Þg þ 15p2 k2 g5 ðp þ 1Þf2ðp
þ pðp þ 1Þ2 ðp þ 2Þgtanhpþ2 s pkg6 ðp þ 1Þðp þ 2Þðp
þ 2Þðp þ 3Þ þ ð3p2 þ 3p þ 2Þg þ 15p2 k2 g5 ðp þ 1Þf2pðp þ 2Þ þ ð3p2 þ 3p þ 2Þg 45p3 ð4p þ 2Þk3 g4 þ 5pðp
þ 3Þfðp þ 4Þðp þ 5Þ þ ½2p2 þ 2ðp þ 2Þ2 þ ðp þ 3Þðp
þ 1Þðp þ 2Þðp þ 3Þg4 kv 30p2 ðp þ 1Þk2 g3 v
þ 4Þgtanhpþ4 s þ pkg6 ðp þ 1Þðp þ 2Þðp þ 3Þðp þ 4Þðp
¼0
þ 5Þtanhpþ6 s þ 15p2 k2 g5 ðp 1Þðp 2Þðp 3Þtanh2p5 s 15p2 k2 g5 ðp 1Þf2ðp 2Þðp 3Þ
pkg6 ðp þ 1Þfðp þ 2Þðp þ 3Þ½2p2 þ 2ðp þ 2Þ2 þ ðp þ 3Þðp
þ ð3p2 3p þ 2Þgtanh2p3 s þ 15p2 k2 g5 fðp 1Þðp
þ 4Þg þ pkg6 ðp þ 1Þfðp þ 1Þðp þ 2Þ½2p2 þ 2ðp þ 2Þ2
2Þðp 3Þ þ 2ðp 1Þð3p2 3p þ 2Þ þ ðp þ 1Þð3p2
þ 4p4 þ pðp 1Þ2 ðp 2Þ þ pðp þ 1Þ2 ðp þ 2Þg
2p1
þ 3p þ 2Þgtanh
2 2 5
s 15p k g fðp þ 1Þðp þ 2Þðp
2
15p2 k2 g5 fðp þ 1Þðp þ 2Þðp þ 3Þ þ 2ðp þ 1Þð3p2 þ 3p
2
þ 3Þ þ 2ðp þ 1Þð3p þ 3p þ 2Þ þ ðp 1Þð3p 3p þ 2Þgtanh
2pþ1
s þ 15p k g ðp þ 1Þf2ðp þ 2Þðp þ 3Þ 2pþ3
þ ð3p þ 3p þ 2Þgtanh 2pþ5
ðp þ 3Þtanh 2Þtanh
2p5
þ 2Þ þ ðp 1Þð3p2 3p þ 2Þg 15p2 k2 g5 fðp 1Þðp
2 2 5
2
þ 1Þðp þ 2Þ þ 2pð3p2 þ 3p þ 2Þ þ ðp þ 1Þð3p2 3p
2 2 5
s 15p k g ðp þ 1Þðp þ 2Þ
2 2 5
þ 2Þg þ 270p4 k3 g4 20pðp þ 1Þðp2 þ 2p þ 2Þg4 kv
2
s þ 15p k g ðp 1Þ ðp
þ 30p2 ð3p þ 1Þk2 g3 v 5pðp þ 1Þkg2 v2
2 2 5
2
s 15p k g ðp 1Þf2ðp 2Þ þ ð3p
¼0
3p þ 2Þgtanh2p3 s þ 15p2 k2 g5 fðp þ 1Þðp 1Þðp þ 2Þgtanh2p1 s 15p2 k2 g5 fðp 1Þðp þ 1Þðp þ 2Þ
þ pðp 1Þ3 ðp 2Þ þ pðp 1Þðp þ 1Þ2 ðp þ 2Þg
þ 2pð3p2 þ 3p þ 2Þ þ ðp þ 1Þð3p2 3p þ 2Þgtanh2pþ1 s
pkg6 fðp þ 1Þ2 ðp þ 2Þ½2p2 þ 2ðp þ 2Þ2 þ 4p4 ðp þ 1Þ
þ 15p2 k2 g5 ðp þ 1Þf2pðp þ 2Þ þ ð3p2 þ 3p
þ pðp þ 1Þðp 1Þ2 ðp 2Þ þ pðp þ 1Þ3 ðp þ 2Þg
þ 2Þgtanh2pþ3 s 15p2 k2 g5 ðp þ 1Þ2 ðp þ 2Þtanh2pþ5 s
þ 15p2 k2 g5 fðp 1Þðp 2Þðp 3Þ þ 2ðp 1Þð3p2 3p
þ 45p3 k3 g4 fðp 1Þtanh3p4 s ð4p 2Þtanh3p2 s
þ 2Þ þ ðp þ 1Þð3p2 þ 3p þ 2Þg þ 15p2 k2 g5 fðp þ 1Þðp
þ 6ptanh3p s ð4p þ 2Þtanh3pþ2 s þ ðp þ 1Þtanh3pþ4 sg
1Þðp 2Þ þ ðp 1Þð3p2 þ 3p þ 2Þ þ 2pð3p2 3p
þ 5pg4 kvfðp 1Þðp 2Þðp 3Þtanhp4 s 4ðp 1Þ
þ 2Þg þ 10p2 g4 kvð3p2 þ 5Þ 45p3 k3 g4 ð4p 2Þ
ðp2 2p þ 2Þtanhp2 sg þ 5pg4 kvf2pð3p2
þ 10p2 kg2 v2 30ð3p 1Þk2 p2 g3 v
þ 5Þtanhp s 4ðp þ 1Þðp2 þ 2p þ 2Þtanhpþ2 s þ ðp
¼0
1Þtanh2p3 s ð3p 1Þtanh2p1 s þ ð3p þ 1Þtanh
s ðp þ 1Þtanh
90k3 g4 540k2 g5 þ 720kg6 ¼ 0
ð26Þ
120g4 kv þ 1380k2 g5 60k2 g3 v 270k3 g4 1680kg6 ¼ 0
ð27Þ
2 2
sg 5pkg v fðp
1Þtanhp2 s 2ptanhp s þ ðp þ 1Þtanhpþ2 sg ¼0
ð25Þ
Solving the above system for p = 1 gives
þ 1Þðp þ 2Þðp þ 3Þtanhpþ4 sg þ 15k2 p2 g3 vfðp 2pþ3
ð24Þ
pkg6 fðp 1Þ2 ðp 2Þ½2p2 þ 2ðp 2Þ2 þ 4p4 ðp 1Þ
2Þ þ ðp 1Þð3p2 þ 3p þ 2Þ þ 2pð3p2 3p
2pþ1
ð23Þ
ð19Þ
200g4 kv þ 1232kg6 þ 120k2 g3 v þ 270k3 g4 1140k2 g5 10kg2 v2 ¼ 0 ð28Þ
A. Bekir, O¨. Gu¨ner
260 300k2 g5 þ 80kg4 v þ 10kg2 v2 272kg6 90k3 g4 60k2 g3 v ¼0
ð29Þ
Solving the above equations yields k g¼ ; 2
k 4 pffiffiffi 1 3 5 2 k v¼ 2 10 g¼
ð30Þ
ð31Þ
Hence, finally, the topological soliton solution to the sixthorder Ramani Eq. (16) is given by uðx; tÞ ¼ k tanhðgðx vtÞÞ ð32Þ where the parameters k and g are all constants and is the velocity of the soliton. 3. Conclusion In this paper, we have derived the topological soliton solutions of the some nonlinear evolution equations. The knowledge of its solutions presented in any form will help to understand better the processes in such systems. To our knowledge, these new solutions have not been reported in former literature, and they may be of significant importance for the explanation of some special physical phenomena. The solitary wave ansatz method can be applied to high-dimensional or coupled NLPDEs in mathematical physics. We hope that the present solutions may be useful in further numerical analysis and may help one to explain some physical phenomena. In future, this equation will be explored further. References [1] Ablowitz MJ, Segur H. Solitons and inverse scattering transform. Philadelphia: SIAM; 1981. 0 [2] Wang ML, Li X, Zhang J. The GG -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys Lett 0 A 2008;372(4):417–23. [3] Bekir A. Application of the GG -expansion method for nonlinear evolution equations. Phys Lett A 2008;372(19):3400–6. [4] He JH, Wu XH. Exp-function method for nonlinear wave equations. Chaos, Solitons Fractals 2006;30:700–8. [5] He JH, Abdou MA. New periodic solutions for nonlinear evolution equations using Exp-function method. Chaos, Solitons Fractals 2007;34(5):1421–9. [6] Abdou MA. The extended F-expansion method and its application for a class of nonlinear evolution equations. Chaos, Solitons Fractals 2007;31:95–104. [7] Zhang JL, Wang ML, Wang YM, Fang ZD. The improved Fexpansion method and its applications. Phys Lett A 2006;350: 103–9. [8] Malfliet W. Solitary wave solutions of nonlinear wave equations. Am J Phys 1992;60:650–4. [9] Malfliet W, Hereman W. The tanh method. I: Exact solutions of nonlinear evolution and wave equations. Phys Scr 1996;54:563–8. [10] Wazwaz AM. The tanh method for travelling wave solutions of nonlinear equations. Appl Math Comput 2004;154(3):713–23. [11] El-Wakil SA, Abdou MA. New exact travelling wave solutions using modified extended tanh-function method. Chaos, Solitons Fractals 2007;31(4):840–52. [12] Fan E. Extended tanh-function method and its applications to nonlinear equations. Phys Lett A 2000;277:212–8.
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Ahmet Bekir is currently Associate Professor of Mathematics-Computer Department, Eskisehir Osmangazi University, Eskisehir, Turkey. His research interests are theory and exact solutions of partial differential equations in mathematical physics. His favorites in mathematics are ODEs, PDEs, fractional differential equations, integral equations, and analytic methods. He has published more than 80 articles journals.
O¨zkan Gu¨ner is currently a PhD student at Eskisehir Osmangazi University, Eskisehir, Turkey. He works as a researcher at Dumlupinar University. His research interests include exact solution of nonlinear differential equations. He has over 10 research articles that published in international journals.
Ain Shams University
Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com
ENGINEERING PHYSICS AND MATHEMATICS
Topological soliton solutions for some nonlinear evolution equations Ahmet Bekir a b
, O¨zkan Gu¨ner
a,*
b
Eskisehir Osmangazi University, Art-Science Faculty, Department of Mathematics-Computer, Eskisehir, Turkey Dumlupınar University, School of Applied Sciences, Department of Management Information Systems, Ku¨tahya, Turkey
Received 12 April 2013; revised 15 July 2013; accepted 10 August 2013 Available online 23 September 2013
KEYWORDS Exact solutions; Topological soliton; (2 + 1)-Dimensional breaking soliton (Calogero) equation; The sixth-order Ramani equation
Abstract In this paper, the topological soliton solutions of nonlinear evolution equations are obtained by the solitary wave ansatz method. Under some parameter conditions, exact solitary wave solutions are obtained. Note that it is always useful and desirable to construct exact solutions especially soliton-type (dark, bright, kink, anti-kink, etc.) envelope for the understanding of most nonlinear physical phenomena. Ó 2013 Production and hosting by Elsevier B.V. on behalf of Ain Shams University.
1. Introduction Nonlinear evolution equations are special classes of the category of partial differential equations (PDEs), which have been studied intensively in past few decades [1]. It is well known that seeking explicit solutions for nonlinear evolution equations, by using different numerous methods, plays a major role in mathematical physics and becomes one of the most exciting and extremely active areas of research investigation for mathematicians and physicists. In fact, when we want to understand the physical mechanism of phenomena in nature, described by nonlinear PDEs, exact solution for the nonlinear PDEs has to * Corresponding author. Tel.: +90 222 2393750; fax: +90 222 2393578. E-mail addresses: [email protected] (A. Bekir), [email protected] (O¨. Gu¨ner). Peer review under responsibility of Ain Shams University.
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be explored. In particular, the soliton solutions play an important role in the study of the models arising from various natural phenomena and scientific fields; for instance, the wave phenomena observed in fluid mechanics, elastic media, optical fibers, nuclear physics, high-energy physics, plasma physics, biology, solid-state physics, chemical kinematics, chemical physics and geochemistry, etc. Most famous models of such equations admitting solitons are, for example, the nonlinear Schro¨dinger equations, the Korteweg–de Vries equations, the Kadomtsev–Petviashvili equations, the Boussinesq equations, and the Zakharov–Kuznetsov equations. A large number of such equations have been studied in these contexts, and numerous analytic and computational effective techniques have been proposed to investigate these types of equations. Some of these methods that have been recently developed 0 are the GG -expansion method [2,3], exponential function method [4,5], Fan’s F-expansion method [6,7], the tanh–sech method [8–10], extended tanh method [11–13], sine–cosine method [14–16], homogeneous balance method [17,18], first integral method [19–21], simplest equation method [22,23], ansatz method [24–27] and many others. There are various
2090-4479 Ó 2013 Production and hosting by Elsevier B.V. on behalf of Ain Shams University. http://dx.doi.org/10.1016/j.asej.2013.08.002
A. Bekir, O¨. Gu¨ner
258 advantages and disadvantages of these modern methods of integrability that includes these methods. Although a closed form soliton solution can be obtained by these techniques, the disadvantage of these methods is that these techniques cannot compute the conserved quantities of nonlinear evolution equations nor it can lay down an expression of the soliton radiation. Nevertheless, the fact that soliton solutions can be obtained is itself a big blessing [28–30]. In this paper, one such method of integration will be used to carry out the integration for some nonlinear evolution equations (NEEs) to obtain the topological soliton solutions.
Now equating the highest exponents of tanh2p+1 s and tanhp+2 s terms in Eq. (4) gives 2p þ 1 ¼ p þ 2 ð5Þ which yields p¼1
ð6Þ
It should be noted that the same value of p is yielded when the exponents pair 2p + 3 and p + 4; 2p 3 and p 2; and 2p 1 and p are equated with each other, respectively. pðp þ 1Þðp þ 2Þðp þ 3Þg31 g2 k þ 6k2 p2 ðp þ 1Þg21 g2 ¼ 0
ð7Þ
4pðp þ 1Þðp2 þ 2p þ 2Þg31 g2 k 6k2 p2 ð3p þ 1Þg21 g2 pðp þ 1Þkvg1 ¼ 0 ð8Þ
2. Ansatz method In this section, the ansatz method will be used to obtain the Calogero and Ramani equations of solutions. The search is going to be for a topological 1-soliton solution, which is also known as a kink solution or a shock wave solution. This will be demonstrated in the following two subsections. For both equations, arbitrary constant coefficients will be considered. 2.1. (2 + 1)-Dimensional breaking soliton (Calogero) equation
2p2 ð3p2 þ 5Þg31 g2 k þ 6k2 p2 ð3p 1Þg21 g2 þ 2p2 kvg1 ¼ 0
ð9Þ
If we put p = 1 in (7)–(9), the system reduces to 24g31 g2 k þ 12k2 g21 g2 ¼ 0
ð10Þ
40g31 g2 k 24k2 g21 g2 2kvg1 ¼ 0
ð11Þ
16g31 g2 k þ 12k2 g21 g2 þ 2kvg1 ¼ 0
ð12Þ
Solving the above equations yields Consider the following nonlinear (2 + 1)-dimensional breaking soliton (Calogero) equation is given uxxxy 2uy uxx 4ux uxy þ uxt ¼ 0
ð1Þ
which was first introduced by Calogero et al. [31]. Mei and Zhang obtained some soliton-like solutions and periodic solutions by using the projective Riccati equation expansion method [32]. Geng and Cao are obtained some N-soliton solutions and algebro-geometric solutions [33]. The topological soliton solution: The topological solitons that are also known as optical solitons are also supported by the NEEs [34–39]. In this case, as it will be seen that for the case of generalized NEEs, the topological solitons will exist under certain restrictions. The subject of topological solitons will be studied for the case of power law nonlinearity only. Thus, the equation of study in this section will be (1). Let us begin the analysis by assuming an ansatz solution of the from [40–45]: uðx; y; tÞ ¼ ktanhp s
ð2Þ
k ¼ 2g1
ð13Þ
v ¼ 4g21 g2
ð14Þ
Hence, finally, the topological 1-soliton solution to (1) is respectively given by ð15Þ uðx; y; tÞ ¼ 2g1 tanh g1 x þ g2 y 4g21 g2 t and where the parameter k is given by (13), the velocity of the solitons is given in (14). 2.2. The sixth-order Ramani equation We consider the sixth-order Ramani equation or the KdV6 equation is given [46]
:uxxxxxx þ 15ux uxxxx þ 15uxx uxxx þ 45u2x uxx 5ðuxxxt þ 3ux uxt þ 3ut uxx Þ 5utt ¼ 0
ð16Þ
where the parameters k, gi (i = 1, 2) are the free parameters and is the velocity of the soliton. The value of the unknown exponent p will be determined during the course of derivation of the soliton solution of (1) From Eqs. (2) and (3), we find ux, uy, uxt, uxx, uxy, uyuxx, uxuxy, uxxxy and substituting these equations into Eq. (1), we obtain
In [47], it was indicated that there are four distinct cases for Eq. (16) to pass the Painleve´ test and was examined for complete integrability by using the Ba¨cklund transformation and Lax pairs. Now, topological soliton solution of this equation will be obtained. The Hirota’s bilinear method is used to determine the two distinct structures of solutions by Wazwaz and Triki in [48]. The topological soliton solution: In this section, we are interested in finding the topological soliton solution, as defined in, [49–54] for the considered sixth-order Ramani Eq. (16).
pg31 g2 kfðp 1Þðp 2Þðp 3Þtanhp4 s 4ðp 1Þðp2 2p
uðx; tÞ ¼ ktanhp s
ð17Þ
and s ¼ gðx vtÞ
ð18Þ
and s ¼ g1 x þ g2 y vt
p2
þ 2Þtanh
ð3Þ
2
p
s þ 2pð3p þ 5Þtanh sg
þ 2p þ 2Þtanh
pþ2
pg31 g2 kf4ðp pþ4
s ðp þ 1Þðp þ 2Þðp þ 3Þtanh
2
þ 1Þðp sg
6k2 p2 g21 g2 fðp 1Þtanh2p3 s ð3p 1Þtanh2p1 s þ ð3p þ 1Þtanh2pþ1 s ðp þ 1Þtanh2pþ3 sg pkvg1 fðp þ 1Þtanhpþ2 s 2ptanhp s þ ðp 1Þtanhp2 sg ¼0
ð4Þ
Here, k, g will be determined free parameters and is the soliton velocity. The unknown index p will be determined in terms of derivation of the exact solution. From Eqs. (17) and (18), it is possible to obtain uxxxxxx, uxuxxxx, utuxx, uxxuxxx, uxxxt, uxuxt, u2x uxx ; utt these terms and substituting these equations into Eq. (16), we obtain
Topological soliton solutions for some nonlinear evolution equations pkg6 ðp 1Þðp 2Þðp 3Þðp 4Þðp 5Þtanhp6 s
in Eq. (19), one gets:
6
2
pkg ðp 1Þðp 2Þðp 3Þfðp 4Þðp 5Þ þ ½2p 2
p4
þ 2ðp 2Þ þ ðp 3Þðp 4Þgtanh
259
3p þ 4 ¼ p þ 6
6
s þ pkg ðp 1Þ
ð20Þ
which yields the following analytical condition:
2
2
fðp 2Þðp 3Þ½2p þ 2ðp 2Þ þ ðp 3Þðp 4Þ
p¼1
þ ðp 1Þðp 2Þ½2p2 þ 2ðp 2Þ2 gtanhp2 s þ pkg6 ðp
It should be noted that the same value of p is yielded when the exponents pair 2p + 1 and p + 2; 3p and 2p + 1; 3p + 2 and p + 4; 2p + 5 and p + 6; 3p 4 and p 2; 2p 5 and p 4; 3p 2 and p and 2p 1 and p are equated with each other, respectively.
1Þf4p4 þ pðp 1Þ2 ðp 2Þ þ pðp þ 1Þ2 ðp þ 2Þgtanhp2 s pkg6 fðp 1Þ2 ðp 2Þ½2p2 þ 2ðp 2Þ2 þ 4p4 ðp 1Þ þ pðp 1Þ3 ðp 2Þgtanhp s pkg6 fpðp 1Þðp þ 1Þ2 ðp þ 2Þ þ ðp þ 1Þ2 ðp þ 2Þ½2p2
pkg6 ðp þ 1Þðp þ 2Þðp þ 3Þðp þ 4Þðp þ 5Þ
þ 2ðp þ 2Þ2 gtanhp s pkg6 f4p4 ðp þ 1Þ þ pðp 2
3
15p2 k2 g5 ðp þ 1Þ2 ðp þ 2Þ 15p2 k2 g5 ðp þ 1Þ2 ðp þ 2Þ
p
þ 1Þðp 1Þ ðp 2Þ þ pðp þ 1Þ ðp þ 2Þgtanh s 6
ð21Þ
þ 45p3 ðp þ 1Þk3 g4 ¼ 0
2
2
þ pkg ðp þ 1Þfðp þ 2Þðp þ 3Þ½2p þ 2ðp þ 2Þ þ ðp pkg6 ðp þ 1Þðp þ 2Þðp þ 3Þfðp þ 4Þðp þ 5Þ þ ½2p2
þ 3Þðp þ 4Þgtanhpþ2 s þ pkg6 ðp þ 1Þfðp þ 1Þðp þ 2Þ ½2p2 þ 2ðp þ 2Þ2 þ 4p4 þ pðp 1Þ2 ðp 2Þ
þ 2ðp þ 2Þ2 þ ðp þ 3Þðp þ 4Þg þ 15p2 k2 g5 ðp þ 1Þf2ðp
þ pðp þ 1Þ2 ðp þ 2Þgtanhpþ2 s pkg6 ðp þ 1Þðp þ 2Þðp
þ 2Þðp þ 3Þ þ ð3p2 þ 3p þ 2Þg þ 15p2 k2 g5 ðp þ 1Þf2pðp þ 2Þ þ ð3p2 þ 3p þ 2Þg 45p3 ð4p þ 2Þk3 g4 þ 5pðp
þ 3Þfðp þ 4Þðp þ 5Þ þ ½2p2 þ 2ðp þ 2Þ2 þ ðp þ 3Þðp
þ 1Þðp þ 2Þðp þ 3Þg4 kv 30p2 ðp þ 1Þk2 g3 v
þ 4Þgtanhpþ4 s þ pkg6 ðp þ 1Þðp þ 2Þðp þ 3Þðp þ 4Þðp
¼0
þ 5Þtanhpþ6 s þ 15p2 k2 g5 ðp 1Þðp 2Þðp 3Þtanh2p5 s 15p2 k2 g5 ðp 1Þf2ðp 2Þðp 3Þ
pkg6 ðp þ 1Þfðp þ 2Þðp þ 3Þ½2p2 þ 2ðp þ 2Þ2 þ ðp þ 3Þðp
þ ð3p2 3p þ 2Þgtanh2p3 s þ 15p2 k2 g5 fðp 1Þðp
þ 4Þg þ pkg6 ðp þ 1Þfðp þ 1Þðp þ 2Þ½2p2 þ 2ðp þ 2Þ2
2Þðp 3Þ þ 2ðp 1Þð3p2 3p þ 2Þ þ ðp þ 1Þð3p2
þ 4p4 þ pðp 1Þ2 ðp 2Þ þ pðp þ 1Þ2 ðp þ 2Þg
2p1
þ 3p þ 2Þgtanh
2 2 5
s 15p k g fðp þ 1Þðp þ 2Þðp
2
15p2 k2 g5 fðp þ 1Þðp þ 2Þðp þ 3Þ þ 2ðp þ 1Þð3p2 þ 3p
2
þ 3Þ þ 2ðp þ 1Þð3p þ 3p þ 2Þ þ ðp 1Þð3p 3p þ 2Þgtanh
2pþ1
s þ 15p k g ðp þ 1Þf2ðp þ 2Þðp þ 3Þ 2pþ3
þ ð3p þ 3p þ 2Þgtanh 2pþ5
ðp þ 3Þtanh 2Þtanh
2p5
þ 2Þ þ ðp 1Þð3p2 3p þ 2Þg 15p2 k2 g5 fðp 1Þðp
2 2 5
2
þ 1Þðp þ 2Þ þ 2pð3p2 þ 3p þ 2Þ þ ðp þ 1Þð3p2 3p
2 2 5
s 15p k g ðp þ 1Þðp þ 2Þ
2 2 5
þ 2Þg þ 270p4 k3 g4 20pðp þ 1Þðp2 þ 2p þ 2Þg4 kv
2
s þ 15p k g ðp 1Þ ðp
þ 30p2 ð3p þ 1Þk2 g3 v 5pðp þ 1Þkg2 v2
2 2 5
2
s 15p k g ðp 1Þf2ðp 2Þ þ ð3p
¼0
3p þ 2Þgtanh2p3 s þ 15p2 k2 g5 fðp þ 1Þðp 1Þðp þ 2Þgtanh2p1 s 15p2 k2 g5 fðp 1Þðp þ 1Þðp þ 2Þ
þ pðp 1Þ3 ðp 2Þ þ pðp 1Þðp þ 1Þ2 ðp þ 2Þg
þ 2pð3p2 þ 3p þ 2Þ þ ðp þ 1Þð3p2 3p þ 2Þgtanh2pþ1 s
pkg6 fðp þ 1Þ2 ðp þ 2Þ½2p2 þ 2ðp þ 2Þ2 þ 4p4 ðp þ 1Þ
þ 15p2 k2 g5 ðp þ 1Þf2pðp þ 2Þ þ ð3p2 þ 3p
þ pðp þ 1Þðp 1Þ2 ðp 2Þ þ pðp þ 1Þ3 ðp þ 2Þg
þ 2Þgtanh2pþ3 s 15p2 k2 g5 ðp þ 1Þ2 ðp þ 2Þtanh2pþ5 s
þ 15p2 k2 g5 fðp 1Þðp 2Þðp 3Þ þ 2ðp 1Þð3p2 3p
þ 45p3 k3 g4 fðp 1Þtanh3p4 s ð4p 2Þtanh3p2 s
þ 2Þ þ ðp þ 1Þð3p2 þ 3p þ 2Þg þ 15p2 k2 g5 fðp þ 1Þðp
þ 6ptanh3p s ð4p þ 2Þtanh3pþ2 s þ ðp þ 1Þtanh3pþ4 sg
1Þðp 2Þ þ ðp 1Þð3p2 þ 3p þ 2Þ þ 2pð3p2 3p
þ 5pg4 kvfðp 1Þðp 2Þðp 3Þtanhp4 s 4ðp 1Þ
þ 2Þg þ 10p2 g4 kvð3p2 þ 5Þ 45p3 k3 g4 ð4p 2Þ
ðp2 2p þ 2Þtanhp2 sg þ 5pg4 kvf2pð3p2
þ 10p2 kg2 v2 30ð3p 1Þk2 p2 g3 v
þ 5Þtanhp s 4ðp þ 1Þðp2 þ 2p þ 2Þtanhpþ2 s þ ðp
¼0
1Þtanh2p3 s ð3p 1Þtanh2p1 s þ ð3p þ 1Þtanh
s ðp þ 1Þtanh
90k3 g4 540k2 g5 þ 720kg6 ¼ 0
ð26Þ
120g4 kv þ 1380k2 g5 60k2 g3 v 270k3 g4 1680kg6 ¼ 0
ð27Þ
2 2
sg 5pkg v fðp
1Þtanhp2 s 2ptanhp s þ ðp þ 1Þtanhpþ2 sg ¼0
ð25Þ
Solving the above system for p = 1 gives
þ 1Þðp þ 2Þðp þ 3Þtanhpþ4 sg þ 15k2 p2 g3 vfðp 2pþ3
ð24Þ
pkg6 fðp 1Þ2 ðp 2Þ½2p2 þ 2ðp 2Þ2 þ 4p4 ðp 1Þ
2Þ þ ðp 1Þð3p2 þ 3p þ 2Þ þ 2pð3p2 3p
2pþ1
ð23Þ
ð19Þ
200g4 kv þ 1232kg6 þ 120k2 g3 v þ 270k3 g4 1140k2 g5 10kg2 v2 ¼ 0 ð28Þ
A. Bekir, O¨. Gu¨ner
260 300k2 g5 þ 80kg4 v þ 10kg2 v2 272kg6 90k3 g4 60k2 g3 v ¼0
ð29Þ
Solving the above equations yields k g¼ ; 2
k 4 pffiffiffi 1 3 5 2 k v¼ 2 10 g¼
ð30Þ
ð31Þ
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Ahmet Bekir is currently Associate Professor of Mathematics-Computer Department, Eskisehir Osmangazi University, Eskisehir, Turkey. His research interests are theory and exact solutions of partial differential equations in mathematical physics. His favorites in mathematics are ODEs, PDEs, fractional differential equations, integral equations, and analytic methods. He has published more than 80 articles journals.
O¨zkan Gu¨ner is currently a PhD student at Eskisehir Osmangazi University, Eskisehir, Turkey. He works as a researcher at Dumlupinar University. His research interests include exact solution of nonlinear differential equations. He has over 10 research articles that published in international journals.