New Application of the (G /G)-Expansion Method to Excite Soliton Structures for Nonlinear Equation

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1 New Application of the /)-Expansion Method to Excite Soliton Structures for Nonlinear Equation Bang-Qing Li ac and Yu-Lan Ma b a Department of Computer Science and Technology Beijing Technology and Business University Beijing PR China b Department of Applied Mathematics Beijing Technology and Business University Beijing PR China c School of Mechanical Electronic and Information Engineering China University of Mining and Technology Beijing PR China Reprint requests to Y.-L. M.; mayl@th.btbu.edu.cn Z. Naturforsch. 65a ); received May / revised October The /)-expansion method is extended to construct non-travelling wave solutions for highdimensional nonlinear equations and to explore special soliton structure excitations and evolutions. Taking an example a new series of the non-travelling wave solutions are calculated for the +)- dimensional asymmetrical Nizhnik-Novikov-Veselov system by using the /)-expansion method. By selecting appropriately the arbitrary functions in the solutions special soliton-structure excitations and evolutions are studied. Key words: /)-Expansion Method; +)-Dimensional Asymmetrical Nihnik-Novikov-Veselov System; Non-Travelling Wave Solution; Soliton Structure Excitation. PACS numbers: 0.30.Jr Yv e. Introduction The dynamics of soliton structures is a fascinating subject for nonlinear evolution equations NEEs) which are drawn from interesting nonlinear physical phenomena. The soliton structures can be considered as a type of the special exact solutions of NEEs. It is a quite difficult but significant task to generate the localized soliton structures for +) spatial and temporal)-dimensional NEEs. Just a few approaches have been mentioned in the literature such as the variable separation method [ ] the Riccati equation mapping method [ 3] and the similarity reduction method [4]. Very recently an approach called the /)- expansion method was proposed to obtain new exact solutions of NEEs [5]. Subsequently the powerful /)-expansion method has been widely used by many such as in [6 34]. The method bases on the homogeneous balance principle and the linear ordinary differential equation LODE) theory. In general the solutions obtained by the /)-expansion method include three types namely hyperbolic function solutions trigonometric function solution and rational function solutions. However the previous works have mainly concentrated on obtaining new exact travelling wave solutions for NEEs. Our goal of this paper is to extend the /)- expansion method to construct non-travelling wave solutions with arbitrary functions which can be used as seed functions to excite localized soliton structures for high-dimensional NEEs. As one application of the method we will consider the well-known +)- dimensional asymmetrical Nizhnik-Novikov-Veselov ANNV) system u t + u x 3v x u 3vu x = 0 ) u x + v y = 0. ) The ANNV system ) ) was initially introduced by Boiti et al. [35] and its several soliton structures were studied respectively by the variable separation method [36] and the Riccati equation mapping method [37]. The organization of this paper is as follows. Section is devoted to describe the process of constructing a new series of the non-travelling wave solutions of the ANNV system by extending the /)-expansion method. On the basis of the solutions a type of the localized soliton structure excitation can be studied / 0 / $ c 00 Verlag der Zeitschrift für Naturforschung Tübingen

2 B.-Q. Li and Y.-L. Ma /)-Expansion Method to Excite Soliton Structures 59 through appropriate selection of the arbitrary functions in Section 3. Finally we conclude in Section 4.. The /)-expansion Method and the Non- Travelling Wave Solutions of the ANNV System For a given +)-dimensional NEE with independent variables xyt and dependent variable u Fuu t u x u y u tt u xt u yt u xy u u yy...)=0 3) the fundamental idea of the /)-expansion method is that the solutions of 3) can be expressed by a polynomial in /) as follows [5]: u = n i=0 a i [ q) q) ] 4) where q = sx + ly Vt is the travelling wave transformation and slv a i i = 0...n) are constants to be determined later q) satisfies the second order LODE as follows: + λ + µ = 0. 5) In order to construct the non-travelling wave solutions with arbitrary function qx yt) for the ANNV system ) ) we suppose its solutions can be express as follows: u = v = n i=0 n + m [ ] q) i a i q) j= i=0 m + j= A j [ q) q) [ ] q) i b i q) B j [ q) q) ] j { ] j { [ ] q) } 6) q) [ ] q) } 7) q) where a i i = 0...n) A j j =...n) b i i = 0...m) B j j =...m) are functions of xyt to be determined later q = qxyt) is an arbitrary function of xyt andq) satisfies the second-order LODE as follows: + µ = 0. 8) Remark : Compared with 4) the travelling wave transformation q = sx + ly Vt is a special case of the arbitrary function q of xyt in 6) and 7). Thus the coefficient a i A j b i B j in 6) and 7) must be functions of x yt. Remark : In 6) and 7) the added term of [ q)/q)] j { [ q)/q)] } / is for obtaining more helpful equations to solve a i A j b i B j. Applying the homogenous balance principle [38] we obtain n = m =. Thus 6) 7) can be converted into ) ) [ ) ] u = a 0 + a + a + A 9) )[ ) ] + A ) ) [ ) ] v = b 0 + b + b + B )[ ) ] + B 0) where a 0 a a A A b 0 b b B B are functions of xyt to be determined later and q is an arbitrary function of x y t. For simplifying the computation we seek for the variable separation solutions of the ANNV system ) ) by taking qx yt)= f xt)+gyt). Substituting 9) 0) into the ANNV system ) ) collecting all terms with the same power of /) together the left-hand sides of the ANNV system ) ) are converted into the polynomials in /). Then setting each coefficient of the polynomials to zero we can derive a set of over-determined partial differential equation for a 0 a a A A b 0 b b B B and q. ) 5 :a q x a b A B = 0 ) ) 4 [ ) 4 :3a b + a b + A B + A B ) + [a q x ) x + q x a x a q x )] = 0 ) 3 [ ) ] :3a B + A b + a B ) ] :A q x a B A b = 0) + A b )+ [ A q x ) x + q x A x A q x ) ] = 0 3) 4)

3 50 B.-Q. Li and Y.-L. Ma /)-Expansion Method to Excite Soliton Structures ) 3 :3q x a 0 b + a b 0 + a b + A B + µa B ) a q t q x a x a q x ) x + a xq x 8µa q 3 x = 0 5) ) [ ) ] :3qx a B + A b + a 0 B + A b 0 ) A q t q x A x A q x ) x 6) + q xa x µa q x ) 4µA q 3 x = 0 ) :3q x a 0 b + a b 0 + µa B + µa B ) a q t q x a x µa q x ) x + x a x a q x )=0 7) )[ ) ] :3qx a 0 B + A b 0 ) A q t q x A x µa q x ) x + xa x A q x )=0 8) ) :3a 0 b + a b 0 + µa B + µa B ) x a t a x µa q x ) + µ[q x a x a q x )] x = 0 9) ) 0 [ ) ] :3a0 B + A b 0 ) x A t A x µa q x ) + µ[q x A x A q x )] x = 0 0) ) 0 :3a 0 b 0 + µa B ) x a 0t a 0x µa q x ) + µ[q x a x µa q x )] x = 0 ) and ) 3 : a q x b q y = 0 ) ) [ ) ] : A q x B q y = 0 3) ) : a x a q x b y + b q y = 0 4) )[ ) ] : 5) ) A x A q x B y + B q y = 0 : a x b y = 0 6) ) 0 [ ) ] : Ax B y = 0 7) ) 0 : a 0x µa q x b 0y + µb q y = 0. 8) Solving ) 8) yields a 0 = x q y b 0 = q t + q x + 3 x 3q x a = A = 0 a = A = q x q y b = B = q x B = b = q a 0 = x q y b 0 = q t + q x + 3 x 3q x 9) a = A = 0 a = A = q x q y b = B = q x B = b = q 30) Substituting 9) 30) and the general solutions of 8) into 9) 0) we can obtain the non-travelling wave solutions for the ANNV system ) ). Case. When µ < 0 the hyperbolic function solutions for the ANNV system ) ) are the following: C sinh +C cosh u = µ f x g y µ f x g y C cosh +C sinh C sinh +C cosh + µ f x g y [ ) C cosh +C sinh 3) C ] C C cosh +C sinh ) v = f t + g t + f x + µ fx 3 3 f x C sinh +C cosh µ f C cosh +C sinh µ fx C sinh +C cosh ) C cosh +C sinh + [ C µ f C C cosh +C sinh ) + µ fx [ C sinh +C cosh C cosh +C sinh C C C cosh +C sinh ) ] ] 3)

4 B.-Q. Li and Y.-L. Ma /)-Expansion Method to Excite Soliton Structures 5 and C sinh +C cosh ) u = µ f x g y µ f x g y C cosh +C sinh C sinh +C cosh [ µ f x g y C cosh +C sinh C ] 33) C C cosh +C sinh ) v = f t + g t + f x + µ fx 3 C sinh +C cosh µ f 3 f x C cosh +C sinh µ f x µ f x C sinh +C cosh ) C cosh +C sinh [ C µ f C C cosh +C sinh ) C sinh +C cosh [ C cosh +C sinh C ] C C cosh +C sinh ) where q = f xt)+gyt) f x 0 f x and g y g t exist and C C 0. Case. When µ > 0 the trigonometric function solutions for the ANNV system ) ) are the following: C sin +C cos ) u 3 = µ f x g y + µ f x g y C cos +C sin C sin +C cos [ µ f x g y C cos +C sin C ] 35) +C C cos +C sin ) v 3 = f t + g t + f x + µ fx 3 C sin +C cos µ f 3 f x C cos +C sin and + µ f x µ f x C sin +C cos ) C cos +C sin + [ C µ f +C C cos +C sin ) C sin +C cos [ C cos +C sin C ] +C C cos +C sin ) C sin +C cos ) u 4 = µ f x g y + µ f x g y C cos +C sin C sin +C cos [ + µ f x g y C cos +C sin C ] 37) +C C cos +C sin ) v 4 = f t + g t + f x + µ fx 3 C sin +C cos µ f 3 f x C cos +C sin + µ f x + µ f x C sin +C cos ) C cos +C sin [ C µ f +C C cos +C sin ) C sin +C cos [ C cos +C sin C ] +C C cos +C sin ) where q = f xt)+gyt) f x 0 f x and g y g t exist and C +C 0 namely C 0andC 0. ] ] ] 34) 36) 38)

5 5 B.-Q. Li and Y.-L. Ma /)-Expansion Method to Excite Soliton Structures a) b) c) d) Fig.. Evolution plots of the solution 3) under the parameters 4) with time: a) t = 0; b) t = 0.6; c) t =.; d) t =.8. Case 3. When µ = 0 the rational function solution for the ANNV system ) ) is the following: ) C u 5 = f x g y 39) C q +C v 5 = f t + g t + f x + 3 f x C C q +C )[ ) ] fx C f C q +C 40) where q = f xt)+gyt) f x 0 f x and g y g t exist and C 0. Comparing the solutions 3) 40) with those in [36] obtained by the variable separation method we find that there are some connections between these solutions. If setting C = C = 0 the solutions 39) and 40) can be degenerated to the solutions 9) and 0) given in [36] respectively. 3. Soliton Structure Excitation Thanks to the arbitrary functions f xt) and gyt) in the solutions 3) 40) it is convenient to excite abundant soliton structures. We take the solution 3) as an example to study the soliton excitations for the ANNV system ) ). For instance we select f xt)=sin x t )exp x t ) gyt)=sin y )exp y 4) ). Substituting 4) into 3) leads to a type of oscillatory dromion soliton structure for the ANNV system ) ). Figure a to d are the evolution plots of the solution 3) with time under the parameters C = C =. µ =. 4) Figure a d are the corresponding projection plots to Figure a d on the xu plane. The amplitude of the soliton structure varies dramatically while the wave shape fluctuates with time. Above we showed the excitation process of a special dromion soliton structure of the solution 3) for the ANNV system ) ). It is clear that other selections of the arbitrary functions f xt) and gy) in 3) may generate rich localized soliton structures. On the other hand the solutions 3) 40) may also be used to excite abundant soliton structures.

6 B.-Q. Li and Y.-L. Ma /)-Expansion Method to Excite Soliton Structures 53 a) b) c) d) Fig.. Corresponding projective plots to Figure on the x u plane: a) t = 0; b) t = 0.6; c) t =.; d) t = Conclusion By extending the /)-expansion method we are able to construct a new series of exact non-travelling wave solutions of the +)-dimensional ANNV system. Furthermore by selecting appropriately the arbitrary function qx yt) included in the solutions one can study various interesting localized soliton excitations. Within our knowledge it is the first attempt to explore the localized soliton excitations through the non-travelling wave solutions by the /)-expansion method. We believe the idea in this paper is also applied to other NEEs in the future. Acknowledgements This work is supported by the Scientific Research Common Program of Beijing Municipal Commission of Education No. KM000000). [] J. Lin and X. M. Qian Phys. Lett. A ). [] S. Y. Lou J. Phys. A: Math. en ). [3] X. Y. Tang and S. Y. Lou Commun. Theor. Phys ). [4] X.M. Qian and S.Y. Lou Z. Naturforsch. 59a ). [5] X. Y. Tang and Z. F. Liang Phys. Lett. A ). [6] S. Y. Lou H. X. Hu and X. Y. Tang Phys. Rev. E ). [7] J. F. Zhang and F. M. Wu Chin. Phys ). [8] C. L. Zheng J. M. Zhu and J. F. Zhang Commun. Theor. Phys ). [9] J. M. Zhu Z. Y. Ma and C. L. Zheng Acta Phys. Sin ). [0] C. L. Zheng H. P. Zhu and L. Q. Chen Chaos Solitons and Fractals ). [] X. J. Lai and J. F. Zhang Z. Naturforsch. 64a 009). [] J. P. Fang C. L. Zheng and L. Q. Chen Commun. Theor. Phys ). [3] J. P. Fang and C. L. Zheng Chin. Phys ).

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