New Variable Separation Excitations of a (2+1)-dimensional Broer-Kaup- Kupershmidt System Obtained by an Extended Mapping Approach

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1 New ariable Separation Ecitations of a (+1)-dimensional Broer-Kaup- Kupershmidt System Obtained by an Etended Mapping Approach Chun-Long Zheng a,b, Jian-Ping Fang a, and Li-Qun Chen b a Department of Physics, Zhejiang Lishui University, Lishui 33, China b Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 7, China Reprint requests to C.-L. Z.; zjclzheng@yahoo.com.cn Z. Naturforsch. 59a, (); received July 6, Using an etended mapping approach, a new type of variable separation ecitation with two arbitrary functions of the (+1)-dimensional Broer-Kaup-Kupershmidt system (BKK) is derived. Based on this ecitation, abundant propagating and non-propagating solitons, such as dromions, rings, peakons, compactons, etc. are found by selecting appropriate functions. PACS: 5.5.Yv, 3.65.Ge Key words: Etended Mapping Approach; (+1)-Dimensional BKK System; Localized Ecitation. 1. Introduction In recent studies on soliton theory many powerful approaches were presented. An important approach is multilinear variable separation [1]. In this way, with a Painlevé-Bäcklund transformation, one can find that some significant types of localized ecitations, such as dromions, rings, compatons, peakons and loop solutions, can simultaneously eist in a (+1)-dimensional soliton model [1 3]. The main reason is that there is a rather common formula, namely u = (a a 1 a 3 a )q y p (a + a 1 p + a q + a 3 pq) (1) with two arbitrary functions p p(,t) and q q(y,t), to describe certain physical quantities of some (+1)- dimensional models. Another important approach is the so-called mapping transformation method []. The basic idea of this algorithm is that for a given nonlinear partial differential equation (NPDE) with independent variables t, 1,,, m, and dependent variables u, P(u,u t,u i,u i j, )=, () where P is in general a polynomial function of its arguments, and the subscripts denote partial derivatives, by using the travelling wave transformation, () possesses the ansatz u = u(ξ ), ξ = m i= k i i, (3) where k i with i =,, m are all arbitrary constants. Substituting (3) into () yields the ordinary differential equation(ode) O(u(ξ ),u(ξ ) ξ,u ξξ, )=. Then u(ξ ) is epanded into a polynomial in g(ξ ) u(ξ )=F(g(ξ )) = n j= a j g(ξ ) j, () where a j are constants to be determined, and n is fied by balancing the linear term of the highest order with the nonlinear term in (). If we suppose g(ξ )= tanh(ξ ), g(ξ )=sech(ξ ) and g(ξ )=sn(ξ ) or g(ξ )= cn(ξ ), respectively, the corresponding approach is usually called the tanh-function method, sech-function method and Jacobian-function method. Although the Jacobian elliptic function method is better known than the tanh-function and sech-function method, the repeated calculations are often tedious. The main idea of the mapping approach is, that g(ξ ) is not assumed to be a specific function, such as tanh, sech, sn, cn, etc., but a solution of a mapping equation such as the Riccati equation (g ξ = g + a ), or a solution of the cubic nonlinear Klein-Gordon equation (g ξ = a g c + a g + a ), or a solution of the general elliptic equation (g ξ = i= a ig i ), where a i,i (1,,3,) are all arbitrary constants. Using the mapping relation () and the solutions of these mapping equations, one can obtain many eplicit and eact travelling wave solutions of () / / 1 91 $ 6. c erlag der Zeitschrift für Naturforschung, Tübingen Download Date /6/18 7:3 AM

2 C.-L. Zheng et al. New ariable Separation Ecitations of a (+1)-dimensional BKK System 913 Now its an interesting question whether all above localized ecitations based on the former multilinear approach can be derived by the latter mapping approach, which is usually used to search for travelling wave solutions. The crucial question is how to obtain solutions of () with certain arbitrary functions. In order to derive new solutions with certain arbitrary functions, we assume solutions of the form [5] with u = n i= α i ()φ i (ω()) (5) φ = σ + φ, (6) where =( = t, 1,,, m ), σ is a constant and the prime denotes differentiation with respect to ω. To determine u eplicitly, one may take the following steps: First, similar to the usual mapping approach, determine n by balancing the highest nonlinear terms and the highest-order partial terms in the given NPDE. Second, substituting (5) and (6) into the given NPDE and collecting coefficients of polynomials of φ, then eliminating each coefficient to derive a set of partial differential equations of α i (i =,1,...,n)and ω. Third, solving the system of partial differential equations to obtain α i and ω. Finally, as (6) possesses the general solution σ tanh( σω) σ <, σ coth( σω) σ <, φ = σ tan( σω) σ >, σ cot( (7) σω) σ >, 1/ω σ =, substituting α i, ω and (7) into (5), one can obtain solutions of the concerned in NPDE.. New ariable Separation Solutions of the (+1)-dimensional BKK System As a concrete eample, we consider the celebrated (+1)-dimensional Broer-Kaup-Kupershmidt(BKK) system [6] H yt H y + (HH ) y + v =, v t + (vh) + v =. (8) The BKK system was used to model nonlinear and dispersive long gravity waves travelling in two horizontal directions on shallow waters of uniform depth. It can also be derived from the celebrated Kadomtsev-Petviashvili(KP) equation by the symmetry constraint [7]. Abundant propagating localized ecitations were derived by Lou [1] with help of a Painlevé-Bäcklund transformation and a multilinear variable separation approach. However, as far as we know, its non-propagating solitons has not been reported. It is worth mentioning that this system has been widely applied in many branches of physics like plasma physics, fluid dynamics, nonlinear optics, etc. So, a good understanding of more solutions of the BKK system (8) is very helpful, especially for coastal and civil engineers to apply the nonlinear water model in a harbor and coastal design. To solve the (+1)-dimensional Broer-Kaup- Kupershmidt system, we consider the following Painlevé-Bäcklund transformation for H and v in (8): H =(ln f ) + H, v =(ln f ) y + v, (9) which can be derived from the standard Painlevé truncated epansion, where the functions H = H (,t) and v = are seed solutions of the BKK system (8). Based on (9) and the seed solutions, we can straightly obtain a simple relation for H and v = H y. Inserting v = H y into (8) yields y (H t + H + HH )=. (1) For simplicity and convenience of discussions, here we take H t + H + HH =. (11) Now we apply the etended mapping approach to (11). By balancing the highest nonlinear term and the highest-order partial term, ansatz (5) becomes H = f (,y,t)+g(,y,t)φ(q(,y,t)), (1) where f f (,y,t),g g(,y,t) and q q(,y,t) are arbitrary functions of {, y,t} to be determined. Substituting (1) together with (6) into (11), and collecting coefficients of polynomials of φ, then setting each coefficient to zero, we have 6gq q y (q + g)=, (13) gq y q t + gq g y + g q y + gq y q + g y q + gq q y + g q y q + fgq q y + g gq y =, (1) Download Date /6/18 7:3 AM

3 91 C.-L. Zheng et al. New ariable Separation Ecitations of a (+1)-dimensional BKK System gq y + g y q + f gq y + fgq y + gq f y +gg y + fg y q + g y q t + 8g q q y σ +g q y + 8gq y q σ + fg q y + g g y + g y q + g t q y + g q y + gq yt =, g y q σ + gq y q t σ + fg y + g f y + g yt +g q q y σ + g y + gq g y σ + f g y +g q y σ + gq q y σ + g gq y σ + fgq y q σ + gf y + gq y q σ =, g y q σ + gq y σ + g t q y σ + g q q y σ + f y + f yt + gq yt σ + fg y q σ + f f y + f gq y σ + fg q y σ + g y q t σ + gq f y σ + fgq y σ + g q y σ + gq q yσ + g y q σ +g q y σ + ff y =. (15) (16) (17) According to (13) (17) and through careful calculations, we finally obtain the following eact solution: g = q, (18) f = q + q t, q (19) q = χ(,t)+ϕ(y), () where χ χ(,t), ϕ ϕ(y) are two arbitrary variable separation functions of {,t} and y, respectively. Now, based on the solutions of (6), one can obtain new types of variable separation ecitations of (8). Case 1. For σ <, we can derive the following solitary wave solutions of (8): H 1 = χ + χ t χ + σχ tanh( σ(χ +ϕ)), (1) v 1 = σχ ϕ y σχ ϕ y tanh ( σ(χ + ϕ), () H = χ + χ t χ + σχ coth( σ(χ +ϕ)), (3) v = σχ ϕ y σχ ϕ y coth ( σ(χ + ϕ), () where χ χ(,t) and ϕ ϕ(y) are two arbitrary functions of the indicated variables. Case. For σ >, we can obtain the following periodic wave solutions of (8): H 3 = χ + χ t χ σχ tan( σ(χ + ϕ)), (5) v 3 = σχ ϕ y σχ ϕ y tan ( σ(χ + ϕ)), (6) H = χ + χ t + σχ cot( σ(χ + ϕ)), (7) χ v = σχ ϕ y σχ ϕ y cot ( σ(χ + ϕ)), (8) with two arbitrary functions χ(,t)and ϕ(y). Case 3. For σ =, we get the following variable separation solution of (8): H 5 = χ + χ t χ + χ χ + ϕ, (9) v 5 = χ ϕ y (χ + ϕ), (3) also with two arbitrary functions χ(,t)and ϕ(y). 3. Some Special Solitons in the (+1)-dimensional BKK System Because of the arbitrariness of the functions χ(,t) and ϕ(y) in the above cases, the physical quantities u and v may possess rich structures. For eample, when χ(,t) = f (k + ct) and ϕ(y) =g(y), where f and g are arbitrary functions of the indicated arguments, then all the solutions of the above cases may show abundant localized and propagating ecitations. Obviously, one of the simplest travelling wave ecitations can be easily obtained when χ(,t)= k+ ct and ϕ(y) = ly, where k,l and c are arbitrary constants. Still based on the derived solutions, we may also derive rich stationary localized solutions, which are not travelling wave ecitations or not propagating waves [8], just as Wu et al. reported about the non-propagating solitons in 198 [9]. For instance, when the arbitrary functions are selected to be χ(,t) =ς() +τ(t) and ϕ(y) =g(y), where ς, τ and g are also arbitrary functions of the indicated arguments, then we can obtain many kinds of non-propagating localized solutions like dromion, ring, peakon, compacton solutions, and so on. Moreover, if τ(t) is considered to be a periodic function or a solution of a chaos system like the Lorenz chaos system [1], then the non-propagating solitons possess periodic or chaotic behaviors. For simplification of the following discussion, we merely analyse some special localized ecitations of solutions v 1 and rewrite () in a simple form, namely v 1 = χ ϕ y σ sec h ( σ(χ + ϕ)), (31) (σ < ). Download Date /6/18 7:3 AM

4 C.-L. Zheng et al. New ariable Separation Ecitations of a (+1)-dimensional BKK System 915 Fig.1a.3.5 Fig.1b y Fig. 1: a) A non-propagating two-dromion structure of the solution v 1 epressed by (31) with the condition (3) at t = π. b) The corresponding evolutional plot related to a) at y = and at the times t = (doted line), t = (dashed line), t = (solid line)..1.5 Fig.a Fig.b 1 1 y y Fig. : a) A standing ring breather structure of the solution v 1 epressed by (31) with the condition (3) at time t =. b) The evolutional contour plot related to a). The value of the contour figure is set =.5 at the times t =, t = or and t = or, respectively from inside to outside Non-propagating Solitons According to the solution v 1 (31), and setting σ = 1, we first discuss its dromion or dromion lattice ecitation, which is one of the significant localized ecitations, localized in all directions. When the simple selections of the functions χ and ϕ are considered to be χ =.1tanh()+.1tanh( + 6)+sin(t), ϕ =.1tanh(y), (3) then we can obtain a non-propagating two-dromion ecitation for the physical quantity v 1 depicted in Figure 1a. The corresponding evolutional plot is presented in Figure 1b. Furthermore, if taking χ = N n= N.1 tanh( + 6n)+sin(t), ϕ = M m= M.1tanh(y + 6m), (33) then we can obtain a (M + 1) (N + 1) dromion lattice ecitation for the physical field v 1, where M and N are arbitrary positive integers. In high dimensions, in addition to the point-like localized coherent ecitations, there may be some other types of physically significant localized ecitations. For instance, in (+1)-dimensional cases, there are some types of ring soliton solutions which are not identically zero at some closed curves and decay eponentially away from the closed curves. Based on the solution v 1 (31) (σ = 1), for eample, when the functions χ and ϕ are simply considered to be χ = + t,ϕ = y, (3) then we can obtain a ring solition ecitation for the physical quantity v 1. Meanwhile, one can find that the Download Date /6/18 7:3 AM

5 916 C.-L. Zheng et al. New ariable Separation Ecitations of a (+1)-dimensional BKK System Fig.3a.3.5 Fig.3b y Fig. 3. a) A non-propagating two-peakon soliton plot of the solution v 1 epressed by (31) with the conditions (37) and (38) at t =. b) The corresponding evolutional plot, related to a) at y = and at times t = 3 (doted line), t = 1 (dashed line) and t = 3 (solid line). Fig.a.5 Fig.b y Fig.. a) A propagating two-dromion structure of the solution v 1 epressed by (31) with condition (39) at t = 3. b) The corresponding evolutional plot related to a) at y = and at times t = 6 (doted line before collision), t = (dashed line in collision), t = 3 (solid line after collision). ring soliton solution is a standing breather ecitation of the (+1)-dimensional BKK system. The related evolutional plots are presented in Figure. Still according to the solution v 1 (31), we can obtain some weakly localized ecitations such as peakon and compaction ecitations [11, 1]. For eample, when selecting χ and ϕ to be some piecewise smooth functions, then we can derive some multipeakon ecitations, say { M Xi (),, χ = τ(t)+ (35) X i ( )+X i (), >, ϕ = 1 + N i=1 i=1 { Yi (y), y, Y i ( y)+y i (), y >, (36) where the functions X i () and Y i (y) are differentiable functions of the indicated arguments and possess the boundary conditions X i (± ) =C ±i,(i = 1,,...,M) and Y i (± )=D ±i,(i = 1,,...,N), where C ±i and D ±i are constants and/or infinity. For instance, when choosing τ = ep(sin(t)) and X 1 =.1ep( + 5),X =.ep( 5), (37) Y 1 =.1ep(y + 1),M =,N = 1, (38) then we can derive a non-propagating two-peakon ecitation for the solution v 1 epressed by (31). The corresponding two-peakon ecitation plot is presented in Figure 3. Similarly, if χ and ϕ are fied to be other types of piecewise smooth functions similar to [1], then we can derive compacton ecitations. Since similar situations have been discussed in [1 3], the related plots are neglected in the present paper. 3.. Propagating Solitons Obviously, in the above cases, if the independent variable is replaced by + ct, since the function Download Date /6/18 7:3 AM

6 C.-L. Zheng et al. New ariable Separation Ecitations of a (+1)-dimensional BKK System 917 Fig.5a. Fig.5b y Fig. 5. a) A propagating two-peakon structure of the solution v 1 epressed by (31) with the conditions X 1 =.1ep(+t),X 1 =.ep( t),y 1 =.1ep(y + 1) in (35) and (36) at t =. b) The corresponding evolutional plot related to a) at y = and at times t = 6 (doted line before collision), t = (dashed line in collision), t = 3 (solid line after collision). Fig.6a 3 Fig.6b y Fig. 6. a) A propagating ring structure of the solution v 1 epressed by (31) with the conditions χ = ( t) +,ϕ = y + at t =. b) The corresponding evolutional contour plot related to a). The value of the contour figure is set =.5 at times t =, t = 3, t = 6, travelling from left to right. 1 y 1 3 χ(,t) is an arbitrary function {,t}, all the above nonpropagating solitons will become propagating solitons. For eample, when the functions χ and ϕ are χ =.1tanh( + t)+.15tanh( t), ϕ =.1tanh(y), (39) then we can obtain the simple travelling two-dromion ecitation for the physical quantity v 1 depicted in Figure a. The corresponding evolutional plot is presented in Figure b. Similarly, the propagating peakons and ring solitons can be derived. Two simple eamples are presented in Figs. 5 and 6, respectively. The corresponding parameters are presented in the figure legends. It is interesting to mention that by the evolutional Figs. b and 5b we have verified the preceding results [1]: for the dromiondromion collision the interaction is completely elastic while for the peakon-peakon interaction it is nonelastic, since the former preserve their shapes and velocities (right velocity along with positive ais c r = 1 and left velocity along with negative ais c l = 1) while the latter change their shapes even though they preserve their velocities after the collision.. Summary and Discussion In summary, by means of an etended mapping approach, the (+1)-dimensional BKK system is successfully solved. Based on the derived new types of variable separation solutions with two arbitrary functions, we have found rich localized ecitations, which are propagating and non-propagating, by selecting the arbitrary functions appropriately. Actually, even according to the solution v 5 (3) are can obtain abundant propagating and non-propagating soli- Download Date /6/18 7:3 AM

7 918 C.-L. Zheng et al. New ariable Separation Ecitations of a (+1)-dimensional BKK System tons. The main reason is that, comparing the common formula (1) with the solution (3), one can easily find that they are essentially equivalent when choosing some parameters appropriately. Therefore, all the localized ecitations based on the common formula (1) can be re-derived from the solution v 5 (3). This paper is only a beginning work. We epect that the mapping approach applied in our present work may be further etended to other nonlinear physical systems. Acknowledgements C.-L. Zheng would like to thank Professor J. F. Zhang for fruitful discussions and helpful suggestions. This work was supported by the Natural Science Foundation of China under Grant No , the Scientific Research Fund of Zhejiang Provincial Education Department under Grant No. 39 and the Natural Science Foundation of Zhejiang Lishui University under Grant Nos. KZ39 and KZ35. [1] X. Y. Tang, S. Y. Lou, and Y. Zhang, Phys. Rev. E 66, 661 (); X. Y. Tang and S. Y. Lou, J. Math. Phys., (3). [] C. L. Zheng and L. Q. Chen, J. Phys. Soc. Japan 73, 93 (); C. L. Zheng and Z. M. Sheng, Inter. J. Mod. Phys. B 17, 7 (3). [3] C. L. Zheng and J. F. Zhang, Chin. Phys. Lett. 19, 1399 (); C. L. Zheng, J. F. Zhang, and Z. M. Sheng, Chin. Phys. Lett., 331 (3); C. L. Zheng et al., Chin. Phys. Lett., 783 (3). [] S. Y. Lou and G. J. Ni, J. Math. Phys. 3, 161 (1989). [5] Z. Lu and H. Zhang, Chaos, Solitons and Fractals 19, 57 (3). [6]. G. Durovsky and E. G. Konopelchenko, J. Phys. A 7, 619 (199); M. Boiti, Inv. Prob. 3, 37 (1987). [7] S. Y. Lou and X. B. Hu, J. Math. Phys. A 38, 61 (1997); S. Y. Lou and X. B. Hu, Commun. Theor. Phys.9, 15 (1998). [8] A. Larraza and S. Putterman, J. Fluid Mech. 18, 3 (198); Y. G. Xu and R. Wei, J. Chin. Phys. Lett. 1, 7 (199); J. R. Yan and Y. P. Mei, Europhys. Lett. 3, 335 (1993). [9] J. R. Wu, R. Keolian, and I. Rudnich, Phys. Rew. Lett. 5, 11 (198). [1] F. W. Lorenz, J. Atomos. Sci., 13 (1963); M. Clerc, P. Coullet and E. Tirapegui, Phys. Rev. Lett. 83, 38 (1999). [11] R. Camassa and D. D. Holm, Phys. Rev. Lett. 71, 1661 (1993). [1] P. Rosenau and J. Hyman, Phys. Rev. Lett. 7, 56 (1993). Download Date /6/18 7:3 AM