FISSION, FUSION AND ANNIHILATION IN THE INTERACTION OF LOCALIZED STRUCTURES FOR THE (2+1)-DIMENSIONAL GENERALIZED BROER-KAUP SYSTEM By

Based on the WTC truncation method and the general variable separation approach (GVSA), we have first found a general solution including three arbitrary functions for the (2+1)-dimensional simplified generalized Broer-Kaup (GBK) system (B=0). A class of double periodic wave solutions is obtained by selecting these arbitrary functions appropriately. The interaction properties of the periodic waves are numerically studied and found to be non-elastic. Limit cases are considered and some new localized coherent structures are obtained, the interaction properties of these solutions reveal that some of them are completely elastic and some are non completely elastic. After that, starting from the (2+1)-dimensional GBK system ( 0 B ≠ ) and using the variable separation approach (VSA) including two arbitrary functions in the general solution, we have constructed by selecting the two arbitrary functions appropriately a rich variety of new coherent structures. The interaction properties of these structures reveal new physical properties like fusion, fission, or both and present mutual annihilation of these solutions as time increasing. The annihilation in this model has found to be rule by the parameter 1 K , when this parameter is taken to be zero, the annihilation disappears in this model and the above mentioned structures recover the solitonic structure properties. INTRODUCTION Early in the study of soliton theory, the main interests of scientists were restricted to the (1+1)-dimensional cases because of the difficulty of finding the physically significant high-dimensional solutions which are localized in all directions. Today, it is found that the study of the (1+1)-dimensional solitons is reaching a certain stage of maturity, the soliton structures and properties of the (1+1)-dimensional integrable nonlinear partial differential equations (NLPDEs) have been now understood very well. However, the study of higherdimensional soliton effects is still in its infancy, the soliton structures in higher spatial dimensions continue to be much more intricate and to attract much more attention. To find some accurate localized coherent soliton solutions in case of (2+1)-dimensions is very important and physically significant because many (2+1)-dimensional NLPDEs are obtained from the physical systems. Thus, other important problem in nonlinear science is to extend lower dimensional integrable models to higher dimensional ones which are very close to the real natural phenomena without lost their integrability. In Ref. [1], Zhang and co-workers investigated the (1+1)-dimensional Broer-Kaup system [2,3] 1 2 1 2 , ( ) , t x x xx t x xx u uu v u v uv v = + − = + (1) Which describes the bi-directional propagation of long waves in shallow water. Then taking into account the Painlevé Property (leading order analysis, resonant points determination and the resonance condition verifications), they derived the following (2+1)-dimensional generalized Broer-Kaup (GBK) system 2 0, 2( ) 4 ( ) 4 ( ) ( 2 ) 0, 0, t xx x x t x xx x xy y yy y y x H H HH U AU BG G GH G A G H B G H C G H U G − − + + + = + + + − + − + − = − = (2) where , , A B C are constant parameters of the system. Zhang et al. discussed the exact explicit solution and propagation in nonlinear model (2), and multi-solitoff solutions, multiple ring type localized solutions, multi-dromion solutions were obtained by the use of the truncated Painlevé expansion and the variable separation approach. However, the variable separation form [(21) in [1]] these authors used to their equation [(20) in [1] ] is very particular, because it is very simplified. It is very natural to ask whether system (2) has other new solutions except those shown in Ref. [1]. The first attempt to answer this question would have been the work by Huang and Zhang [4], they used a new intermediate transformation and a variable-coefficient projective Riccati equation method to solve their so-called new (2+1)-dimensional simplified GBK system, which is a reduced form of (2) when 0 B = . They obtained new families of exact soliton-like solutions, then by imposing some conditions on these families of soliton-like solutions, some new exact solitary waves were given. The same question remained unsolved, since these authors solved a very specify system (2) with 0 B = . On the other hand, in the study of the (2+1)-dimensional models, two kinds of “variable separating” procedure have been established. The first method called the “formal variable separation approach” (FVSA) [5], or equivalently the symmetry constraints or nonlinearization of the Lax-Pairs [6-8]. The independent variables of a reduced field in FVSA have not totally been separated though the reduced field satisfies some lower-dimensional equations. The second type of variable separation method has been established by Lou [9], he has proposed a multi-linear variable separation approach to search for exact solutions of the higher-dimensional especially (2+1)-dimensional NLPDEs. By solving the multi-linear form of these NLPDEs and introducing a prior ansatz, some special types of exact solutions can be obtained from two (1+1)-dimensional variable separated fields. This variable separation approach (VSA) is successfully applied to many (2+1)-dimensional integrable models such as the Davey Stewartson (DS) equation [10], the Nizhnik-Novikov-Veselov (NNV) equation [10], the dispersive long wave equation (DLWE) [11], the Broer-Kaup-Kupershmidt (BKK) system [12], the Burgers equation [13], the Maccari system [14], the general (N+M)components Ablowitz-Kaup-Newell-Segur (AKNS) system [15]. Because the formula of the field includes some arbitrary functions, abundant coherent structures such as, the solitoffs, dromions, lumps, ring-solitons, peakons, compactons, breathers and instantons are given. More recently, Tang and Lou have proposed a more general variable separation approach (GVSA) for several (2+1)-dimensional integrable models [13,16,17]. In this letter, we use the WTC truncation method and VSA (for 0 B ≠ ) or GVSA (for 0 B = ) to present another features of the (2+1)-dimensional GBK system (2). We are interested in the interactions of the coherent structures (solitons). As we know that soliton supplies good applied prospects in many fields of natural science such as plasmas, hydrodynamics, nonlinear optics, fiber optics, solid state physics, and the interactive property of soliton plays an important role in developing of many applications. Therefore, the study of the interactive property of soliton for integrable models is more significance. For this aim, will start as one of us did in two recent papers [18,19], by the interaction properties of Jacobi elliptic waves, then the interaction of solitons are obtained as special case of the Jacobi elliptic waves. The structure of the letter is elucidated as follows. In section 2, a summary of the properties of Jacobi elliptic function and that of the troncature of Painlevé expansion are given. In Section 3, a general solution including two arbitrary functions is obtained for the (2+1)-dimensional simplified GBK system ( 0 B = ) by means of WTC truncation method. By choosing appropriately these arbitrary functions, we study interaction properties of the periodic wave and the one of solitons. Section 4, is devoted to the study of the interaction of some other new localized structures to the (2+1)-dimensional GBK system (2). Some conclusion and discussion are given in the last section. 2. Summary of the properties of Jacobi elliptic functions and that of the troncature of the Painlevé expansion The exact periodic wave solutions in terms of Jacobi elliptic functions for NLPDEs attract considerable interest [20-22] due to the elegant properties of the elliptic functions. The Jacobi elliptic functions ( | ), ( | ) sn sn m cn cn m ξ ξ ξ ξ = = and ( | ), dn dn m ξ ξ = where (0 1) m m ≤ ≤ is the modulus of the elliptic function, are double periodic and possess properties of trigonometric functions namely, 2 2 2 2 2 2 1, 1, ( ) ' , ( ) ' , ( ) ' . sn cn dn m sn sn cn dn cn sn dn dn m cn sn ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ + = + = = = − = − When 0, m → the Jacobi elliptic functions degenerate into the trigonometric functions, i.e. sin , cos , 1. sn cn dn ξ ξ ξ ξ ξ → → → When 1, m → the Jacobi elliptic functions degenerate into the hyperbolic functions, i.e. tanh , sech , sech . sn cn dn ξ ξ ξ ξ ξ ξ → → → Detailed explanations about Jacobi elliptic functions can be found in Refs. [23,24]. The other nine Jacobi elliptic functions are all expressible in terms of , sn cn ξ ξ and dnξ [23]. To conveniently solve the system (2), differentiating the first equation of this system with respect to y once and substituting the third equation of (2) into the resulting equation, we obtain the simplified form of(2) as follows ( 2 ) 0, 2( ) 4 ( ) 4 ( ) ( 2 ) 0. t xx x y xx x y t x xx x xy y y y H H HH G AG BG G GH G A G H B G H C G H − + + + + = + + + − + − + − = (3) It is interesting to see that let 2 y G H = , Eq.(3) is decoupled into 2( ) 2 2 0. ty x y xxy xy yy H HH H AH BH + + + + = (4) Then following the idea of WTC [25], we begin with the Painlevé expansion of Eq.(4) truncated at the constant term 1 0 1, H H H φ − = + (5) where ( , , ) x y t φ φ ≡ is the singular manifold variable, 0 H and 1 H the functions of arguments x, y and t. The substitution of (5) into (4) leads to five equations depending of the different powers of φ . At the powers 4 φ − and 0 φ , the following results 0 , x H φ = (6) 1 1 1 1 1 1 ( 2 2 2 ) 0, t x xx x y y H H H H AH BH + + + + = (7) are obtained respectively. By choosing 1 1( , ) H H x t ≡ , Eq. (7) is satisfied automatically. After that, using (6) into the three remaining equations, the following results are obtained