Lax pair, Bäcklund transformation and multi-soliton solutions for the Boussinesq–Burgers equations from shallow water waves

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Abstract

Under investigation in this paper is the set of the Boussinesq–Burgers (BB) equations, which can be used to describe the propagation of shallow water waves. Based on the binary Bell polynomials, Hirota method and symbolic computation, the bilinear form and soliton solutions for the BB equations are derived. Bäcklund transformations (BTs) in both the binary-Bell-polynomial and bilinear forms are obtained. Through the BT in the binary-Bell-polynomial form, a type of solutions and Lax pair for the BB equations are presented as well. Propagation characteristics and interaction behaviors of the solitons are discussed through the graphical analysis. Shock wave and bell-shape solitons are respectively obtained for the horizontal velocity field u and height v of the water surface. In both the head-on and overtaking collisions, the shock waves for the u profile change their shapes, which denotes that the collisions for the u profile are inelastic. However, the collisions for the v profile are proved to be elastic through the asymptotic analysis. Our results might have some potential applications for the harbor and coastal design.

Introduction

With the development of nonlinear science, nonlinear evolution equations (NLEEs) have been used as the models to describe some physical phenomena in fluid mechanics, plasma waves, solid state physics, chemical physics, etc. [1], [2], [3], [4]. In order to understand the mechanisms of those physical phenomena, it is necessary to explore their solutions and properties [5], [6], [7], [8], [9], [10]. Solutions for the NLEEs can not only describe the designated problems, but also give more insights on the physical aspects of the problems in the related fields [11], [12], [13], [14]. For example, the nonlinear wave phenomena observed in fluid dynamics, plasma and optical fibers can be illustrated by the bell-shape (sech profile) and kink-shape (tanh profile) solutions [15]. Methods to derive the solutions for the NLEEs have been proposed, such as the inverse scattering transformation (IST) method [16], [17], Hirota method [18], [19], Darboux transformation (DT) [20], Bäcklund transformation (BT) [21] and algebra-geometric method [22], [23]. Among them, the Hirota method is a direct approach for deriving the soliton solutions through the dependent-variable transformation and formal parameter expansion [24], [25], [26]. Besides, this method is also helpful to investigate the integrable properties of the NLEEs, e.g., the BT and Lax pair [18], [27]. Key step for the Hirota method is to derive the bilinear form through the proper dependent-variable transformations [28], [29], [30].

In this paper, the Boussinesq–Burgers (BB) equations [15], [31], [32], [33], [34], [35], [36], [37], [38],ut=-2uux+12vx,vt=12uxxx-2(uv)x,which describe the propagation of shallow water waves, will be considered, where x and t respectively represent the normalized space and time, the subscripts denote the derivatives, u(x, t) is the horizontal velocity field (at the leading order it is the depth-averaged horizontal field) and v(x, t) denotes the height of the water surface above a horizontal bottom. Via the gauge transformation of the spectral problem, DT with multi-parameter for Eqs. (1a), (1b) has been derived [31]. IST integrability for Eqs. (1a), (1b) has been investigated [32]. The multi-phase periodic solutions for Eqs. (1a), (1b) have been obtained [15], [33], the Whitham theory of modulations has been applied to the problem of the decay of an initial discontinuity [34], [35], and a quasiclassical description of soliton trains arising from a large initial pulse has also been developed [36]. The traveling wave solutions for Eqs. (1a), (1b) have been obtained via the extended homogeneous balance method [37].

However, to our knowledge, the analytic properties such as the soliton solutions, BT and Lax pair for Eqs. (1a), (1b) have not been studied via the binary Bell polynomials. With the help of the binary Bell polynomials [39], [40], [41], Hirota method [18], [19] and symbolic computation [11], [12], [13], [14], this paper will be organized as follows: In Section 2, concepts and formulae about the binary Bell polynomials will be introduced. In Section 3, the bilinear form and multi-soliton solutions for Eqs. (1a), (1b) will be presented. Section 4 will give the BTs in both the binary-Bell-polynomial and bilinear forms. According to the BT in the binary-Bell-polynomial form, a type of solutions and Lax pair, different from that in Ref. [31], will be performed as well. Section 5 will concentrate on two types of interactions of the solitons, i.e., the head-on and overtaking interactions. Conclusions will be addressed in Section 6.

Section snippets

Binary Bell polynomials

With the assumption that w is a C function of x and wn=xnw(x), the Bell polynomials presented in Refs. [39], [40], [41] are as follows:Ynx(w)Yn(wx,,wnx)=e-w(x)xnew(x)(n=1,2,).Similarly, if w = w(x1,  , xn) is a C function with multi-variables, the following polynomials [39], [40], [41]Yn1x1,,nlxl(w)Yn1,,nl(wr1x1,,rlxl)=e-wx1n1xlnlew,are the multi-dimensional Bell polynomials, in which we denote that wr1x1,,rlxl=x1r1xlrlw,(r1=0,,n1;;rl=0,,nl). The binary Bell polynomials take

Binary-Bell-polynomial form

In this section, with the help of the binary Bell polynomials, the bilinear form for Eqs. (1a), (1b) will be given. For Eqs. (1a), (1b), its invariance under the scale transformationsxλx,tλ2t,uλ-1u,vλ-2vshows that u and v have the dimensions −1 and −2, respectively. Therefore, two dimensionless fields p and q can be introduced by setting u = c px and v = d qxx, where p and q are the functions of x and t with c and d as the dimensionless parameters to be determined. The equations for p and q can

BT

The BT [24], [25], [26] provides a way of constructing new solutions from known ones for the soliton equations. In this section, based on Eqs. (10a), (10b), we will obtain the BTs in both the binary-Bell-polynomial and bilinear forms for Eqs. (1a), (1b).

We consider thatQ1=Yt(p,q)+cY2x(p,q)-Yt(p,q)+cY2x(p,q),Q2=4cYxt(p,q)+Y3x(p,q)-4cYxt(p,q)+Y3x(p,q),where (p′, q′) and (p, q) both satisfy Eqs. (10a), (10b). Assuming p = ln (f′/g′) and q = ln(fg′) and introducing the following relations:v1=ln

Analysis and discussions

In this section, Fig. 1, Fig. 2, Fig. 3, Fig. 4 are depicted to describe the propagation characteristics and interaction behaviors of the solitons via Solutions (15a), (15b) and (17a), (17b).

Fig. 1(a) shows the motion of the shock wave for the horizontal velocity field u, and Fig. 1(b) displays the bell-shape soliton for the height v of the water surface. We also observe that the shock wave and bell-shape soliton maintain their velocities and directions during the propagation.

Seen from Fig. 2,

Conclusions

In this paper, with the help of the binary Bell polynomials, Hirota method and symbolic computation, the BB equations [Eqs. (1a), (1b)], which describe the propagation of the shallow water waves, have been investigated. We have obtained the bilinear form, multi-soliton solutions, BT and Lax pair for Eqs. (1a), (1b).

Based on the above results, we conclude that:

  • 1.

    In virtue of the binary Bell polynomials, Binary-Bell-Polynomial Form (10a), (10b) and Bilinear Form (11b) have been obtained for Eqs.

Acknowledgments

We express our sincere thanks to Dr. H. Q. Zhang and other members of our discussion group for their valuable suggestions. This work has been supported by the National Natural Science Foundation of China under Grant No. 60772023, by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02, by the Supported Project (No. SKLSDE-2010ZX-07) and Open Fund (No. SKLSDE-2011KF-03) of the State Key Laboratory of Software Development Environment, Beijing

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