相对论量子力学和场论中的非线性Klein-Gordon方程的Backlund变换

I. Introduction
The nonlinear Klein-Gordon (K-G) equation arises in relativistic quantum mechanics and field theory [1], plus other fields, to model such nonlinear phenomena as the propagation of dislocations in crystals and the behavior of elementary particles, and propagation of fluxons in Josephson junctions [2, 3]. It plays an important role in mathematical physics [4], and has attracted much attention in studying solitons and condensed matter physics [4], in investigating the interaction of solitons in a collisionless plasma, the recurrence of initial states, and in examining the nonlinear wave equations [5]. In recent years, there has been an increasing interest in the study of the nonlinear K-G equation [6]-[12]. For instance, the decomposition method is studied in Ref. [4]; the soliton solutions of coupled nonlinear K-G equations are
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B¨ acklund Transformations for the Nonlinear Klein-Gordon Equation in Relativistic Quantum Mechanics and Field Theory
Xiao-Ge Xu1,2,3 ∗, Yi-Tian Gao2 and Guang-Mei Wei1,2 1. School of Science, Beijing University of Aeronautics and Astronautics, Beijing 100083, China 2. Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China 3. Beijing Information Technology Institute, Beijing 100101,China
II. B¨ acklund transformations and symbolic computations
We investigate the characteristic-coordinate K-G equation φξτ φξτ = F (φ) , = G(φ) , (1) (2)
where F (φ), G(φ) are nonlinear functions. The functions F (φ), G(φ) can take many forms, such as sin φ and sinh φ that characterize the Sine-Gordon equation and Sinh-Gordon equation. These equations appear in many fields such as the propagation of fluxons in Josephson junctions 2
∗
E-mail address for XGX: xxg@biti.edu.cn
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derived in Ref. [9]; the standing waves for the nonlinear K-G equations with nonnegative potentials are concerned in Ref. [11]; the K-G equal scalar and vector potential is considered in Ref. [12]. The nonlinear K-G equation is a nonlinear partial differential equation (PDE). The nonlinear PDEs are encountered in particle physics, plasma and fluid dynamics, statistical mechanics, solid state physics, protein dynamics, laser and fiber optics [1]- [13]. Various effective methods have been developed to solve the nonlinear PDEs like the inverse scattering transformation [14], Darboux transformation [15], B¨ acklund transformation (BT) [16], Hirota’s direct method [17], balancing-act algorithm [18], hyperbolic function expansion method [19], standard and extended truncated Painlev´ e analysis [20], etc. The BT is one of the powerful tools for studying nonlinear PDEs. Different methods have been suggested for the construction of BTs [21]-[24]. The BTs, originated in the study of surfaces of constant negative curvature, are a system of equations relating the solution of a given equation either to another solution of the same equation, or to a solution of another equation. They lead to the construction of an infinite number of conserved quantities and provide exact solutions for the nonlinear equations [25]-[32]. The nonlinear iterative principle from BTs converts the problem of solving nonlinear PDE to purely algebraic calculations [33]-[35]. For this token, there has been considerable interest in the search for the BT of the nonlinear evolution equations in recent years [36]. For example, a BT for the Korteweg-de Vries (KdV) equation and a BT for the potential KdV equation are given in Ref. [37]. A restricted BT is studied in Ref. [38], and the B¨ acklund correspondence is given in Ref. [39]. The gauge transformation interpretation of BTs is given in Ref. [40], and several examples for this interpretation are worked out in Ref. [41]. The BTs for the Sawada-Kotera (SK) and Kaup-Kupershmidt (KK) equations are constructed in Ref. [25]. Despite these advances, several interesting questions remain to be investigated. In this paper, we give several different general forms of BTs of the nonlinear K-G equation according to the different conditions. As special cases of our results, we obtain an auto-BT of the Sine-Gordon equation φξτ = sin φ, an auto-BT of the Sinh-Gordon equation φξτ = sinh φ and a BT from the Liouville equation φξτ = eφ to the linear wave equation φξτ = 0, etc. These results are the same as the previously published results.
Abstract The nonlinear Klein-Gordon equation which arises from relativistic quantum mechanics and field theory can model many nonlinear phenomena and plays an important role in mathematical physics. The B¨ acklund transformations are powerful tools for studying nonlinear partial differential equations. In this paper, we obtain the general forms of B¨ acklund transformations of the nonlinear Klein-Gordon equation with the corresponding conditions. Our studies could be applicable to some other classes of nonlinear partial differential equations. PACS numbers: 05.45.Yv; 02.30.Jr; 03.65.Pm