Tham khảo tài liệu 'nonlinear dynamics part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Exact Nonlinear Dynamics in Spinor Bose-Einstein Condensates 43 Here 1 and 1 2 take their values in 2 X 2 matrices. The complex number k is the spectral parameter. I is the 2 X 2 unit matrix. The 2 X 2 matrix Q plays a role as a potential function in this linear system. According to Tsuchida Wadati 1998 N-soliton solution of eq. 72 with eq. 73 is expressed as Q x f n--n S 1 rhe 1 n e 2 _ nNe N . 75 where the 2N X 2N matrix S is given by 76 Here we have introduced the following parameterizations 77 Xj Xj x t kjX ikjt - Cj 78 The 2 X 2 matrices Hj normalized to unity in a sense of the square norm l H 2 ựz2a p I 1 1 2 I 7 2 _ 1 79 must take the same form as Q from their definition. We call them polarization matrices which determine both the populations of three components 1 0 -1 within each soliton and the relative phases between them. The complex constants kj denote discrete eigenvalues each of which determines a bound state by the potential Q. Cj are real constants which can be used to tune the initial displacements of solitons. It is worth noting that all x and t dependence is only through the variables Xj x t . As we shall see in Sec. 6 the real part of Xj x t represents the coordinate for observing soliton-j s envelope while the imaginary part of it represents the coordinate for observing soliton-j s carrier waves. The same procedure can be performed for nonvanishing boundary conditions Ieda et al. 2007 which is relevant to formation of spinor dark solitons Uchiyama et al. 2006 . Equation 72 is a completely integrable system whose initial value problems can be solved via for example the ISM Tsuchida Wadati 1998 leda et al. 2007 . The existence of the r-matrix for this system guarantees the existence of an infinite number of conservation laws which restrict the dynamics of the system in an essential way. Here we show explicit forms of some conserved quantities . total number total spin magnetization total momentum and total energy. total number Ny Ị dxn x t 80
