B(?)cklund Transformations And Structures For The Integrable Discretization Of The NLS～+ Equation

In this paper, we give the Backlund transformations and discuss its structures to some extent for the integrable discretization of the NLS1" equation:The transformation gives a way to construct solutions to the system(1.3), and it also does to the equation(3.1), which contribute to reconstruct solutions to the system(1.3). In this paper, In order to attain the transformation (Theorems. 2), we use a special method, that is, the one of Lax pair. So we first find out the Lax pair of the system(1.3) (Lemma3.l). Through the system(1.3)'s equivalence to the Lax representation, we finish the proof of Theorems.2. Moreover, we give the standardization of the transformation (Theorems. 3, Corollary3.4).As far as the discussion about the structures is concerned, to some extent, it may be said to be an application to the Backlund transformation: Fix a solution qn to the system(l.3), construct a group of solutions Qn different from qn through the Backlund transformation, and then prove that Qn is convergent to qn. In the section 4, it is vital for us to find out the fixed solution n to the equation(3.1) in the Theorem3.2, which is completed in the Lemma4.2 and Theorem4.3.LemmaS.l : The Lax pair of system (1.3) takes the formwhereandin which z = ei h , and system (3.1) gives the "Lax representation" of the discrete NLS+ equation (1.3)Theorems. 2 : (Backlund transformation): Fix a solution fin to the system (3.1) at (qn,z+} , and use (j)n define a 2 x 2 matrix n bywhereandFurther , definethen , if n solves system (3.1) at (qn, z+) , n will solve system (3.1) at (Qn, z+) . Moreover, by the compatability of system (3.1) , both qn and Qn solve the discrete NLSf equation (1.3).Lemma4.2 : Let a eigenvalue and the corresponding eigenfunction located at z of Ln be ( ) , where (p0 = ( )T , then 0 is compatable with eq.(3.1b) , that is , 0 may be the solution to eq.(3.lb) .Theorem4.3 : Let the two eigenvalues and the corresponding eigenfunctions located at z of Ln be ( ) and ( ) , and both 0 and 0 are the solutions to eq.(3.1b) . Furthermore, let , and , where k1,k2 are arbitrary real(complex) numbers , then n is the solution to eq(3.1) .