THE BÄCKLUND TRANSFORMATION BETWEEN A COMMON SOLUTION OF BOTH LINEAR EQUATIONS IN THE LAX PAIR AND THE SOLUTION OF THE ASSOCIATED INITIAL-BOUNDARY VALUE PROBLEM FOR NONLINEAR EVOLUTION EQUATIONS ON THE HALF-LINE

Abstract

We present a way to obtain a Bäcklund transformation (BT) from two linear equations on the half-line in the Lax pair. By the unique solvability of the inverse scattering problem, there exists a one-to-one correspondence between the scattering data of the scattering problem and the solution (recovered potential) of the associated initial-boundary value problem (IBVP) for a nonlinear evolution equation (NLEE) on the half-line, and conversely. Further, a common solution is constructed from two overdetermined equations in the Lax pair. Then, a one-to-one correspondence between the common solution and the solution of this IBVP is established, and conversely. Hence, a BT is obtained from these one-to-one correspondences. The constructed BT transforms the common solution of both equations in the Lax pair into the solution (recovered potentials) of the associated IBVP, and conversely. In the way presented above, the BTs for the solutions of IBVPs for NLEEs on the half-line, such as the attractive nonlinear Schrödinger (NLS), modified Korteweg-de Vries (KdV) equations, sine-Gordon and sinh-Gordon equations, and the KdV equation with dominant surface tension are constructed.