A Class of Solutions of Initial-boundary Value Problems for the Korteweg-de Vries Equation on the Interval and Bäcklund Transformations

The initial-boundary value problems (IBVPs) for the Korteweg-de Vries (KdV) equation on the interval (0, q), where q is a large positive number, are solved by the Inverse Scattering Method (ISM). By virtue of the appropriate setup of the associated scattering problem (SP), the fundamental equation in this inverse SP (ISP) is reduced to a system of linear algebraic equations. The known function in this equation describes only the discrete spectrum of the SP. The solution of this system completely describes the whole family of recovered potentials y(x, t) in the linear Schrödinger equation (LSEq). By the unique solvability of the ISP, the recovered potential y(x, t) is a solution of the considered IBVP. A class of various scattering data (SD) sets is derived from various pairs of prescribed initial and boundary conditions. A class G of solutions of IBVPs on the interval (0, q) is constructed from the class of various SD sets of SPs. The class of Bäcklund transformations (BTs) linking each solution of the IBVP in the class G with its corresponding common solution of linear equations of the Lax pair is derived.