Abstract
One of the basic problems about the inverse scattering transformfor solving a completely integrable nonlinear evolutions equationis to demonstrate that the Jost solutions obtained from theinverse scattering equations of Cauchy integral satisfy the Laxequations. Such a basic problem still exists in the procedure ofderiving the dark soliton solutions of the NLS equation in normaldispersion with non-vanishing boundary conditions through theinverse scattering transform. In this paper, a pair of Jostsolutions with same analytic properties are composed to be a2×2 matrix and then another pair are introduced to be itsright inverse confirmed by the Liouville theorem. As they are both2×2 matrices, the right inverse should be the left inversetoo, based upon which it is not difficult to show that these Jostsolutions satisfy both the first and second Lax equations. As aresult of compatibility condition, the dark soliton solutionsdefinitely satisfy the NLS equation in normal dispersion withnon-vanishing boundary conditions.
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