The nonlinear Schrödinger equation on the half-line

The initial-boundary value problem (ibvp) for the cubic nonlinear Schrödinger (NLS) equation on the half-line with data in Sobolev spaces is analysed via the formula obtained through the unified transform method and a contraction mapping approach. First, the linear Schrödinger (LS) ibvp with initial and boundary data in Sobolev spaces is solved and the basic space and time estimates of the solution are derived. Then, the forced LS ibvp is solved for data in Sobolev spaces on the half-line $[0, \infty )$ for the spatial variable and on an interval $[0, T]$, $0<T<\infty$, for the temporal variable by decomposing it into a free ibvp and a forced ibvp with zero data, and its solution is estimated appropriately. Furthermore, using these estimates, well-posedness of the NLS ibvp with data $(u(x,0), u(0,t))$ in $H_x^s(0,\infty )\times H_t^{(2s+1)/4}(0,T)$, $s>1/2$, is established via a contraction mapping argument. In addition, this work places Fokas' unified transform method for evolution equations into the broader Sobolev spaces framework.