Initial-Boundary Value Problems for Linear and Soliton PDEs
  Evolution PDEs for dispersive waves are considered in both linear and nonlinear integrable cases, and initial-boundary value problems associated with them are formulated in spectral space. A method of solution is presented, which is based on the elimination of the unknown boundary values by proper restrictions of the functional space and of the spectral variable complex domain. Illustrative examples include the linear Schroedinger equation on compact and semicompact n-dimensional domains and the nonlinear Schroedinger equation on the semiline. 
A. Degasperis, S. V. Manakov and P. M. Santini