This talk will present a method to solve initial-boundary value problems (IBVPs) for linear and integrable nonlinear discrete evolution equations. The method, which is an extension of the inverse scattering transform, yields an integral representation of the solution based on the simultaneous spectral analysis of the two parts of the Lax pair. For integrable nonlinear equations, the method allows one to characterize the spectral data, identify special classes of boundary conditions that are called linearizable, and to obtain the soliton solutions. The method is also advantageous for linear equations, however, since it can be applied to solve IBVPs for any equation and with complicated boundary conditions. The present method is the discrete analogue of the one developed by A.S. Fokas to solve IBVPs for linear and integrable nonlinear evolution equations [2, 3]. In this talk we will demonstrate the method by solving explicitly the IBVPs for the discrete version of the linear and integrable nonlinear Schrodinger equations [1].
References:
1. G. Biondini and G. Hwang, Initial-boudnary value problems for discrete
evolution equations: discrete linear Schrodinger and integrable discrete
nonlinear Schrodinger equations, to appear in Inv. Probl.
2. A. S. Fokas, On the integrability of certain linear and nonlinear
partial differential equations, J. Math. Phys., 41, (2000), 4188-4237.
3. A. S. Fokas, A. R. Its and L.-Y. Sung, The nonlinear Schrodinger
equation on the half-line, Nonlinearity, 18, (2005), 1771-1822.