Initial-boundary value problems for discrete evolution equations: discrete linear Schrodinger and integrable discrete nonlinear Schrodinger equations

Details Initial-boundary value problems for discrete evolution equations: discrete linear Schrodinger and integrable discrete nonlinear Schrodinger equations Initial-boundary value problems for discrete evolution equations: discrete linear Schrodinger and integrable discrete nonlinear Schrodinger equations

2008-10-07

arxiv.org/abs/0810.1300v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

41 pages, 3 figures, to appear in Inverse Problems

Scientific paper

10.1088/0266-5611/24/6/065011

We present a method to solve initial-boundary value problems for linear and integrable nonlinear differential-difference evolution equations. The method is the discrete version of the one developed by A. S. Fokas to solve initial-boundary value problems for linear and integrable nonlinear partial differential equations via an extension of the inverse scattering transform. The method takes advantage of the Lax pair formulation for both linear and nonlinear equations, and is based on the simultaneous spectral analysis of both parts of the Lax pair. A key role is also played by the global algebraic relation that couples all known and unknown boundary values. Even though additional technical complications arise in discrete problems compared to continuum ones, we show that a similar approach can also solve initial-boundary value problems for linear and integrable nonlinear differential-difference equations. We demonstrate the method by solving initial-boundary value problems for the discrete analogue of both the linear and the nonlinear Schrodinger equations, comparing the solution to those of the corresponding continuum problems. In the linear case we also explicitly discuss Robin-type boundary conditions not solvable by Fourier series. In the nonlinear case we also identify the linearizable boundary conditions, we discuss the elimination of the unknown boundary datum, we obtain explicitly the linear and continuum limit of the solution, and we write down the soliton solutions.

Affiliated with

Also associated with

No associations

Rating

Initial-boundary value problems for discrete evolution equations: discrete linear Schrodinger and integrable discrete nonlinear Schrodinger equations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Initial-boundary value problems for discrete evolution equations: discrete linear Schrodinger and integrable discrete nonlinear Schrodinger equations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Initial-boundary value problems for discrete evolution equations: discrete linear Schrodinger and integrable discrete nonlinear Schrodinger equations will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-29185