On the discrete nonlinear Schrödinger equation with PT-Symmetry

The purpose of this thesis is to develop the inverse scattering method for the nonlocal semi-discrete nonlinear Schrödinger equation (known as Ablowitz-Ladik equation) with parity-time symmetry proposed in Ablowitz and Musslimani’s paper. This includes the eigenfunctions (Jost solutions) of the associated Lax pair, the scattering data and the fundamental analytic solutions. In addition, we study the spectral properties of the associated discrete Lax operator. Based on the formulated Riemann-Hilbert problem, we derive the one- and two-soliton solutions to the nonlocal Ablowitz-Ladik equation. Finally, we prove the completeness relation for the associated Jost solutions. Based on this, we derive the expansion formula over the Jost solutions is evaluated. This allows interpreting the inverse scattering method as a generalised Fourier transform. We derive the dressing method based on the seed solution to the discrete nonlinear Schrödinger equation. Explicit relations are obtained amongst the spectrum problem associated with the expansion over the negative and positive power of the eigenvalues. We show a general formula for the Riemann-Hilbert problem based dressing method in terms of the Lax representation associated with a given nonlinear equation. Next, we study square barrier potentials for the Ablowitz-Ladik like of the discrete nonlinear Schrödinger equation and a certain class of integrable systems of multi-component generalisation of the Manakov model. We are interested in conditions distinguishing blow up and not blow up solutions. From considering single and double excitations as initial conditions, we conjecture the following: 1) if the Lax operator has no spectrum outside nor inside the unit circle, there is no blow up; 2) when it does, mirror symmetric initial conditions are sufficient, but not necessary, for bounded solutions; 3) to obtain bounded solutions, each spectrum outside the unit circle needs (but is not sufficient) to have a reciprocal counterpart on the inside. Numerical method used to evaluate the eigenvalues of the Ablowitz-Ladik problem. Numerical simulations are also presented, illustrating our analytical results.