1 Department of Informatics and Mathematical Modeling, Technical University of Denmark
It is generally very difficult to solve nonlinear systems, and such systems often possess chaotic solutions. In the rare event that a system is completely solvable, it is said to integrable. Such systems never have chaotic solutions. Using the Inverse Scattering Transform Method (ISTM) two particular configurations of the Discrete Self-Trapping (DST) system are shown to be completely solvable. One of these systems includes the Toda lattice in a certain limit. An explicit integration is carried through for this Near-Toda lattice. The Near-Toda lattice is then generalized to include singular boundary terms, while at the same time retaining the integrability. When quantizing products of momentum p and position q an ambiguity arises. This is discussed in detail and the need for choosing a particular ordering is shown. The Symmetric Ordering rule, which is equivalent to Weyl's rule, is considered in detail. Explicit formulae for quantizing arbitrary functions of p and q are derived. When the basis functions are chosen as eigenfunctions of the harmonic oscillator, explicit formulae are obtained for the matrix elements of the Hamiltonian. Properties of the solutions to the radially symmetric two-dimensional defocusing Nonlinear Schroedinger (NLS) equation are studied analytically and numerically. It is found that no bound states exist. When the initial condition is a dark ring on a background of finite amplitude, the ring initially shrinks until the curvature effects become dominant, forcing the ring to expand to infinity with constant velocity.