Spatially periodic initial value problem for completely integrable partial differential equations
Loading...
Authors
Wolski, Zachery
Issue Date
2025-12
Type
Electronic thesis
Thesis
Thesis
Language
en_US
Keywords
Mathematics
Alternative Title
Abstract
This thesis addresses the periodic problem of the focusing nonlinear Schrödinger equation and the derivative nonlinear Schrödinger equation. It is split into two parts as follows: In part one the inverse spectral theory for a non-self-adjoint one-dimensional Dirac operator associated periodic potentials via a Riemann-Hilbert problem approach is formulated. The resulting formalism is used to solve the initial value problem for the focusing nonlinear Schrödinger equation. A uniqueness theorem for the solutions of the Riemann-Hilbert problem is established, which provides a new method for obtaining the potential from the spectral data. The formalism applies for both finite- and infinite-genus potentials. As in the defocusing case, the formalism shows that only a single set of Dirichlet eigenvalues is needed in order to uniquely reconstruct the potential of the Dirac operator and the corresponding solution of the focusing NLS equation.
In part two the semiclassical limit of the derivative nonlinear Schrodinger equation with periodic initial conditions is studied analytically and numerically. The spectrum of the associated scattering problem for a certain class of initial conditions, referred to as periodic single lobe potentials, is numerically computed, and it is shown that the spectrum becomes confined to the real and imaginary axes or the spectral parameter in the semiclassical limit. A formal Wentzel-Kramers-Brillouin expansion is computed for the scattering eigenfunctions, which allows one to obtain asymptotic expressions for the number, location and size of the spectral bands and gaps. The results of these calculations suggest that, in the semiclassical limit, all excitations in the spectrum becomes asymptotic solitons. Finally, the analytical results are shown
to agree very well with those obtained from direct numerical computations.
Description
December2025
School of Science
School of Science
Full Citation
Publisher
Rensselaer Polytechnic Institute, Troy, NY