Shifted nonlocal integrable equations and their soliton solutions

Abstract

One of the approaches to construct integrable equations is using consistent nonlocal reductions. In this thesis, it is showed that shifted nonlocal reductions are special cases of the shifted discrete scale symmetry transformations through the integrable nonlinear Schrödinger (NLS), modified Korteweg-de Vries (MKdV), and Maccari systems.

All consistent shifted nonlocal reductions for these systems are obtained. For the NLS and MKdV systems, by using nonlocal shifted reductions, shifted nonlocal integrable equations are derived. For the 5-component Maccari system, new integrable two-place and four-place shifted nonlocal Maccari systems and Maccari equations are obtained.

In addition to that one-soliton solutions of the NLS, MKdV, and Maccari systems and their shifted nonlocal reductions are found by Hirota method. Particular examples with corresponding graphs are also given in this thesis.