Nonlocal integrable systems and covariant Hodograph transformations

报告题目:Nonlocal integrable systems and covariant Hodograph transformations

报告摘要:An integrable PT-symmetric system called nonlocal nonlinear Schrödinger (NLS) equation was proposed by Ablowitz and Musslimani [PRL-2013-No.064105]. It turns out that such nonlocal type integrable equations are derived from unreduced 2-component systems by using nonlocal reductions. In this talk we introduce a reduction approach to getting solutions of these nonlocal type integrable equations from the known solutions of those unreduced systems. Examples include semi-discrete NLS equation, and several nonlocal hierarchies with in the AKNS scheme. We also introduce other recent progress from other researchers such as formal transformation between local and nonlocal equations and reduction from matrix systems on half line.

We also introduce covariant hodograph transformations of nonlocal integrable systems. Short pulse (SP) equations serve as examples. First we describe connections between the first member in the AKNS negative hierarchy (AKNS(-1)) and several known integrable physical models, including Pedlosky's finite-amplitude baroclinic wave system, Konno-Oono system and its generalizations, SP equation and its complex and multi-component versions. These systems either are the AKNS(-1) system or can be derived from the system through suitable reductions and hodograph transformations. These connections can be extended to multi-component case. With this preparation we come to nonlocal reductions of the multi-component AKNS(-1) system and short pulse systems. In particular, we present nonlocal hodograph transformations between them.



报告人: 张大军教授(上海大学)