Geometric solitons of Hamiltonian flows on manifolds

TL;DR

This paper introduces the concept of geometric solitons for Hamiltonian flows on manifolds, revealing their dependence on domain and target space isometries, and provides explicit examples including magnetic curves and surfaces of revolution.

Contribution

It proposes the new notion of geometric solitons for Hamiltonian flows and demonstrates their properties with concrete examples on various manifolds.

Findings

Geometric solitons are determined by domain and target space isometries.

Examples include magnetic curves as Schrödinger solitons.

Explicit geometric KdV solitons are constructed on surfaces of revolution.

Abstract

It is well-known that the LIE(Locally Induction Equation) admit soliton-type solutions and same soliton solutions arise from different and apparently irrelevant physical models. By comparing the solitons of LIE and Killing magnetic geodesics, we observe that these solitons are essentially decided by two families of isometries of the domain and the target space respectively. With this insight, we propose the new concept of geometric solitons of Hamiltonian flows on manifolds, such as geometric Schr\"odinger flows and KdV flows for maps. Moreover, we give several examples of geometric solitons of the Schr\"odinger flow and geometric KdV flow, including magnetic curves as geometric Schr\"odinger solitons and explicit geometric KdV solitons on surfaces of revolution.