Geometric solitons of Hamiltonian flows on manifolds

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.