Geometric solitons of Hamiltonian flows on manifolds

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.Comment: 20 pages, 3 figure

Magnetic curves in Sasakian manifolds

“…Further on, Barros et al in [9] showed that these curves are solitons of the localized induction equation (LIE). A more global and geometric view on the connection between solitons of LIE and magnetic curves are provided in [29].…”

Vortex structures for some geometric flows from pseudo-Euclidean spaces

“…Recently, from the viewpoint of differential geometry C. Song, X. Sun and Y. Wang [15] propose a notion "geometric soliton", which is concerning special solutions for some geometric flows and can be regarded as a geometric generalization of the classical solitary wave solutions. In the present paper, we also use the similar method to study (1)- (4).…”

“…We have transformed all problems to two-dimensional elliptic cases which are suitable variances of (15). Furthermore, we can choose real constants c i 's and ω i 's such that µ is a small positive constant, which gives a small parameter ε > 0 by the relation µ = ε or µ = ε| log ε|.…”

Harmonic maps with potential from $\mathbb{R}^2$ into $S^2$

“…For details we refer to [4]. Such a special class of periodic solutions can also be called as "geometric solitons" in [11,12]. For the case M ≡ R 2 and H(u) ≡ G(d(u)), where d(u) denotes the geodesic distance from u ∈ S 2 to the north pole P = (0, 0, 1), we would like to consider the equivariant solutions to (6.1) written by u(x, t) = (sin h(r) cos(mθ + wt), sin h(r) sin(mθ + wt), cos h(r)), (…”