arXiv subjects
Publications and source records attributed to Youde Wang.
By a perturbation approach, we construct geometric solitons with various vortex structures(vortex pairs, vortex rings) for some geometric flows(Wave maps, Shrödinger flows) from Minkowski spaces to ${\mathbb S}^2\subset\R^3$.
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 Schrödinger flows and KdV flows for maps. Moreover, we give several examples of geometric solitons of the Schrödinger flow and geometric KdV flow, including magnetic curves as geometric Schrödinger solitons and explicit geometric KdV solitons on surfaces of revolution.
We define a class of geometric flows on a complete Kähler manifold to unify some physical and mechanical models such as the motion equations of vortex filament, complex-valued mKdV equations, derivative nonlinear Schrödinger equations etc. Furthermore, we consider the existence for these flows from $S^1$ into a complete Kähler manifold and prove some local and global existence results.
In this paper, we introduce a new notion named as Schrödinger soliton. So-called Schrödinger solitons are defined as a class of special solutions to the Schrödinger flow equation from a Riemannian manifold or a Lorentzian manifold $M$ into a Kähler manifold $N$. If the target manifold $N$ admits a Killing potential, then the Schrödinger soliton is just a harmonic map with potential from $M$ into $N$. Especially, if the domain manifold is a Lorentzian manifold, the Schrödinger soliton is a wave map with potential into $N$. Then we apply the geometric energy method to this wave map system, and obtain the local well-posedness of the corresponding Cauchy problem as well as global existence in 1+1 dimension. As an application, we obtain the existence of Schrödinger soliton of the hyperbolic Ishimori system.
We will give a weak energy identity for Sacks-Uhlenbeck approximation of harmonic maps and calculate the length of the necks.
Let $f$ be a positive smooth function on a close Riemann surface (M,g). The $f-energy$ of a map $u$ from $M$ to a Riemannian manifold $(N,h)$ is defined as $$E_f(u)=\int_Mf|\nabla u|^2dV_g.$$ In this paper, we will study the blow-up properties of Palais-Smale sequences for $E_f$. We will show that, if a Palais-Smale sequence is not compact, then it must blows up at some critical points of $f$. As a sequence, if an inhomogeneous Landau-Lifshitz system, i.e. a solution of $$u_t=u\timesτ_f(u)+τ_f(u),\s u:M\to S^2$$ blows up at time $\infty$, then the blow-up points must be the critical points of $f$.