Related papers: On Schr\"odinger maps from $T^1$ to $S^2$

It is shown that the Schrodinger equation can be cast in the form of two coupled real conservation equations, in Euclidean spacetime in the free case and in a five-dimensional Eisenhart geometry in the presence of an external potential.…

We present some general results for the time-dependent mass Hamiltonian problem with H=-{1/2}e^{-2\nu}\partial_{xx} +h^{(2)}(t)e^{2\nu}x^2. This Hamiltonian corresponds to a time-dependent mass (TM) Schr\"odinger equation with the…

The spectral properties of the Schr\"odinger operator $T_ty= -y''+q_ty$ in $L^2(\R)$ are studied, with a potential $q_t(x)=p_1(x), x<0, $ and $q_t(x)=p(x+t), x>0, $ where $p_1, p$ are periodic potentials and $t\in \R$ is a parameter of…

We investigate the canonical Scheidegger Model of river network morphology for the case of convergent and divergent underlying topography, by embedding it on a cone. We find two distinct phases corresponding to few, long basins and many,…

We consider the gradient flow of the Ambrosio-Tortorelli functional at fixed $\epsilon>0$, proving existence, uniqueness and $L^2 _t (H_x ^2) \cap L^\infty _t (H^1 _x) \cap H^1 _t (L^2 _x) $ regularity in dimension 2. In particular we…

Let $M$ be a closed Riemannian manifold with a parallel 1-form $\Omega$. We prove two theorems about the curve shortening flow in $M$. One is that the {\csf} $\ct$ in $M$ exists for all $t$ in $[0, \infty)$, if it satisfies $\Omega(T)\geq…

We study joining rigidity in the class of von Neumann flows with one singularity. They are given by a smooth vector field $\mathcal{X}$ on $\mathbb T^2\setminus \{a\}$, where $\mathcal{X}$ is not defined at $a\in \mathbb T^2$. It follows…

We consider the one-dimensional nonlinear Schr\"odinger equation with a nonlinearity of degree $p>1$. We exhibit measures on the space of initial data for which we describe the non trivial evolution by the linear Schr\"odinger flow and we…

We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics $\sigma_\e$ collapsing to…

We consider It\^o SDE $\d X_t=\sum_{j=1}^m A_j(X_t) \d w_t^j + A_0(X_t) \d t$ on $\R^d$. The diffusion coefficients $A_1,..., A_m$ are supposed to be in the Sobolev space $W_\text{loc}^{1,p} (\R^d)$ with $p>d$, and to have linear growth;…

Suppose $(M^i_t)_{t\in [0,T)}$, $i=1,2$, are two mean curvature flows in $\mathbb{R}^{n+1}$ encountering a multiplicity one compact singularity at time $T$, in such a manner that for every $k$, the Hausdorff distance between the two flows,…

Given any $K>0$, we construct two equivalent $C^2$ flows, one of which has positive topological entropy larger than $K$ and admits zero as the exponential growth of periodic orbits, in contrast, the other has zero topological entropy and…

The two-sphere valued wave map flow on a Lorentzian domain R x Sigma, where Sigma is any flat two-torus, is studied. The Cauchy problem with initial data tangent to the moduli space of holomorphic maps Sigma -> S^2 is considered, in the…

We consider steady solutions to the incompressible Euler equations in a two-dimensional channel with rigid walls. The flow consists of two periodic layers of constant vorticity separated by an unknown interface. Using global bifurcation…

We study the Hessian of the solutions of time-independent Schr\"odinger equations, aiming to obtain as large a class as possible of complete Riemannian manifolds for which the estimate $C(\frac 1 t +\frac {d^2}{t^2})$ holds. For this…

The paper deals with the problem of stability for the flow of the 1D Burgers equation on a circle. Using some ideas from the theory of positivity preserving semigroups, we establish the strong contraction in the $L^1$ norm. As a…

In this paper, we obtain pointwise convergence of solutions to the Schrodinger equation along a class of curves in $\mathbb{R}^{2}$ by the polynomial partitioning.

In this short note, we show a uniqueness result of the energy solutions for the Cauchy problem of Schrodinger flow in the whole space $R^n$ provided there is a smooth solution in the energy class.

We establish several quantitative stability estimates for optimal transport maps between non-degenerate densities on uniformly convex domains for the quadratic cost. Under H\"older regularity assumptions, we prove Lipschitz $L^2$…

In this paper, we show the existence and uniqueness of local regular solutions to the initial-Neumann boundary value problem of Schr\"{o}dinger flow from a smooth bounded domain $\Omega\subset\mathbb{R}^3$ into $\mathbb{S}^2$(namely…