Related papers: A Short Primer on the Half-Wave Maps Equation

We consider the half-wave maps equation $$ \partial_t \vec{S} = \vec{S} \wedge |\nabla| \vec{S}, $$ where $\vec{S}= \vec{S}(t,x)$ takes values on the two-dimensional unit sphere $\mathbb{S}^2$ and $x \in \mathbb{R}$ (real line case) or $x…

We consider the half-wave maps equation $$ \partial_t \mathbf{u} = \mathbf{u} \times |D| \mathbf{u} $$ for $\mathbf{u} : \mathbb{R} \times \mathbb{T} \to \mathbb{S}^2$, where $\mathbb{T}=\mathbb{R}/2 \pi \mathbb{Z}$ is the one-dimensional…

Half-wave maps appear in the physics literature as the continuum limit of Calogero-Moser spin systems. We obtain a uniqueness result for the Half-Wave Maps equation in dimension $d \ge 3$ in the natural energy class with $\mathbb{H}^2$…

In this article we initiate the study of 1+ 2 dimensional wave maps on a curved spacetime in the low regularity setting. Our main result asserts that in this context the wave maps equation is locally well-posed at almost critical…

In this paper, we study the solitary wave and the Cauchy problem for Half-wave-Schr\"{o}dinger equations in the plane. First, we show the existence and orbital stability of the ground states. Secondly, we prove that traveling waves exist…

The half-wave maps equation is a nonlocal geometric equation arising in the continuum dynamics of Haldane-Shashtry and Calogero-Moser spin systems. In high dimensions $n\geq4$, global wellposedness for data which is small in the critical…

We formulate the half-wave maps problem with target $S^2$ and prove global regularity in sufficiently high spatial dimensions for a class of small critical data in Besov spaces.

We consider the energy-critical half-wave maps equation $$\partial_t \mathbf{u} + \mathbf{u} \wedge |\nabla| \mathbf{u} = 0$$ for $\mathbf{u} : [0,T) \times \mathbb{R} \to \mathbb{S}^2$. We give a complete classification of all traveling…

We study the energy-critical half-wave maps equation: \[ \partial_t \mathbf{u} = \mathbf{u} \times |D| \mathbf{u} \] for $\mathbf{u} : [0, T) \times \mathbb{R} \to \mathbb{S}^2$. Our main result establishes the global existence and…

In this article, we study the well-posedness of the energy-critical half-wave maps equation (HWM) in dimension $1$. The half-wave maps equation emerges from the continuum limit of the Haldane Shastry spin chains and has been shown to arise…

We establish the existence of weak global solutions of the half-wave maps equation with the target $S^2$ on $\mathbb{R}^{1+1}$ with large initial data in $\dot{H}^1 \cap \dot{H}^{\frac{1}{2}}(\mathbb{R})$. We first prove the global…

We prove that the energy-critical half-wave maps equation \[ \partial_t \mathbf{S} =\mathbf{S} \times |\nabla| \mathbf{S}, \quad (t,x) \in \mathbb{R} \times \mathbb{T} \] arises as an effective equation in the continuum limit of completely…

We consider the energy supercritical wave maps from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave…

We analyze the size limits of coupled map lattices with diffusive coupling at the crossover of low-dimensional to high-dimensional chaos. We investigate the existence of standing-wave-type periodic patterns, within the low-dimensional…

We prove uniqueness of solutions to the wave map equation in the natural class, namely $ (u, \partial_t u) \in C([0,T); \dot{H}^{d/2})\times C^1([0,T); \dot{H}^{d/2-1})$ in dimensions $d\geq 4$. This is achieved through estimating the…

We prove that the half-wave maps problem on $\mathbb{R}^{4+1}$ with target $S^2$ is globally well-posed for smooth initial data which are small in the critical $l^1$ based Besov space. This is a formal analogue of the result for wave maps…

In this paper we introduce the channel of energy argument to the study of energy critical wave maps into the sphere. More precisely, we prove a channel of energy type inequality for small energy wave maps, and as an application we show that…

Extending an argument by Shatah and Struwe we obtain uniqueness for solutions of the half-wave map equation in dimension $d \geq 3$ in the natural energy class.

We investigate the Cauchy problem for the half wave Schr\"odinger equation in the energy space. We derive the local well-posedness in the energy space for the odd power type nonlinearities under certain additional assumption for the initial…

This paper focuses on the study of semilinear fractional diffusion-wave equations in the context of critical nonlinearities. Firstly, we address the issue of local well-posedness for the problem, examine spatial regularity, and the…