نحن نستعرض الحالة الحالية للنتائج حول معادلة الخرائط النصف الموجية في المجال $mathbb {R} ^ d $ مع الهدف $mathbb {S} ^ 2 $. بالخصوص، نركز على الحالة الحرارية الحرجة $ d = 1 $، حيث نناقش تصنيف الموجات المنفردة السفر وبناء زوج لاكس مع تداعياته (على سبيل المثال، الثبات من الحلول العقلانية والكثير من أنظمة الاحتفاظ على مستوى من المساحات Besov المتساوية). إضافة إلى ذلك، نؤكد أيضًا على الحالة المتزامنة في البعد الأول. وأخيرًا، نضع قائمة ببعض المشاكل المفتوحة للأبحاث المستقبلية.
We review the current state of results about the half-wave maps equation on the domain $mathbb{R}^d$ with target $mathbb{S}^2$. In particular, we focus on the energy-critical case $d=1$, where we discuss the classification of traveling solitary waves and a Lax pair structure together with its implications (e.,g.~invariance of rational solutions and infinitely many conservation laws on a scale of homogeneous Besov spaces). Furthermore, we also comment on the one-dimensional space-periodic case. Finally, we list some open problem for future research.
We consider the half-wave maps equation $$ partial_t vec{S} = vec{S} wedge | abla| vec{S}, $$ 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 $x in mathbb{T}$ (periodi
We consider the energy-critical half-wave maps equation $$partial_t mathbf{u} + mathbf{u} wedge | abla| mathbf{u} = 0$$ for $mathbf{u} : [0,T) times mathbb{R} to mathbb{S}^2$. We give a complete classification of all traveling solitary waves with fin
The KPII equation is an integrable nonlinear PDE in 2+1 dimensions (two spatial and one temporal), which arises in several physical circumstances, including fluid mechanics where it describes waves in shallow water. It provides a multidimensional gen
A new method for the solution of initial-boundary value problems for textit{linear} and textit{integrable nonlinear} evolution PDEs in one spatial dimension was introduced by one of the authors in 1997 cite{F1997}. This approach was subsequently exte
We consider the quartic focusing Half Wave equation (HW) in one space dimension. We show first that that there exist traveling wave solutions with arbitrary small $H^{frac 12}(R)$ norm. This fact shows that small data scattering is not possible for (