论文标题
calogero- moser衍生物非线性schrödinger方程
The Calogero--Moser Derivative Nonlinear Schrödinger Equation
论文作者
论文摘要
我们研究calogero-moser衍生nls方程$$ i \ partial_t u +\ partial_ {xx} u +（d +| d |）（| u |^2）u = 0 $$在hardy-sobolev space $ h^s $ h^s _ +（\ mathbb {r}）$中，适用于$ s> 0 $ 0 $ s> 0 $ s> 0 $ s> 0。通过为此$ l^2 $ - 临界方程使用宽松对结构，我们证明了$ s \ geq 1 $的全球适应性和带有亚临界或关键$ l^2 $ -mass $ \ |的初始数据。 u_0 \ | _ {l^2}^2 \ leq2π$。此外，我们证明了基础状态的独特性，还将所有行进的孤独波分类。最后，我们详细研究了多索顿解决方案$ u（t）$的类别，我们证明它们以下面的强烈意义表现出能量级联反应，以至于$ \ | u（t）\ | _ {h^s} \ sim_s | t | t | t |^{2s} $ to \ to \ to \ to \ to \ pm \ pm \ for a compp for avery $ s for aunver $ s $ s $ s> 0 $ s> 0 $ s>> 0 $ s>> 0 $ s>> 0 $ s>> 0 $ s>> 0 $ s>> 0 $ s>> 0 $ s>> 0 $。 \ end {摘要}
We study the Calogero--Moser derivative NLS equation $$ i \partial_t u +\partial_{xx} u + (D+|D|)(|u|^2) u =0 $$ posed on the Hardy-Sobolev space $H^s_+(\mathbb{R})$ with suitable $s>0$. By using a Lax pair structure for this $L^2$-critical equation, we prove global well-posedness for $s \geq 1$ and initial data with sub-critical or critical $L^2$-mass $\| u_0 \|_{L^2}^2 \leq 2 π$. Moreover, we prove uniqueness of ground states and also classify all traveling solitary waves. Finally, we study in detail the class of multi-soliton solutions $u(t)$ and we prove that they exhibit energy cascades in the following strong sense such that $\|u(t)\|_{H^s} \sim_s |t|^{2s}$ as $t \to \pm \infty$ for every $s > 0$. \end{abstract}
