论文标题
衍生物非线性schrödinger方程的潜在良好理论
Potential well theory for the derivative nonlinear Schrödinger equation
论文作者
论文摘要
我们考虑衍生类型的以下非线性schrödinger方程：\ begin {equination} i \ partial_t u +\ partial_x^2 U +i +i | \ in \ mathbb {r}。 \ end {equation}如果$ b = 0 $，则该方程被称为据等效形式的众所周知的衍生化非线性schrödinger方程（DNLS），这是质量关键且完全可集成的。该方程可以被视为DNL的通用方程，同时保留了质量关键和哈密顿结构。对于DNL，众所周知，如果初始数据$ u_0 \ in H^1（\ Mathbb {r}）$满足质量条件$ \ | u_0 \ | _ {l^2}^2 <4π$，相应的解决方案是全局且有限的。在本文中，我们首先在\ mathbb {r} $中的一般$ b \的方程式上建立质量条件，该方程完全对应于DNLS的$4π$ - 质量条件，然后从潜在良好理论的角度来表征它。我们看到质量阈值值给出了孤子产生的潜在井结构的转折点。特别是，我们的DNL结果给出了$4π$质量条件和代数孤子的表征。
We consider the following nonlinear Schrödinger equation of derivative type: \begin{equation}i \partial_t u + \partial_x^2 u +i |u|^{2} \partial_x u +b|u|^4u=0 , \quad (t,x) \in \mathbb{R}\times\mathbb{R}, \ b \in\mathbb{R}. \end{equation} If $b=0$, this equation is known as a gauge equivalent form of well-known derivative nonlinear Schrödinger equation (DNLS), which is mass critical and completely integrable. The equation can be considered as a generalized equation of DNLS while preserving mass criticality and Hamiltonian structure. For DNLS it is known that if the initial data $u_0\in H^1(\mathbb{R})$ satisfies the mass condition $\| u_0\|_{L^2}^2 <4π$, the corresponding solution is global and bounded. In this paper we first establish the mass condition on the equation for general $b\in\mathbb{R}$, which is exactly corresponding to $4π$-mass condition for DNLS, and then characterize it from the viewpoint of potential well theory. We see that the mass threshold value gives the turning point in the structure of potential wells generated by solitons. In particular, our results for DNLS give a characterization of both $4π$-mass condition and algebraic solitons.
