We consider the following nonlinear Schrödinger equation of derivative type:

If $b=0$, this equation is known as a standard derivative nonlinear Schrödinger equation (DNLS), which is mass-critical and completely integrable. The equation above 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 $u_0\in H^1\left(ℝ\right)$ satisfies the mass condition $\parallel u_0{\parallel }_{L^2}^2<4\pi$, the corresponding solution is global and bounded. In this paper we first establish the mass condition on the equation above for general $b\in ℝ$, which corresponds exactly to the $4\pi$-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 the $4\pi$-mass condition and algebraic solitons.

我们考虑以下导数类型的非线性Schrödinger方程：

如果 $b=0$，该方程被称为标准导数非线性Schrödinger方程（DNLS），它对质量至关重要，并且可以完全积分。上面的方程式可以被视为DNLS的广义方程式，同时保留了质量临界性和哈密顿结构。对于DNLS，已知如果初始数据${ü}_0\in H^{\mathrm{1个}}（ℝ）$ 满足群众条件 $\parallel {ü}_0{\parallel }_{{\mathrm{大号}}^{\mathrm{2个}}}^{\mathrm{2个}}<4\pi$，相应的解决方案是全局的和有界的。在本文中，我们首先根据上述方程式建立质量条件$b\in ℝ$，它完全对应于 $4\pi$DNLS的质量条件，然后从势阱理论的角度对其进行表征。我们看到质量阈值给出了孤子产生的势阱结构的转折点。特别是，我们对DNLS的结果给出了$4\pi$质量条件和代数孤子。