论文标题
代数孤子对非线性schrödinger方程的稳定性：变异方法
Stability of algebraic solitons for nonlinear Schrödinger equations of derivative type: variational approach
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 | \ mathbb {r}。 \ end {equation}如果$ b = 0 $，则该方程是众所周知的衍生范围非线性schrödinger（DNLS）方程的量规等效形式。 DNLS方程的孤子曲线具有与立方Quintic非线性的一定双功率椭圆方程。方程式中的Quintic非线性仅影响椭圆方程中五重指数前面的系数，因此额外的非线性是自然的，因为保留DNLS方程的孤子曲线的扰动。如果$ b> - \ frac {3} {16} $，则该方程式具有代数孤子和指数衰减的孤子。在本文中，我们通过各种方法研究孤子的稳定性，并证明如果$ b <0 $，包括代数孤子在内的所有孤子在能量空间中都是稳定的。稳定的代数孤子的存在显示了一个有趣的数学示例，因为在相应的双重功率NLS的背景下稳定的代数孤子是不知道的。
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 a gauge equivalent form of well-known derivative nonlinear Schrödinger (DNLS) equation. The soliton profile of the DNLS equation satisfies a certain double power elliptic equation with cubic-quintic nonlinearities. The quintic nonlinearity in the equation only affects the coefficient in front of the quintic term in the elliptic equation, so the additional nonlinearity is natural as a perturbation preserving soliton profiles of the DNLS equation. If $b>-\frac{3}{16}$, the equation has algebraic solitons as well as exponentially decaying solitons. In this paper we study stability properties of solitons by variational approach, and prove that if $b<0$, all solitons including algebraic solitons are stable in the energy space. The existence of stable algebraic solitons shows an interesting mathematical example because stable algebraic solitons are not known in the context of the corresponding double power NLS.
