论文标题
黑色孤子的轨道稳定性，用于五重总杆虫方程
Orbital stability of the black soliton for the quintic Gross-Pitaevskii equation
论文作者
论文摘要
在这项工作中，获得了一个空间尺寸中的五重总杆虫方程的黑色孤子溶液的轨道稳定性的严格证明。我们首先建立并显示明确的黑色和深色孤子解决方案，并证明，通过使用与黑白孤子扰动有关的一些正交条件，相应的金茨堡 - 兰道能量是强制性的。从隐式函数定理中推导出满足所需正交条件的黑色和深色孤子周围的合适扰动。实际上，这些扰动涉及速度足够小的暗孤子和某些比例因素，从其空间衍生物的显式表达中提出。我们还能够通过估计其时间的增长来控制调制参数沿Quintic Gross-Pitaevskii流的演变。作为一个直接的结果，我们还以较小的速度间隔证明了深色孤子的轨道稳定性。
In this work, a rigorous proof of the orbital stability of the black soliton solution of the quintic Gross-Pitaevskii equation in one spatial dimension is obtained. We first build and show explicitly black and dark soliton solutions and we prove that the corresponding Ginzburg-Landau energy is coercive around them by using some orthogonality conditions related to perturbations of the black and dark solitons. The existence of suitable perturbations around black and dark solitons satisfying the required orthogonality conditions is deduced from an Implicit Function Theorem. In fact, these perturbations involve dark solitons with sufficiently small speeds and some proportionality factors ari\-sing from the explicit expression of their spatial derivative. We are also able to control the evolution of the modulation parameters along the quintic Gross-Pitaevskii flow by estimating their growth in time. As a direct consequence, we also prove the orbital stability of the dark soliton in a small speed interval.
