论文标题
零能量自相似的解决方案描述了非线性Schrodinger方程中的奇异性形成n = 3
Zero Energy Self-Similar Solutions Describing Singularity Formation In The Nonlinear Schrodinger Equation In Dimension N=3
论文作者
论文摘要
在尺寸n = 3中，立方非线性schrodinger方程的溶液变得奇异，即在空间点它们在有限的时间内吹到无穷大。 1972年，扎哈罗夫（Zakharov）著名地研究了立方非线性Schrodinger方程中有限的时间奇异性形成，是血浆中Langmuir波的空间崩溃的模型，这是宇宙中观察到的最丰富的物质形式。 Zakharov假设（NLS）吹动解决方案是自相似的，并且径向对称，并且可以通过相关的自相似，复杂的普通微分方程（ODE）的溶液来建模奇异性形成。参数a> 0出现在ode中，而因变量q，满足（q（q（0），q'（0））=（q_ {0}，0），其中q（0）> 0。将Zakharov模型放在牢固的数学基础上的根本重要步骤是在n = 3时证明，是否存在值a> 0和q_ {0}> 0，以使Q也满足物理上重要的“零能量”的整体约束。自1972年以来，这仍然是一个空旷的问题。在这里，我们通过证明每个a> 0和q（0）> 0来解决此问题，Q满足`零能源的积分约束。
In dimension N=3 the cubic nonlinear Schrodinger equation has solutions which become singular, i.e. at a spatial point they blow up to infinity in finite time. In 1972 Zakharov famously investigated finite time singularity formation in the cubic nonlinear Schrodinger equation as a model for spatial collapse of Langmuir waves in plasma, the most abundant form of observed matter in the universe. Zakharov assumed that (NLS) blow up of solutions is self-similar and radially symmetric, and that singularity formation can be modeled by a solution of an associated self-similar, complex ordinary differential equation~(ODE). A parameter a>0 appears in the ODE, and the dependent variable, Q, satisfies (Q(0),Q'(0))=(Q_{0},0), where Q(0)>0. A fundamentally important step towards putting the Zakharov model on a firm mathematical footing is to prove, when N=3, whether values a>0 and Q_{0}>0 exist such that Q also satisfies the physically important `zero-energy' integral constraint. Since 1972 this has remained an open problem. Here, we resolve this issue by proving that for every a>0 and Q(0)>0, Q satisfies the the `zero-energy' integral constraint.
