arXiv 0908.02462009-08-03math-phmath.MP

具有对称双势阱的非线性Schrodinger方程的万有临界幂

Universal critical power for nonlinear Schrodinger equations with symmetric double well potential

Andrea Sacchetti

备注：: 8 pages with 2 figures
DOI：: 10.1103/PhysRevLett.103.194101

摘要：

本文考虑具有对称双阱势的非线性薛定谔方程的定态。这些定态可能会分叉的强度的非线性项的增加，我们观察到两个不同的图片取决于非线性功率的值：一个简单的节叉分叉，和一对夫妇的鞍点，不稳定的分支崩溃的逆节叉分叉。本文证明了在半经典极限下，或者当两个势阱之间的势垒足够大时，当非线性功率小于一个临界值（2）时，总是发生第一类分岔;相反，当非线性功率大于这个临界值时，总是观察到第二种情形。值得注意的是，这样一个临界值是一个普适常数，因为它不依赖于双阱势的形状。

英文摘要：

Here we consider stationary states for nonlinear Schrodinger equations with symmetric double well potentials. These stationary states may bifurcate as the strength of the nonlinear term increases and we observe two different pictures depending on the value of the nonlinearity power: a simple pitch-fork bifurcation, and a couple of saddle points which unstable branches collapse in an inverse pitch-fork bifurcation. In this paper we show that in the semiclassical limit, or when the barrier between the two wells is large enough, the first kind of bifurcation always occurs when the nonlinearity power is less than a critical value (2); in contrast, when the nonlinearity power is larger than such a critical value then we always observe the second scenario. The remarkable fact is that such a critical value is an universal constant in the sense that it does not depends on the shape of the double well potential.

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英文摘要：

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