非线性问题从小变量极限到大变量极限的自相似外推
Self-similar extrapolation of nonlinear problems from small-variable to large-variable limit
摘要：
复杂的物理问题通常借助于微扰理论来解决，从而得到以小参数幂的渐近级数形式的解。然而，有限的，甚至大的参数值往往是主要的物理利益。本文描述了一种方法，它仅从非线性问题的小变量展开式的知识来预测其解的大变量行为。该方法是基于自相似逼近理论的自相似因子逼近。后者可以很好地近似一大类函数，有理函数，无理函数和超越函数。该方法通过统计物理和凝聚态物理的几个例子来说明，其中自相似预测可以与可用的大变量行为进行比较。结果表明，该方法允许在大变量的解决方案的行为时，只知道几个小变量的扩展。证明了逼近式的数值收敛性。
英文摘要：
Complicated physical problems usually are solved by resorting to perturbation theory leading to solutions in the form of asymptotic series in powers of small parameters. However, finite, and even large values of the parameters often are of the main physical interest. A method is described for predicting the large-variable behavior of solutions to nonlinear problems from the knowledge of only their small-variable expansions. The method is based on self-similar approximation theory resulting in self-similar factor approximants. The latter can well approximate a large class of functions, rational, irrational, and transcendental. The method is illustrated by several examples from statistical and condensed matter physics, where the self-similar predictions can be compared with the available large-variable behavior. It is shown that the method allows for finding the behavior of solutions at large variables when knowing just a few terms of small-variable expansions. Numerical convergence of approximants is demonstrated.
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