Nonlinear Transition Layers—The Second Painleve Transcendent
Nonlinear Transition Layers—The Second Painleve Transcendent
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非线性过渡层——第二个Painlevel超越
DOI：
10.1002/sapm1977573247
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发表时间：
1977
影响因子：
2.7
通讯作者：
R. Haberman
中科院分区：
文献类型：
--
作者：
R. Haberman
研究了一大类具有慢变系数的弱非线性二阶常微分方程解的存在性。我们证明了在从振荡行为到指数衰减行为的转变过程中，标准的双时间微扰解是无效的。在所有情况下，非线性过渡层都弥补了这一困难，其前导阶特性由一个特殊的非线性微分方程描述，称为第二Painlevé超越方程(本质上是一个非线性艾里方程)。匹配渐近展开法给出了所需的连接公式。第二个Painlevé超越解还提供了另外两种类型的转换：(1)在弱非线性解(振荡或指数衰减)和特殊的完全非线性解之间，以及(2)在这些特殊的非线性解之间。这些特解有三种：(A)缓慢变化的稳定平衡解，(B)“爆炸”解，以及(C)依赖于快和慢尺度的解(从不稳定的零平衡解出现)。
We investigate a large class of weakly nonlinear second‐order ordinary differential equations with slowly varying coefficients. We show that the standard two‐timing perturbation solution isnotvalid during the transition from oscillatory to exponentially decaying behavior. In all cases this difficulty is remedied by anonlinear transition layer, whose leading‐order character is described by one special nonlinear differential equation known as the second Painlevé transcendent (in essence a nonlinear Airy equation). The method of matched asymptotic expansions yields the desired connection formula. The second Painlevé transcendent also provides two other types of transitions: (1) between weakly nonlinear solutions (either oscillatory or exponentially decaying) and special fully nonlinear solutions, and (2) between two of these special nonlinear solutions. These special solutions are of three: different kinds: (a) slowly varying stable equilibrium solutions, (b) “exploding” solutions, and (c) solutions depending on both the fast and slow scales (which emerge from the unstable zero equilibrium solution).