Bounds for the nonlinear Schrödinger approximation of the Fermi–Pasta–Ulam system
Bounds for the nonlinear Schrödinger approximation of the Fermi–Pasta–Ulam system
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Fermi-Pasta-Ulam 系统的非线性薛定谔近似的界限
DOI：
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发表时间：
2010
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通讯作者：
G. Schneider
中科院分区：
文献类型：
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作者：
G. Schneider
我们证明了Fermi-Pasta-Ulam （FPU）系统中底层振荡波包缓慢变化的小振幅包络的演化可以用非线性Schrödinger方程近似描述。与其他晶格方程相比，该问题已在现有文献中得到解决，由于特征值曲线在波数k = 0处消失，FPU系统具有非平凡二次共振。误差估计的证明是基于范式变换和误差函数的波数相关标度。
We prove that the evolution of a slowly varying envelope of small amplitude of an underlying oscillating wave packet in the Fermi–Pasta–Ulam (FPU) system can be described approximately by the nonlinear Schrödinger equation. In contrast to other lattice equations for which this question has been addressed in the existing literature, the FPU system possesses a nontrivial quadratic resonance due to the curves of eigenvalues which vanish at the wave number k = 0. The proof of the error estimates is based on normal form transforms and a wave number-dependent scaling of the error function.