非线性演化下的波包保持
Wavepacket preservation under nonlinear evolution
摘要：
我们研究非线性系统的双曲（在更广泛的意义上）PDE的在整个d维空间描述波的传播与初始数据的形式的一个有限的总和波包称为多波包。该问题涉及两个小参数β和ρ，其中：（i）（1/β）是描述波包空间扩展的因子;（ii）（1/ρ）是描述发展方程的线性部分与其非线性部分相比的相对大小的因子。对于一个广泛的小参数和时间间隔足够长的强非线性效应，我们证明了多波包下的非线性演化。特别地，所涉及的波包的相应波矢量和带数是“守恒量”。我们还证明了一个多波包的演化是由一个适当构造的系统的包络方程与普遍的非线性，在简单的情况下变成众所周知的非线性薛定谔或耦合模式方程的高精度描述。通过对原始非线性进行一定的时间平均来获得通用非线性。这可以看作是对有限维非线性振动系统的平均方法的推广，推广到线性部分具有连续谱的一般平移不变偏微分方程系统.
英文摘要：
We study nonlinear systems of hyperbolic (in a wider sense) PDE's in entire d-dimensional space describing wave propagation with the initial data in the form of a finite sum of wavepackets referred to as multi-wavepackets. The problem involves two small parameters beta and rho where: (i) (1/beta) is a factor describing spatial extension of the wavepackets; (ii) (1/rho) is a factor describing the relative magnitude of the linear part of the evolution equation compared to its nonlinearity. For a wide range of the small parameters and on time intervals long enough for strong nonlinear effects we prove that multi-wavepackets are preserved under the nonlinear evolution. In particular, the corresponding wave vectors and the band numbers of involved wavepackets are "conserved quantities". We also prove that the evolution of a multi-wavepacket is described with high accuracy by a properly constructed system of envelope equations with a universal nonlinearity which in simpler cases turn into well-known Nonlinear Schrodinger or coupled modes equations. The universal nonlinearity is obtained by a certain time averaging applied to the original nonlinearity. This can be viewed as an extension of the well known averaging method developed for finite-dimensional nonlinear oscillatory systems to the case of a general translation invariant PDE systems with the linear part having continuous spectrum.
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