Soliton approach to the noisy Burgers equation: Steepest descent method
Soliton approach to the noisy Burgers equation: Steepest descent method
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处理噪声 Burgers 方程的孤子方法：最速下降法
DOI：
10.1103/physreve.57.4943
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发表时间：
1997
影响因子：
2.4
通讯作者：
H. Fogedby
中科院分区：
文献类型：
--
作者：
H. Fogedby
利用函数形式的Martin-Siggia-Rose技术对一维空间中的噪声Burgers方程进行了分析。在正则表达式中，通过在渐近非摄动弱噪声极限下的最小作用原理可以获得其形态和标度行为。得到了局部斜率场和噪声场的耦合鞍点场方程，取代了有噪声的Burgers方程，得到了非线性局域孤子解和扩展的线性扩散模式解，描述了生长界面的形态。规范形式化和最小作用原理也将动量、能量和作用与孤子-扩散模式构型联系起来，从而为噪声引起的波动提供了选择标准。在路径积分的“量子力学”表示中，对应于路径积分中不同路径的噪声涨落被解释为“量子涨落”，而由具有无间隙色散$E\ensuremath{\propto}{P}^{3/2}$的“量子孤子”和具有光谱间隙的“量子扩散模式”的朗道型准粒子气体所代表的生长形态。最后，根据斜率相关性的“量子光谱表示”，从启发式的角度讨论了缩放特性。动态指数$z=3/2$由无间隙孤子色散定律给出，而粗糙度指数$\ensuremath{\zeta}=1/2$由谱表示中形状因子的规则性给出。尺度函数的启发式表达式由谱表示给出，其形式类似于带指数的lsamvy飞行的概率分布 $z.$
The noisy Burgers equation in one spatial dimension is analyzed by means of the Martin-Siggia-Rose technique in functional form. In a canonical formulation the morphology and scaling behavior are accessed by means of a principle of least action in the asymptotic nonperturbative weak noise limit. The ensuing coupled saddle point field equations for the local slope and noise fields, replacing the noisy Burgers equation, are solved yielding nonlinear localized soliton solutions and extended linear diffusive mode solutions, describing the morphology of a growing interface. The canonical formalism and the principle of least action also associate momentum, energy, and action with a soliton-diffusive mode configuration and thus provide a selection criterion for the noise-induced fluctuations. In a ``quantum mechanical'' representation of the path integral the noise fluctuations, corresponding to different paths in the path integral, are interpreted as ``quantum fluctuations'' and the growth morphology represented by a Landau-type quasiparticle gas of ``quantum solitons'' with gapless dispersion $E\ensuremath{\propto}{P}^{3/2}$ and ``quantum diffusive modes'' with a gap in the spectrum. Finally, the scaling properties are discussed from a heuristic point of view in terms of a ``quantum spectral representation'' for the slope correlations. The dynamic exponent $z=3/2$ is given by the gapless soliton dispersion law, whereas the roughness exponent $\ensuremath{\zeta}=1/2$ follows from a regularity property of the form factor in the spectral representation. A heuristic expression for the scaling function is given by a spectral representation and has a form similar to the probability distribution for L\'evy flights with index $z.$