Resonantly interacting, weakly nonlinear hyperbolic waves.II. Several space variables
Resonantly interacting, weakly nonlinear hyperbolic waves.II. Several space variables
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DOI：
10.1002/sapm1986753187
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发表时间：
1986-12
影响因子：
2.7
通讯作者：
J. K. Hunter;A. Majda;R. Rosales
中科院分区：
文献类型：
--
作者：
J. K. Hunter;A. Majda;R. Rosales
我们提出了一种系统渐近理论，用于在单个空间变量中共振相互作用的弱非线性双曲波。该理论包括 J. Hunter 和 J. B. Keller 最近开发的一般双曲系统的非共振相互作用波理论，该理论专门针对单个空间变量。然而，我们也能够处理双曲系统中发生共振时的一般情况。当双曲系统具有至少三个方程并且例如规定小幅度周期性初始数据时，这种共振是典型情况。在描述单个空间变量中可压缩流体流动的 3 × 3 系统的重要物理示例中，作者开发的共振渐近理论产生了一对无粘性 Burgers 方程作为极限方程，它们通过线性积分算子耦合，已知核通过熵波的初始数据定义。 （在一般情况下，我们给出了许多新的条件，保证给定的双曲系统在规定的初始数据下不共振，以及其他新的结构条件，这意味着共振发生。）作者将在其他地方开发一种处理多个空间变量中共振相互作用波的方法及其应用。
We present a systematic asymptotic theory for resonantly interacting weakly nonlinear hyperbolic waves in a single space variable. This theory includes as a special case the theory of nonresonant interacting waves for general hyperbolic systems developed recently by J. Hunter and J. B. Keller, when specialized to a single space variable. However, we are also able to treat the general situation when resonances occur in the hyperbolic system. Such resonances are the typical case when the hyperbolic system has at least three equations and when, for example, small-amplitude periodic initial data are prescribed. In the important physical example of the 3 × 3 system describing compressible fluid flow in a single space variable, the resonant asymptotic theory developed by the authors yields, as limit equations, a pair of inviscid Burgers equations coupled through a linear integral operator with known kernel defined through the initial data for the entropy wave. (In the general case we give many new conditions guaranteeing nonresonance for a given hyperbolic system with prescribed initial data, as well as other new structural conditions which imply that resonance occurs.) A method for treating resonantly interacting waves in several space variables, together with applications, will be developed by the authors elsewhere.