The Influence of Modulational Instability on Energy Exchange in Coupled Sine-Gordon Equations

We consider a two-component system of coupled sine-Gordon equations, particular solutions of which represent a continuum generalization of periodic energy exchange in a system of coupled pendulums. Weakly nonlinear solutions describing periodic energy exchange between waves traveling in the two components are governed, depending on the length scale of the amplitude variation, either by two nonlocally coupled nonlinear Schrödinger equations, with different transport terms due to the group velocity, or by a model that is nondispersive to the leading order. Using both asymptotic analysis and numerical simulations, we show that the effects of dispersion significantly influence the structure of these solutions, causing modulational instability and the formation of localized structures but preserving the pattern of energy exchange between the components.     

一维Chaplygin气体欧拉方程组中波的相互作用

研究一维Chaplygin气体欧拉方程组中波的相互作用.方程组的波包含接触间断和在密度变量以及内能变量上同时具有狄拉克函数的狄拉克激波.根据这些波的不同组合,问题被分成了7种情形.通过详细地构造每种情形的整体解,获得了各种波相互作用的完整结果.特别地,对于一类初值,两个接触间断相互作用后,产生了一个狄拉克激波;然而,对于另外一类初值,一个狄拉克激波与一个接触间断相互作用后,狄拉克激波消失.这些都是波相互作用中非常特别的现象.