Degasperis-Procesi方程中的峰-反峰相互作用
Peakon-antipeakon interactions in the Degasperis-Procesi Equation
摘要：
峰是非线性波方程的奇异类孤子解，其动力学可以用常微分方程（ODE）来研究。Degasperis-Procesi方程（DP）是可积偏微分方程的一个重要例子，它在峰子区表现出波的破碎，从而将波的破碎解释为粒子的机械碰撞。本文建立了一个研究DP峰子碰撞的一般形式，并以三个碰撞峰子为例进行了详细的研究。结果表明，峰子只能成对碰撞，不允许三重碰撞，碰撞时产生激波峰子。我们还表明，峰子-反峰子对的初始配置与伴随的非自伴边值问题的光谱性质是非平凡相关的。特别是，如果峰子或反峰子在相对于剩余的反峰子或峰子的一侧聚成一束，则频谱是真实的和简单的。即使频谱是在一般复杂的存在性，在任何一个时间方向的动态的整体解决方案是暗示现实的边值问题的频谱。
英文摘要：
Peakons are singular, soliton-like solutions to nonlinear wave equations whose dynamics can be studied using ordinary differential equations (ODEs). The Degasperis-Procesi equation (DP) is an important example of an integrable PDE exhibiting wave breaking in the peakon sector thus affording an interpretation of wave breaking as a mechanical collision of particles. In this paper we set up a general formalism in which to study collisions of DP peakons and apply it, as an illustration, to a detailed study of three colliding peakons. It is shown that peakons can collide only in pairs, no triple collisions are allowed and at the collision a shockpeakon is created. We also show that the initial configuration of peakon-antipeakon pairs is nontrivially correlated with the spectral properties of an accompanying non-selfadjoint boundary value problem. In particular if peakons or antipeakons are bunched up on one side relative to a remaining antipeakon or peakon then the spectrum is real and simple. Even though the spectrum is in general complex the existence of a global solution in either time direction dynamics is shown to imply the reality of the spectrum of the boundary value problem.
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