Clarke Simon, Grimshaw Roger, Miller Peter, Pelinovsky Efim, Talipova Tatiana
Department of Mathematics and Statistics, Monash University, Clayton, Victoria, Australia.
Chaos. 2000 Jun;10(2):383-392. doi: 10.1063/1.166505.
We consider the evolution of an initial disturbance described by the modified Korteweg-de Vries equation with a positive coefficient of the cubic nonlinear term, so that it can support solitons. Our primary aim is to determine the circumstances which can lead to the formation of solitons and/or breathers. We use the associated scattering problem and determine the discrete spectrum, where real eigenvalues describe solitons and complex eigenvalues describe breathers. For analytical convenience we consider various piecewise-constant initial conditions. We show how complex eigenvalues may be generated by bifurcation from either the real axis, or the imaginary axis; in the former case the bifurcation occurs as the unfolding of a double real eigenvalue. A bifurcation from the real axis describes the transition of a soliton pair with opposite polarities into a breather, while the bifurcation from the imaginary axis describes the generation of a breather from the continuous spectrum. Within the class of initial conditions we consider, a disturbance of one polarity, either positive or negative, will only generate solitons, and the number of solitons depends on the total mass. On the other hand, an initial disturbance with both polarities and very small mass will favor the generation of breathers, and the number of breathers then depends on the total energy. Direct numerical simulations of the modified Korteweg-de Vries equation confirms the analytical results, and show in detail the formation of solitons, breathers, and quasistationary coupled soliton pairs. Being based on spectral theory, our analytical results apply to the entire hierarchy of evolution equations connected with the same eigenvalue problem. (c) 2000 American Institute of Physics.
我们考虑由修正的科特韦格 - 德弗里斯方程描述的初始扰动的演化，该方程的三次非线性项系数为正，从而能够支持孤子。我们的主要目的是确定能够导致孤子和/或呼吸子形成的情形。我们利用相关的散射问题并确定离散谱，其中实特征值描述孤子，复特征值描述呼吸子。为了便于分析，我们考虑各种分段常数初始条件。我们展示了复特征值如何通过从实轴或虚轴的分岔产生；在前一种情况下，分岔发生在双实特征值的展开过程中。从实轴的分岔描述了具有相反极性的孤子对向呼吸子的转变，而从虚轴的分岔描述了从连续谱产生呼吸子的过程。在我们考虑的初始条件类别中，单一极性（正或负）的扰动只会产生孤子，孤子的数量取决于总质量。另一方面，具有两种极性且质量非常小的初始扰动有利于呼吸子的产生，此时呼吸子的数量取决于总能量。对修正的科特韦格 - 德弗里斯方程的直接数值模拟证实了分析结果，并详细展示了孤子、呼吸子和准静态耦合孤子对的形成。基于谱理论，我们的分析结果适用于与相同特征值问题相关的整个演化方程层次。（c）2000美国物理学会。
