Besley J A, Miller P D, Akhmediev N N
Optical Sciences Centre, Research School of Physical Sciences and Engineering, Australian National University, Canberra, Australian Capital Territory, Australia.
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 2000 Jun;61(6 Pt B):7121-33. doi: 10.1103/physreve.61.7121.
We use multiscale perturbation theory in conjunction with the inverse scattering transform to study the interaction of a number of solitons of the cubic nonlinear Schrödinger equation under the influence of a small correction to the nonlinear potential. We assume that the solitons are all moving with the same velocity at the initial instant; this maximizes the effect each soliton has on the others as a consequence of the perturbation. Over the long time scales that we consider, the soliton amplitudes remain fixed, while their center of mass coordinates obey Newton's equations with a force law for which we present an integral formula. For the interaction of two solitons with a quintic perturbation term we present more details since symmetries - one related to the form of the perturbation and one related to the small number of particles involved - allow the problem to be reduced to a one-dimensional one with a single parameter, an effective mass. The main results include calculations of the binding energy and oscillation frequency of nearby solitons in the stable case when the perturbation is an attractive correction to the potential and of the asymptotic "ejection" velocity in the unstable case. Numerical experiments illustrate the accuracy of the perturbative calculations and indicate their range of validity.
我们运用多尺度微扰理论并结合逆散射变换，来研究在非线性势受到小修正影响的情况下，立方非线性薛定谔方程多个孤子之间的相互作用。我们假设孤子在初始时刻都以相同速度运动；由于微扰，这使得每个孤子对其他孤子的影响最大化。在我们所考虑的长时间尺度上，孤子振幅保持不变，而它们的质心坐标遵循牛顿方程，其力定律我们给出了一个积分公式。对于具有五次微扰项的两个孤子的相互作用，我们给出了更多细节，因为对称性——一个与微扰形式相关，另一个与所涉及的粒子数量少有关——使得问题可以简化为具有单个参数（有效质量）的一维问题。主要结果包括计算当微扰是对势的吸引性修正时稳定情况下附近孤子的结合能和振荡频率，以及不稳定情况下的渐近“射出”速度。数值实验说明了微扰计算的准确性并指出了它们的有效范围。
