Ablowitz-Musslimani非线性薛定谔方程孤子动力学的变分近似
Variational approximations of soliton dynamics in the Ablowitz-Musslimani nonlinear Schrödinger equation
摘要：
研究了由Ablowitz和Musslimani提出的可积非局部非线性薛定谔方程，它被认为是具有宇称-时间（$\mathcal{PT}$）对称自感生势的方程的一个特例.我们考虑运动孤子的动力学（包括碰撞）。在分析上，我们开发了一个集体的协调方法的基础上变分方法，并研究其在系统中的适用性。数值计算表明，单个运动孤子可以通过原点并衰减，也可以在原点被捕获并在有限时间内爆破。使用标准孤子解，变分近似可以很好地捕获动力学，包括有限时间爆破，即使该解与实际爆破孤子解相距较远。在两个孤子相互运动的情况下，我们表明，它们之间可以有一个质量转移，除了波散射。我们还证明了散焦非线性可以支持亮孤子。
英文摘要：
We study the integrable nonlocal nonlinear Schrödinger equation proposed by Ablowitz and Musslimani, that is considered as a particular example of equations with parity-time ($\mathcal{PT}$) symmetric self-induced potential. We consider dynamics (including collisions) of moving solitons. Analytically we develop a collective coordinate approach based on variational methods and examine its applicability in the system. We show numerically that a single moving soliton can pass the origin and decay or be trapped at the origin and blows up at a finite time. Using a standard soliton ansatz, the variational approximation can capture the dynamics well, including the finite-time blow up, even though the ansatz is relatively far from the actual blowing-up soliton solution. In the case of two solitons moving towards each other, we show that there can be a mass transfer between them, in addition to wave scattering. We also demonstrate that defocusing nonlinearity can support bright solitons.
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