一类具有平衡损耗-增益的广义矢量非局部非线性薛定谔方程的可解极限
Solvable Limits of a class of generalized Vector Nonlocal Nonlinear Schrödinger equation with balanced loss-gain
AI总结：
本文研究一类具有平衡损耗-增益和线性耦合的矢量非局部非线性薛定谔方程，通过矩的时间演化和非幺正变换等方法给出可解条件与多种精确解。
摘要：
我们考虑一类一维矢量非局部非线性薛定谔方程（VNNLSE），其处于外部复势中，并具有时间调制的平衡损耗-增益（BLG）以及薛定谔场各分量之间的线性耦合（LC），非线性强度依赖于时空坐标。在一定条件下，该系统允许拉格朗日表述和哈密顿表述。研究表明，尽管哈密顿密度为复值，但总功率、$\cal{PT}$-对称哈密顿量、波包宽度及其增长速度等各种动力学变量均为实值。我们研究了具有或不具有哈密顿表述的一般 VNNLSE 的精确可解性。在第一部分中，我们研究了类似于斯托克斯变量空间积分的矩的时间演化，并找到了时间有界解存在的条件。在第二部分中，我们使用一个非幺正变换，随后进行坐标变换，将 VNNLSE 映射到各种可解方程。当非幺正变换退化为赝幺正变换时，该极限情形完全不需要坐标变换。精确解在时间上有界的条件与通过矩的时间演化研究得到的条件相同。文中给出了 VNNLSE 的各种精确解。
英文摘要：
We consider a class of one dimensional Vector Nonlocal Non-linear Schrödinger Equation (VNNLSE) in an external complex potential with time-modulated Balanced Loss-Gain(BLG) and Linear Coupling(LC) among the components of Schrödinger fields, and space-time dependent nonlinear strength. The system admits Lagrangian and Hamiltonian formulations under certain conditions. It is shown that various dynamical variables like total power, $\cal{PT}$-symmetric Hamiltonian, width of the wave-packet and its speed of growth, etc. are real-valued despite the Hamiltonian density being complex-valued. We study the exact solvability of the generic VNNLSE with or without a Hamiltonian formulation. In the first part, we study time-evolution of moments which are analogous to space-integrals of Stokes variables and find condition for existence of solutions which are bounded in time. In the second part, we use a non-unitary transformation followed by a coordinate transformation to map the VNNLSE to various solvable equations. The cordinate transformation is not required at all for the limiting case when non-unitary transformation reduces to pseudo-unitary transformation. The exact solutions are bounded in time for the same condition which is obtained through the study of time-evolution of moments. Various exact solutions of the VNNLSE are presented.
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