论文标题
具有平衡损失增益的一类广义矢量非局部非线性schrödinger方程的可解决极限
Solvable Limits of a class of generalized Vector Nonlocal Nonlinear Schrödinger equation with balanced loss-gain
论文作者
论文摘要
我们考虑一类一级矢量非局部非线性schrödinger方程（vnnlse）在外部复合电势中，具有时间调节的平衡损失增益（BLG）和线性耦合（LC）和Schrödinger领域的组成部分，以及时空依赖性非线性依赖性非线性强度。该系统在某些条件下承认Lagrangian和Hamiltonian配方。结果表明，尽管汉密尔顿密度复杂得出，但各种动力学变量，例如总功率，$ \ cal {pt} $ - 对称的哈密顿式，波包及其生长速度等。我们研究具有有或没有哈密顿配方的通用VNNLSE的确切可溶性。在第一部分中，我们研究了类似于stokes变量的空间融合的矩的时间进化，并找到了存在时间界定的溶液的条件。在第二部分中，我们使用非自动变换，然后使用坐标转换来将VNNLSE映射到各种可求解方程。对于限制案例，当非自动转化减少为伪自动转化时，根本不需要绳索转换。确切的溶液在相同的条件上及时限制，这是通过研究矩的时间进化而获得的。提出了Vnnlse的各种精确解决方案。
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.
