一个新的修正非线性薛定谔方程的精确解析解：晶格上的量子孤波
Exact analytical solution of a novel modified nonlinear Schrödinger equation: solitary quantum waves on a lattice
摘要：
提出了一种新的修正的非线性薛定谔方程。通过一个行波解，将方程转化为一个非线性常微分方程，然后精确解析求解。孤立子的解决方案的特点是在波形和波速方面，这些属性的依赖关系，在方程中的参数进行了详细说明。它被发现，一些参数设置产生独特的波形，而其他产量退化，与两个不同的波形，每组参数值。参数空间的单波形和双波形区域被识别。还发现，每个波形具有两种传播模式，具有共享的方向性，但不同的速度。最后，该方程被证明是一个模型的量子力学激子，如电子，通过集体振荡的平面晶格与激子相互作用的传播。孤子解的物理意义进行了讨论。
英文摘要：
A novel modified nonlinear Schrödinger equation is presented. Through a travelling wave ansatz, the equation is transformed into a nonlinear ODE which is then solved exactly and analytically. The soliton solution is characterised in terms of waveform and wave speed, and the dependence of these properties upon parameters in the equation is detailed. It is discovered that some parameter settings yield unique waveforms while others yield degeneracy, with two distinct waveforms per set of parameter values. The uni-waveform and bi-waveform regions of parameter space are identified. It is also found that each waveform has two modes of propagation with shared directionality but distinct speeds. Finally, the equation is shown to be a model for the propagation of a quantum mechanical exciton, such as an electron, through a collectively-oscillating plane lattice with which the exciton interacts. The physical implications of the soliton solution are discussed.
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