Exact analytical solution of a novel modified nonlinear Schrödinger equation: solitary quantum waves on a lattice

A novel modified nonlinear Schr¨odinger equation is presented. Through a travelling wave ansatz, the equation is 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 uniwaveform and bi-waveform regions of parameter space are identified. 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.