Phase properties of several nonlinear optical waves described by rational solutions

The complete set of eigenstates in one type of N-multiple quantum wells

“…Namely, the phase jump at one node is ±π, where ± is determined by the approach directions. Therefore each node of the eigenstate corresponds to the merging of a pair of monopoles with inverse charges [63][64][65]. These results uncover an interesting relation between the phase jump π at each node and the quantized magnetic flux of Dirac monopoles.…”

“…In fact, the extension is a natural process due to that the roots of n(x) = 0 are generally complex. Inspired by [63][64][65][66], we also analytically extend the function F x d dx = f ( ) to be F(z) to investigate whether there are some topological vector potential on the complex plane. We define a vector potential e e…”

“…A , for which there is a quantized charge on each singularity. Based on the Mittag-Leffler theorem [67], A can be expanded as [63][64][65][66] x x y y…”

“…Each singularity in the vector potential forms a monopole field with an elementary π quantized magnetic flux, which corresponds to a monopole with a charge of 1/2 [63][64][65][66]. But for the bound states and the first scattering state, the wave function becomes a real function, which makes it difficult to fix the phase.…”

“…3.2. Dirac monopole fields for eigenstates in a simple finite square well Noting that Dirac monopole field on the extended complex plane was uncovered by investigating phases of nonlinear waves [63][64][65], the above discussions suggest that the monopole field also exist for eigestates in usual quantum systems. We would like to choose a simple finite square well to support this point more directly, since the above solutions are derived from the nonlinear Schrödinger equation.…”