Soliton, rational, and periodic solutions for the infinite hierarchy of defocusing nonlinear Schrödinger equations

Analysis of short-pulse propagation in positive dispersion media, e.g., in optical fibers and in shallow water, requires assorted high-order derivative terms. We present an infinite-order "dark" hierarchy of equations, starting from the basic defocusing nonlinear Schrödinger equation. We present generalized soliton solutions, plane-wave solutions, and periodic solutions of all orders. We find that "even"-order equations in the set affect phase and "stretching factors" in the solutions, while "odd"-order equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are complex. There are various applications in optics and water waves.