The steepest descent method for Riemann-Hilbert problems (a nonlinear analog of the classical steepest descent method for the evaluation of integrals) is the method of choice for the solution of integrable systems in asymptotic regimes, such as small dispersion or long time. We propose the study of the second and higher breaking of the solution of the semiclassical focusing nonlinear Schroedinger equation (NLS), through a new development of the steepest descent method. The method will allow the deformation of strings of poles in the complex plane (they are solutions of the Zakharov-Shabat eigenvalue problem) in similar ways and for similar purposes that continuous contours are deformed. The discreteness of the poles will be retained, contrary to our previous approach, that condensed the poles to their continuum limit, obtaining the first break, but leading to serious obstacles in treating the second break. We will also tackle the derivation of the NLS waves in a spatial half-line, that result from a time-periodic driver at the boundary point. The long-standing challenge here is that the method of inverse scattering method is natural for solving evolution equations. Confronted with a boundary value problem, the method demands overdetermined data on one side. We propose to overcome the difficulty by matching the wavetrain to the driver, through a boundary layer constructed of multiphase waves and solitons (breathers). This is consistent with the linear limit of the problem. In the second part of the proposal, we continue our code development and analytic model development on the transmission and guidance of light through photonic crystals near resonance (see below).
In both parts of the proposal, we address waves in natural and in man-made materials, in particular, the generation, propagation, breaking and extraordinary behaviour under resonant conditions of these waves.The physical and technological counterparts of the analysis are mainly in optics (linear and nonlinear). A central object of our analysis is the focusing nonlinear Schroedinger equation (NLS), which appears dominantly in nonlinear optical transmission, together with its many variants. NLS is notoriously unstable, exhibiting the spontaneous break-up of wave-trains and the formation of new ones. While the mechanism of the first break is understood from our previous work, subsequent breaks and the type of wave-trains that eventually emerge are challenging open questions. These questions constitute one of the two focal points of our proposal. The second focal point of the proposal addresses electromagnetic (EM) waves, in particular light and its interaction with manufactured materials, known as photonic crystals (PC). These small-scale materials, metallic or non-metallic, have a periodic or repetitive geometry (e.g. film or plate perforated at the nodes of a periodic lattice). Monochromatic light, incident upon the PC, must adjust to the refractive index inside each of the PC constituent materials. When the PC geometry is structured appropriately, light can be manipulated to respond taking novel and technologically exploitable modes. Examples are the guiding, blocking and even trapping of light, as well as extraordinary transmission through PC films. The first commercial products involving two-dimensionally periodic photonic crystals are already available in the form of photonic-crystal fibers (PCF). Because of its ability to confine light in hollow cores or with confinement characteristics not possible in conventional optical fiber, PCF is now finding applications in fiber-optic communications, fiber lasers, nonlinear devices, high-power transmission, highly sensitive gas sensors, and other areas. Our effort is two-fold. We will continue our development of numerical codes that are dedicated to PC structures and are appropriate for overcoming the difficulties of calculation near resonance. We have verified that such codes significantly outperform commercial EM codes. We will implement these codes to optimize transmission in perforated films and to experiment with novel PC geometries. We will also continue our development of simplified mathematical models that capture analytically the essence of phenomena when a PC operates in the nonlinear regime.
