We consider the small time semi-classical limit for nonlinear Schrödinger equations with defocusing, smooth, nonlinearity. For a super-cubic nonlinearity, the limiting system is not directly hyperbolic, due to the presence of vacuum. To overcome this issue, we introduce new unknown functions, which are defined nonlinearly in terms of the wave function itself. This approach provides a local vers...

§1.1. The method of characteristics §1.2. Examples of propagation of singularities using progressing waves §1.3. Group velocity and the method of nonstationary phase §1.4. Fourier synthesis and rectilinear propagation §1.5. A cautionary example in geometric optics §1.6. The law of reflection §1.6.1. The method of images §1.6.2. The plane wave derivation §1.6.3. Reflected high frequency wave pac...

This paper studies high frequency solutions of nonlinear hyperbolic equations for time scales at which diiractive eeects and nonlinear eeects are both present in the leading term of approximate solutions. The key innovation is the analysis of rectiication eeects, that is the interaction of the nonoscillatory local mean eld with the rapidly oscillating elds. The main results prove that in the li...

We demonstrate unrestricted superlensing in a triangular twodimensional hotonic crystal. We investigate simple two-point light sources maged by a slab lenses made of this photonic crystal, and show that the efraction of light follows simple rules of geometric optics with the Snell'slaw efraction at each interface, and an effective isotropic refractive index n= -1 for light propagating inside th...

We construct a unified operator framework for quantum holonomies generated from bosonic systems. For system whose Hamiltonian is bilinear in the creation and annihilation operators, we find holonomy group determined only by set of selected orthonormal modes obeying stronger version adiabatic theorem. This photon-number independent description offers deeper insight as well computational advantag...

The aim of this work is to develop and apply a mathematical apparatus based on nonlinear operators for solving problems geometric optics, namely the construction images objects in systems thin lenses. problem constructing image point lens was considered, basis which concept lensing operator defined. properties were investigated. model an lenses folded together investigated, it became possible e...