Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations

The study of the Schrödinger equation is one of the main objects of quantum physics. The evolution of a group of identical particles interacting with each other in ultra-cold states, in particular, Bose-Einstein condensates, is described via Hartree approximation, to an excellent degree of accuracy by nonlinear Schrödinger equations. The equation also arises in many fields of physics. For instance, when we describe the propagation of light in some nonlinear optical materials, the nonlinear Schrödinger equations in nonlinear optics are reduced from Maxwell's equations.