We demonstrate the possibility of generation of coherent radiation with tunable frequencies higher than the frequency of the driving field $ u _{d}$ in a nonlinear medium utilizing the difference combination resonance that occurs when $ u _{d}$ match
es the difference of the frequencies of the two generated fields $omega _{1}$ and $omega _{2}$. We find that such a resonance can appear in materials which have opposite signs of refractive index at $omega _{1}$ and $omega _{2}$. It can also occur in positive refractive index materials with strong anomalous dispersion if at one of the generated frequencies the group and phase velocities are opposite to each other. We show that the light amplification mechanism is equivalent to a combination resonance in a system of two coupled parametric oscillators with the opposite sign of masses. Such a mechanism holds promise for a new kind of light source that emits coherent radiation of tunable wavelengths by an optical parametric amplification process with the frequency higher than $ u_{d}$.
After reviewing the problematic behavior of some previously suggested finite interval spatial operators of the symmetric Riesz type, we create a wish list leading toward a new spatial operator suitable to use in the space-time fractional differential
equation of anomalous diffusion when the transport of material is strictly restricted to a bounded domain. Based on recent studies of wall effects, we introduce a new definition of the spatial operator and illustrate its favorable characteristics. We provide two numerical methods to solve the modified space-time fractional differential equation and show particular results illustrating compliance to our established list of requirements, most important to the conservation principle and the second law of thermodynamics.
