Nonlinear Klein-Gordon equation pulsons with a fractional power potential

The study of three-dimensional localized solutions of the nonlinear Klein-Gordon equation was conducted. The article describes new features of these solutions that distinguish them from other solutions. The consequences of such properties, which can be experimentally verified, were studied. The case where the Klein-Gordon equation is non-linear while solution amplitude tends to zero leads to the solutions localization in a spherically symmetric case. The breather-like spherically symmetric solutions show the constancy of the oscillations fast mode frequency and are three-dimensional objects. Such Lorentz-invariant breather-like solutions with motions will be spatially modulated in the direction of motion as well as de Broglie wave. For these solutions we reviewed a "soliton" interference pattern that complements conventional Young's corpuscular and wave interference patterns on two slits. The experiment scheme to test such "soliton" interference pattern was created. Since such oscillating generations tend to be localized, at collision of such two objects, they can also be localized in one state. Due to the spatial modulation of the colliding states, the motion direction of the resulting state will depend on the directions of the colliding states, on their phase and frequency oscillations. When taking into account the localization of such oscillating states, the "interference" pattern will be significantly different from the unlimited waves linear interference pattern. The characteristic feature of "soliton" interference is in the disappearance of the interference pattern on the screen when the distance between the slits is bigger than the value determined by the characteristic dimensions of the "soliton".