Functional differential equations of pointwise type, periodic and bounded solutions. Solutions of traveling waves type for infinite dimensional dynamic systems. What is common?

Existence of solutions of the functional differential equations and existence of solutions of traveling wave type (solitonic solutions) for infinite dimensional dynamic systems are problems with common nature. At the same time, the solutions of traveling wave type, invariant with respect to complex shift of phase space, correspond to the periodic solutions of the functional-differential equations. The solutions of traveling wave type with the bounded orbits of the shift operator on phase space correspond to the bounded solutions. In this way conditions for existence of bounded and periodic solutions of new type, which consider asymptotic properties of solutions of the differential equations, occur.