Elementary particles are particles that are not considered as a combination of simpler particles and exist in a free state. In the process of interaction with other particles, they should behave as a whole. The indivisibility of elementary particles does not mean that they have no internal structure or additional properties. The indivisibility of elementary particles does not mean that they have no internal structure or additional properties. The elementary particles of modern physics do not satisfy the rigorous definition of elementarity. Most of them are composite systems according to modern concepts. The fundamental possibilities of constructing a mathematical model of the formation of the structure and basic properties of matter in the process of the creation and development of the Universe, using special protoparticles of small size and small mass are discussed. If the basis of processes in the Universe is motion, then one can proceed from the basic laws of physics, statistical and thermodynamic equilibrium. With the formation of the Universe, in addition to transverse electromagnetic waves, traveling longitudinal oscillations can occur. These oscillations are included in the composition of electromagnetic waves, endowing all together the term wave-particle duality. Gravitational waves can also be longitudinal. However, longitudinal waves are sometimes expressed as solitary waves. The collective displacement of quasiparticles, the wave function of which satisfies the Schrödinger equation of quantum mechanics, is considered. The solution of this equation is found in the form of a wave function of a spherical longitudinal wave, which can be represented in a special form of a soliton solution.