“Nanoscale physics”   
Nanoscale excitations of solids in the form of axially symmetric solitons
L.V. Shmeleva, A.D. Suprun
 Department of Theoretical Physics, Faculty of Physics, Taras Shevchenko National University of Kyiv. Volodymyrska Street, 64/13, Kyiv-01601, Ukraine. 
E-mail: lshmel@univ.kiev.ua 
Soliton excitations in solids represent the development of the concept of traditional electronic excitations (injected electrons, excitons) or excitations of another physical nature (magnons, polarons, etc.). The main feature that distinguishes soliton excitations from similar traditional excitations is the appearance of distributed amplitude, not only in space but also in time. In this case, the spatial distribution of the amplitude of soliton excitations has dimensions not exceeding several nanometers. Although traditionally excitations are considered as spatially delocalized.
Auto-localized spatial distribution of excitation arises due to nonlinearity. This nonlinearity is due to the interaction of excitation with the lattice.
The general conditions of soliton formation in the case when disturbance of equilibrium arises in the crystal, caused by excitation are considered. In particular, new aspects of the nonlinear Schrödinger equation (NSE) with cubic nonlinearity were considered. It is known that NSE with cubic nonlinearity has analytical solutions on the class of hyperbolic functions only for the spatially one-dimensional case.
It turned out that there is a physically consistent case when it is possible to perform exact separation of variables in a spatially three-dimensional NSE in terms of generalized functions. It is shown that the soliton has nanoscale axially symmetric spindle-shaped spatial form. The soliton shape can vary from strongly elongated along the dynamic direction to strongly oblate. It depends on the values of the specific solution parameters. Also there are parameters values in which the excitation has an almost spherical shape. Transverse size and velocity of the soliton is significantly influenced to its shape. Variation these parameters allows of control the spatial form of solitonic excitation. 
The conducted research has practical interest because they allow of the use of soliton excitation for control the processes of directed point transfer of energy or charge.