Abstract:
The Toda lattice is a non-linear evolution equation describing an infinite system of masses on a line that interacts through an exponential force. The paper analyzes the construction of soliton solution for the q-Toda lattice in the two-dimensional case. For this purpose, the equation of motion is taken and the transformation of the dependent variable is used to convert the nonlinear equation into a bilinear form, which is written as the Hirota polynomial. As one of the most effective methods for constructing multisoliton solutions of integrable nonlinear evolution equations, Hirota method is applicable to a wide class of equations, including nonlinear differential, nonlinear differential-difference equations. Using the Hirota method, the bilinear form was obtained for the two-dimensional q-Toda lattice on the basis of which the q-soliton solution was found. The dynamics of the q-soliton solution for two-dimensional q-Toda lattice is presented. Note that the soliton is conserved due to the equilibrium between the action of the nonlinear environment with dispersion. In addition, the soliton behaves like a particle: does not collapse when interacting with each other or other disturbances, while maintaining the structure and continues to move. This quality has the ability to use when transferring data or information over long distances with virtually no interference. In addition, the study of the Toda lattice and the application to it of different methods in different dimensions allows one to proceed to the understanding of such complex terms as matrix models that can be used to describe different physical systems.
