Energy transfer mediated by Holstein-type interactions in two space dimensions

We study the two-dimensional extension of the energy or charge transfer problem in the long-wave limit along a mechanical lattice of oscillators for the interplay between linear longitudinal and periodic on-site interactions in the Holstein approach [S. Reza-Mejía and L. A. Cisneros-Ake, Wave Motion 130, 103382 (2024)0165-212510.1016/j.wavemoti.2024.103382]. It is found that the coupling between the linear Schrödinger and sine-Gordon equations, governing the modeling, holds coupled localized and traveling solutions radially symmetric in shape, contrary to the well-known decoupled case where solutions collapse in finite time. We then use the variational description of the problem to find the parameters regime where localization and further propagation take place. For the steady-state scenario, we find a decreasing and oscillatory branch of solutions for the wave function's power, as depending on the frequency of the phase, and then obtain an extension of the Vakhitov-Kolokolov stability criterion that predicts the bifurcation points. We validate these findings by solving the reduced coupled system in the stationary state by means of the Newton method and the Townes soliton condition in the polar coordinates. Finally, we extend our variational approach to the moving case to find the appropriate model and wave parameters where radial motion and the existence of a critical ratio for the strength between longitudinal and on-site interactions take place. These findings are numerically confirmed by solving the full coupled system with the help of the pseudospectral method.