Nonlinear excitations in two-dimensional molecular structures with impurities

We study the nonlinear dynamics of electronic excitations interacting with acoustic phonons in two-dimensional molecular structures with impurities. We show that the problem is reduced to the nonlinear Schr\"odinger equation with a varying coefficient. The latter represents the influence of the impurity. Transforming the equation to the noninertial frame of reference coupled with the center of mass we investigate the soliton behavior in the close vicinity of the impurity. With the help of the lens transformation we show that the soliton width is governed by an Ermakov-Pinney equation. We also investigate bound state of the soliton with impurity and show that in addition to the radially symmetric bound state a dipolelike state can exist if the number of excitations in the soliton does not exceed some threshold value. We study both the equilibrium states and the dynamics of the dipolelike excitations. Analytical results are in good agreement with numerical simulations of the nonlinear Schr\"odinger equation.