Transverse linear stability of line solitons for 2D Toda

The $2$-dimensional Toda lattice ($2$D Toda) is a completely integrable semi-discrete wave equation with the KP-II equation in its continuous limit. Using Darboux transformations, we prove the linear stability of $1$-line solitons for $2$D Toda of any size in an exponentially weighted space. We prove that the dominant part of solutions to the linearized equation around a $1$-line soliton is a time derivative of the $1$-line soliton multiplied by a function of time and transverse variables. The amplitude is described by a $1$-dimensional damped wave equation in the transverse variable, as is the case with the linearized KP-II equation.