Traveling-Wave Solutions of the Schwarz–Korteweg–de Vries Equation in $2+1$ Dimensions and the Ablowitz–Kaup–Newell–Segur Equation Through Symmetry Reductions

One of the more interesting solutions of the $(2+1)$-dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation is the soliton solutions. We previously derived a complete group classification for the SKdV equation in $2+1$ dimensions. Using classical Lie symmetries, we now consider traveling-wave reductions with a variable velocity depending on the form of an arbitrary function. The corresponding solutions of the $(2+1)$-dimensional equation involve up to three arbitrary smooth functions. Consequently, the solutions exhibit a rich variety of qualitative behaviors. In particular, we show the interaction of a Wadati soliton with a line soliton. Moreover, via a Miura transformation, the SKdV is closely related to the Ablowitz–Kaup–Newell–Segur (AKNS) equation in $2+1$ dimensions. Using classical Lie symmetries, we consider traveling-wave reductions for the AKNS equation in $2+1$ dimensions. It is interesting that neither of the $(2+1)$-dimensional integrable systems considered admit Virasoro-type subalgebras.