Bäcklund transformation with two pseudo-potentials for a coupled KdV system

We give a Bäcklund transformation with two pseudo-potentials for a coupled KdV system, which can be derived from a single complex KdV equation. By applying the Bäcklund transformation to a trivial seed solution, we construct soliton and periodic solutions. We show that each soliton solution is invariant on a circle and each periodic solution is invariant on a closed curve, which approaches a cardioid or a circle, depending on how to choose the parameters. We also prove that the Bäcklund transformation admits a nonlinear superposition principle, which allows us to discuss nonlinear interactions of these solutions.