Multi-component generalizations of the Hirota–Satsuma coupled KdV equation

In this paper, we consider multi-component generalizations of the Hirota–Satsuma coupled Korteweg–de Vries (KdV) equation. By introducing a Lax pair, we present a matrix generalization of the Hirota–Satsuma coupled KdV equation, which is shown to be reduced to a vector Hirota–Satsuma coupled KdV equation. By using Hirota’s bilinear method, we find a few soliton solutions to the vector Hirota–Satsuma coupled KdV equation in a symmetric case. Finally, in this symmetric case, we give a multi-soliton solution expressed by pfaffians and prove it by pfaffian techniques.