Method for finding soliton solutions of the nonlinear Schrödinger equation

After discussing a possible form of the transformation from the wave-function of the Lax equations of the nonlinear Schr\"odinger (NLS) equation relating to the (n-1)-soliton solution to that relating to the n-soliton solution, an explicit solution of it is found for general n. This is then used to obtain soliton solutions of the NLS equation and solutions of the Lax equations. The usual B\"acklund transformation for the NLS equation is deduced simply. The Zakharov-Shabat equations of the inverse scattering method are also deduced simply, but without the restriction of λⱼ in the upper half-plane. It is shown that the solutions of the NLS equation are regular when no two of λ₁,{λ}₁,...,λN,{λ}N are coincident.