On the applicability of the variation of action method to some one-field solitons

The applicability of the widely used method of variation of action is discussed by investigating the transverse stability of the kink solution of the nonlinear Schrödinger equation and the bell solution of the Kadomtsev–Petviashvili equation. An exact calculation shows that neither instability nor stability can be predicted correctly by the variation of action method. The reasons for that defect are discussed and the correct stability regions are presented.