Uncertainty principle for the nonlinear waves of the Korteweg–de Vries equation

The solutions of KdV studied in Kovalyov [Kovalyov M. On a class of solutions of KdV. J Differ Equations 1951;15:309–60; Kovalyov M. Basic motions of the Korteweg–de Vries equation, Nonlinear Anal Theory Methods Appl 1998;31(5/6):599–619; Kovalyov M. Modulating properties of harmonic breather solutions of KdV. J Phys A: Math Gen 1998;31:5117–28.] exhibit wave–particle behavior albeit somewhat different from the wave–particle behavior of quantum mechanics. Yet the balance between the wave and particle properties of these solutions is governed by an uncertainty principle remarkably similar to that of quantum mechanics.