Physical relation between quantum mechanics and solitons on a thin elastic rod

The 2×2-matrix-valued first-order linear differential equation appears when we solve the modified Korteweg–de Vries (MKdV) equation. This linear differential equation is regarded as a fictitious quantum equation. On the other hand, it is known that the dynamics of an elastic rod is governed by the MKdV equation. In this paper, after we construct a Dirac equation on an elastic rod embedded into (2+1)-dimensional space-time, we show that this linear differential equation is naturally introduced through this Dirac equation. Then we can explain the reason why the classical nonlinear differential equation is associated physically with quantum mechanics. In other words, we prove that fictitious quantum mechanics related to the soliton is real quantum mechanics on the soliton as a base space. We also argue that the Berry phase of the Dirac particle is related to the Lax pair ${\mathit{L}}_{\mathrm{\tau }}$=i[L,B].