Hirota's virtual multisoliton solutions of $N=2$ supersymmetric Korteweg–de Vries equations

We prove that for $a=1$ or $a=4$, the $N=2$ supersymmetric Korteweg–de Vries (super-KdV) equations obtained by Mathieu admit Hirota's $n$-supersoliton solutions, whose nonlinear interaction does not produce any phase shifts. For initial profiles that cannot be distinguished from a one-soliton solution at times $t\ll0$, we reveal the possibility of a spontaneous decay and transformation into a solitonic solution with a different wave number within a finite time. This paradoxical effect is realized by the completely integrable $N=2$ super-KdV systems if the initial soliton is loaded with other solitons that are virtual and become manifest through the $\tau$-function as time increases.