Hirota Bilinear Formalism and Supersymmetry

Extending the gauge-invariance principle for τ functions of the standard bilinear formalism to the supersymmetric case, we define N = 1 supersymmetric Hirota operators. Using them, we bilinearize SUSY nonlinear evolution equations. The super-soliton solutions and extension to SUSY sine-Gordon are also discussed. As a quite strange paradox it is shown that the Lax integrable SUSY KdV of Manin-Radul-Mathieu equation does not possess N super-soliton solution for N ≥ 3 for arbitrary parameters. Only for a particular choice of them the N super-soliton solution exists.