Weakly localized and supersymmetric structures in nonlinear evolution equations

Several nonlinear differential-difference and supersymmetric systems were studied. Following the method of Ablowitz and Satsuma and some nonlinear superposition formulas the structure of complex rational and quasi-soliton solutions for Toda lattice was studied. It was conjectured that Toda and Boussinesq systems possess a large class of common rational solutions. For the generalized Volterra systems using Hirota method and the procedure of Ablowitz and Satsuma we studied the interaction of nonsingular rational solutions. It was shown that they interact elastically with no phase shift. Generalizing the gauge invariance principle of the τ function to the supersymmetric case, we have constructed supersymmetric Hirota bilinear operators. Using them, we studied supersymmetric KdV-type equations. The solutions for multiple collisions of super-solitons were derived. (author)