Bilinear approach to N = 2 supersymmetric KdV equations

Abstract

The N = 2 supersymmetric KdV equations are studied within the framework of Hirota bilinear method. For two such equations, namely N = 2, a = 4 and N = 2, a = 1 supersymmetric KdV equations, we obtain the corresponding bilinear formulations. Using them, we construct particular solutions for both cases. In particular, a bilinear Bäcklund transformation is given for the N = 2, a = 1 supersymmetric KdV equation.

New Bilinear Bäcklund Transformation and Lax Pair for the Supersymmetric Two‐Boson Equation

“…It has been shown that these supersymmetric integrable systems possess bi-Hamiltonian structure, Painlevé property, infinite many symmetries, Darboux transformation, Bäcklund transformation, super soliton solutions [26][27][28][29][30][31][32][33][34][35][36][37]. However, application of Hirota's bilinear method to supersymmetric equations has been launched only in recent years [38][39][40][41][42][43][44]. The systematic super bilinearization method of supersymmetric equations was introduced by Carstea [39].…”

“…The key idea is to extend the Hirota's bilinear operator to supersymmetric case. Despite this bilinearization of supersymmetric equations, up to now the standard multisoliton solutions still cannot be constructed [38][39][40][41][42][43][44]. Recently, we have further generalized Nakamura's method to construct explicitly quasi-periodic solutions of the supersymmetric equations [45,46].…”

Soliton and Similarity Solutions of Ν = 2, 4 Supersymmetric Equations

“…Recently, the formalism was adapted to N = 2 extensions [4,7,8] by splitting the equation into two N = 1 equations, one fermionic and one bosonic. Our approach consists of treating the equation as a N = 2 extension without splitting it, but imposing chirality conditions.…”

“…The study of N = 2 supersymmetric (SUSY) extensions of nonlinear evolution equations has been largely studied in the past [1][2][3][4][5][6][7][8] in terms of integrability conditions and solutions. Such extensions are given as a Grassmann-valued partial differential equation with one dependent variable A(x, t; θ 1 , θ 2 ) which is assumed to be bosonic to get nontrivial extensions.…”

“…Equation (41) can easily be viewed as a generalization of a N = 2 equation. Indeed, setting θ 3 = θ 4 = 0 and Γ = 1 √ 2 A in Equation (41), we retrieve the SmKdV Equation(8).To construct virtual solitons of N = 2 SUSY extensions, we have considered chiral superfields.…”