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

Abstract

We produce soliton and similarity solutions of supersymmetric extensions of Burgers, Korteweg-de Vries and modified KdV equations. We give new representations of the τ -functions in Hirota bilinear formalism. Chiral superfields are used to obtain such solutions. We also introduce new solitons called virtual solitons whose nonlinear interactions produce no phase shifts.

Bäcklund–Darboux transformations and discretizations of N = 2 a = −2 supersymmetric KdV equation

“…From v [12] = v [21] , γ [12] = γ [21] and (20), after some tedious calculations we obtain for SKdV −2 equation the following nonlinear superposition formula v [12]…”

“…which is a semi-discrete or differential-difference system. To find a difference-difference system, we define for any field variable u u ≡ u n,m , u [1] ≡ u n+1,m , u [2] ≡ u n,m+1 , u [12] ≡ u n+1,m+1 , then the nonlinear superposition formula (21)…”

“…Then, for SKdV 4 equation, a class of solutions was calculated and for SKdV 1 equation, a bilinear Bäcklund transformation was obtained. To the best of our knowledge, a proper Hirota bilinear representation of the SKdV −2 equation is still missing, although some attempts have been made by Delisle and Hussin [20] [21] and certain interesting solutions have been obtained. Also, any form of Bäcklund transformation is not known to this equation.…”

Classical and SUSY solutions of the Boiti–Leon–Manna–Pempinelli equation

“…We refer the readers to [12,13,15,16,17,18,19,21,23,24] for more details on the construction of supersymmetric multisoliton solutions.…”

A N = 2 extension of the Hirota bilinear formalism and the supersymmetric KdV equation

“…The study of exact solutions of completely integrable supersymmetric systems is of current interest in modern mathematical physics research. In particular, the N =2 supersymmetric extension of the Korteweg-de Vries (KdV) equation [1] has been largely studied in terms of integrability conditions, exact solutions and symmetry group structures [2,3,4,5,6,7,8,9,10,16]. The equation is described by a bosonic superfield A defined on the superspace R 2|2 [11] of local coordinates (x, t, θ 1 , θ 2 ).…”