Bäcklund-Darboux Transformations and Discretizations of Super KdV Equation

A Supersymmetric AKNS Problem and Its Darboux‐Bäcklund Transformations and Discrete Systems

“…The simplest nontrivial case corresponds to m 11 = 1, m 33 = 0 and p 33 = 1, which solve (14). Then (11) and the first equation of (13) lead to n 13 = α, n 31 = −β [1] , n 32 = −n 23 .…”

“…Then from (17), we have (Dα)(Dβ [1] ) − p 11 (1 − αβ [1] ) = p 2 1 , which gives p 11 = (1 + αβ [1] ) (Dα)(Dβ [1] ) − p 2 1 , where p 1 is a constant of integration. Summarizing above discussion, we obtain a Darboux transformation of elementary type, named as eDT.…”

“…In this way, one obtains [3] − α)(β [3] − β), n 13 = α − α [3] − α , n 31 = β − β [3] + β , [3] − β) α [3] − α , [3] − α) β [3] − β , p 12 = D(α [3] − α) − β [3] − β h + k 1 D(αα [3] ) + 2h D(α [3] − α)αβ, [3] ) + 2h D(β [3] − β)βα, 22 = 11 + α [3] − α β [3] − β , 11 = 2h (Dβ [3] )(Dα) − (Dα [3] )(Dβ)…”

Symbolic Representation and Classification of Supersymmetric Evolutionary Equations

“…These supersymmetric integrable prototypes have been extensively studied in the past four decades, and shown to have various novel features, such as fermionic nonlocal conserved densities [18], nonunique roots of Lax operator [19], and odd Hamiltonian structures [20]. Recently, discrete integrable systems on Grassmann algebras [21][22][23][24], as well as Grassmann extensions of Yang-Baxter maps [25], have been established due to the development of Darboux-Bäcklund transformations of super and/or supersymmetric integrable equations.…”

Novel Traveling Hyperbolic Function Solution for Evolution Equation

“…Nonlinear equations are very important in describe evolution phenomena in engineering and physics. People have investigated many useful methods to obtain the evolution solutions: the inverse scattering method [1][2][3], the Hirota's bilinear method [4][5][6], the Darboux and Backlund transformation [7,8]. Sine-cosine method [9,10].…”