Supersymmetric Sawada-Kotera Equation: Bäcklund-Darboux Transformations and Applications

Abstract

In this paper, we construct a Darboux transformation and the related Bäcklund transformation for the supersymmetric Sawada-Kotera (SSK) equation.The associated nonlinear superposition formula is also worked out. We demonstrate that these are natural extensions of the similar results of the Sawada-Kotera equation and may be applied to produce the solutions of the SSK equation. Also, we present two semi-discrete systems and show that the continuum limit of one of them goes to the SKK equation.

A study on Caudrey–Dodd–Gibbon–Sawada–Kotera partial differential equation

“…There are several other special methods, closely related Bernoulli sub‐equation method, to obtain solutions of NLPDEs in various existing works 5–7 . Also, for Bäcklund transformation, 8 a generalized Bernoulli sub‐ODE method to some NLPDEs, 9 Bernoulli sub‐equation method, a modified $$$ \mathit{\exp}\left(-\Omega \left(\xi \right)\right) $$$‐expansion function method and their improved versions, see references, 10–15 sine‐Gordon expansion method, 16 the first integral method has been used to construct traveling‐wave solutions of the Cahn–Allen equation, 17 the exp‐function method has been used to solve the fifth‐order Caudrey–Dodd–Gibbon equation, 18 Lie symmetry and sub‐equation methods have been used to solve the time fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation, 19 a Crum transformation was used to get new properties of the Caudrey–Dodd–Gibbon–Sawada–Kotera equation and its $$$ \lambda $$$‐modified equation, 20 Bäcklund transformation, 21 the NLPDEs are solved by using the function $$$ U $$$ 22 . Alternative $$$ \left({G}^{\prime }/G\right) $$$‐expansion method has been used to analyze solution of fifth‐order Caudrey–Dodd–Gibbon–Sawada–Kotera equation, 23 the Lie symmetry method, 24 the generalized exponential rational function method, 25 $$$ q $$$‐homotopy analysis transform method, 26 homotopic technique, 27 the reduced differential transform method and local fractional series expansion method, 28 the blended homotopy techniques pertaining to the Sumudu transform, 29 the shooting method, 30 the traveling‐wave solutions with use of the Kohlrausch–Williams–Watts function, 31 the fractal traveling‐wave transformation, 32 a new factorization technique, 33 the local fractional Riccati dif...…”

“…real solution(21) are plotted in FigureCaseFrom system associated with Equation(11), one gets new coefficients as…”

Integrable semi-discretisation of the Drinfel’d–Sokolov hierarchies

“…One successful example is possibly the class of equations of A (2) 2 -type (also known as the BC 1 -type in the literature). In this class the BT and the permutability are fully understood [29,39,51,52] for the Sawada-Kotera (SK), Kaup-Kupershmidt (KK) and Tzitzeica equations, but were not interpreted as discrete equations (which was illustrated by the SK equation) until very recently, see [35]. It is also worth mentioning that there exist different integrable discretisations of the Drinfel'd-Sokolov hierarchies, see e.g.…”

Integrable semi-discretisation of the Drinfel'd--Sokolov hierarchies

“…The successful example is possibly the class of equations of A (2) 2 -type (also known as the BC 1 -type in the literature). In this class the BT and the permutability are fully understood [29,39,51,52] for the Sawada-Kotera (SK), Kaup-Kupershmidt (KK) and Tzitzeica equations, but were not interpreted as discrete equations (which was illustrated by the SK equation) until very recently, see [35]. It is also worth mentioning that there exist different integrable discretisations of the Drinfel'd-Sokolov hierarchies, see e.g.…”