B\"acklund-Darboux Transformations for Super KdV Type Equations

Lingling Xue; Qing Ping Liu; Shasha Wang

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B\"acklund-Darboux Transformations for Super KdV Type Equations

Lingling Xue, Shasha Wang, Qing Ping Liu

Exactly Solvable and Integrable Systems · Submitted 29 Oct 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim By proper reductions, we are able not only to recover the previously known BDTs for Kupershimdt's sKdV and smKdV equations, but also to obtain the BDTs for the Geng-Wu's sKdV/smKdV and Hu's smKdV equations.

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By introducing a Miura transformation, we derive a generalized super modified Korteweg-de Vries (gsmKdV) equation from the generalized super KdV (gsKdV) equation. It is demonstrated that, while the gsKdV equation takes Kupershmidt's super KdV (sKdV) equation and Geng-Wu's sKdV equation as two distinct reductions, there are also two equations, namely Kupershmidt's super modified KdV (smKdV) equation and Hu's smKdV equation, which are associated with the gsmKdV equation. By analyzing the flows within the gsKdV and gsmKdV hierarchies, we specifically derive the first negative flows associated with both hierarchies.We then construct a number of B\"acklund-Darboux transformations (BDTs) for both the gsKdV and gsmKdV equations, elucidating the interrelationship between them. By proper reductions, we are able not only to recover the previously known BDTs for Kupershimdt's sKdV and smKdV equations, but also to obtain the BDTs for the Geng-Wu's sKdV/smKdV and Hu's smKdV equations. As applications, we construct some exact solutions for those equations. Since all flows of the sKdV or smKdV hierarchy share the same spatial parts of spectral problem, thus these Darboux matrices and spatial parts of BTs are applicable to any flow of those hierarchies.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	By introducing a Miura transformation, we derive a generalized super modified Korteweg-de Vries (gsmKdV) equation from the generalized super KdV (gsKdV) equation.	contribution 0.62	0.95	0.15	asserted 1.00
s1	It is demonstrated that, while the gsKdV equation takes Kupershmidt's super KdV (sKdV) equation and Geng-Wu's sKdV equation as two distinct reductions, there are also two equations, namely Kupershmidt's super modified KdV (smKdV) equation and Hu's smKdV equation, which are associated with the gsmKdV equation.	result 0.91	0.65	0.18	asserted 1.00
s2	By analyzing the flows within the gsKdV and gsmKdV hierarchies, we specifically derive the first negative flows associated with both hierarchies.We then construct a number of B\"acklund-Darboux transformations (BDTs) for both the gsKdV and gsmKdV equations, elucidating the interrelationship between them.	contribution 0.67	0.96	0.18	asserted 1.00
s3	By proper reductions, we are able not only to recover the previously known BDTs for Kupershimdt's sKdV and smKdV equations, but also to obtain the BDTs for the Geng-Wu's sKdV/smKdV and Hu's smKdV equations.	result 0.73	0.92	0.27	asserted 0.93
s4	As applications, we construct some exact solutions for those equations.	contribution 0.51	0.95	0.10	asserted 1.00
s5	Since all flows of the sKdV or smKdV hierarchy share the same spatial parts of spectral problem, thus these Darboux matrices and spatial parts of BTs are applicable to any flow of those hierarchies.	interpretation 0.81	0.41	0.17	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.87 · evidence from proof or derivation 1.00.

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Original abstract

By introducing a Miura transformation, we derive a generalized super modified Korteweg-de Vries (gsmKdV) equation from the generalized super KdV (gsKdV) equation. It is demonstrated that, while the gsKdV equation takes Kupershmidt's super KdV (sKdV) equation and Geng-Wu's sKdV equation as two distinct reductions, there are also two equations, namely Kupershmidt's super modified KdV (smKdV) equation and Hu's smKdV equation, which are associated with the gsmKdV equation. By analyzing the flows within the gsKdV and gsmKdV hierarchies, we specifically derive the first negative flows associated with both hierarchies.We then construct a number of B\"acklund-Darboux transformations (BDTs) for both the gsKdV and gsmKdV equations, elucidating the interrelationship between them. By proper reductions, we are able not only to recover the previously known BDTs for Kupershimdt's sKdV and smKdV equations, but also to obtain the BDTs for the Geng-Wu's sKdV/smKdV and Hu's smKdV equations. As applications, we construct some exact solutions for those equations. Since all flows of the sKdV or smKdV hierarchy share the same spatial parts of spectral problem, thus these Darboux matrices and spatial parts of BTs are applicable to any flow of those hierarchies.

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Therefore, one may show that [ref], [ref], [ref] coincide with the results obtained in [ref] .

B\"acklund-Darboux Transformations and Discretizations of Super KdV Equation · Bäcklund–Darboux transformations for gsKdV equation · 1.00
The first equation is the temporal Miura transformation which is exactly the one appeared in [ref] .

Negative flows of generalized KdV and mKdV hierarchies and their gauge-Miura transformations · The flows of the gsKdV and gsmKdV hierarchies · 0.83

Cites as background 14 papers

Bäcklund–Darboux transformations (BDTs) were established to generate associated integrable discrete systems for super nonlinear Schrödinger equations [ref] and gsKdV equations [ref] .

B\"acklund-Darboux Transformations and Discretizations of Super KdV Equation · Introduction · 0.91
Bäcklund–Darboux transformations (BDTs) were established to generate associated integrable discrete systems for super nonlinear Schrödinger equations [ref] and gsKdV equations [ref] .

Integrable Discretisations for a Class of Nonlinear Schrodinger Equations on Grassmann Algebras · Introduction · 0.91
With the development of theory of integrable systems, numerous classical systems including KdV and mKdV equations are extended to super or supersymmetric (see [ref] and references therein).

Grassmann extensions of Yang-Baxter maps · Introduction · 0.99

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