We solve the Gardner deformation problem for the N = 2 supersymmetric a = 4 Korteweg–de Vries equation [P. Mathieu, “Supersymmetric extension of the Korteweg–de Vries equation,” J. Math. Phys. 29(11), 2499–2506 (1988)]. We show that a known zero-curvature representation for this super-equation yields the system of new nonlocal variables such that their derivatives contain the Gardner deformation for the classical KdV equation.

1.
C. S.
Gardner
, “
Korteweg–de Vries equation and generalizations. I. A remarkable nonlinear substitution
,”
J. Math. Phys.
9
,
1202
1204
(
1968
).
2.
P.
Labelle
and
P.
Mathieu
, “
A new supersymmetric Korteweg–de Vries equation
,”
J. Math. Phys.
32
(
4
),
923
927
(
1991
).
3.
P.
Mathieu
, “
Open problems for the super KdV equations
,” in
Bäcklund and Darboux Transformations. The Geometry of Solitons
,
CRM Proceedings and Lecture Notes
Vol.
29
, edited by
A.
Coley
,
D.
Levi
,
R.
Milson
,
C.
Rogers
, and
P.
Winternitz
(
AMS
,
2001
), pp.
325
334
;
4.
V.
Hussin
,
A. V.
Kiselev
,
A. O.
Krutov
, and
T.
Wolf
, “
N = 2 supersymmetrica = 4-Korteweg–de Vries hierarchy derived via Gardner's deformation of Kaup–Boussinesq equation
,”
J. Math. Phys.
51
(
8
),
083507
(
2010
);
e-print arXiv:0911.2681 [nlin.SI].
5.
I. M.
Gelfand
and
L. A.
Dikiĭ
, “
Asymptotic properties of the resolvent of Sturm-Liouville equations, and the algebra of Korteweg-de Vries equations
,”
Russ. Math. Surveys
30
(
5
),
77
113
(
1975
).
6.
G.
Wilson
, “
On two constructions of conservation laws for Lax equations
,”
Q. J. Math. Oxford Ser.
32
(
128
),
491
512
(
1981
).
7.
M.
Roelofs
, “
Prolongation structures of supersymmetric systems
,” Ph.D. dissertation (
University of Twente
,
1993
).
8.
I. S.
Krasil'shchik
and
P. H. M.
Kersten
,
Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations
(
Kluwer
,
Dordrecht
,
2000
).
9.
A. V.
Kiselev
, “
Algebraic properties of Gardner's deformations for integrable systems
,”
Theor. Math. Phys.
152
(
1
),
963
976
(
2007
);
10.
M.
Marvan
, “
On the horizontal gauge cohomology and non-removability of the spectral parameter
,”
Acta Appl. Math.
72
,
51
65
(
2002
).
11.
T.
Miva
,
M.
Jimbo
, and
E.
Date
,
Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras
(
Cambridge University Press
,
Cambridge
,
2000
).
12.
A. V.
Bocharov
,
A. M.
Verbovetsky
,
A. M.
Vinogradov
 et al., in
Symmetries and Conservation Laws for Differential Equations of Mathematical Physics
, edited by
I. S.
Krasil'shchik
and
A. M.
Vinogradov
(
AMS
,
Providence, RI
,
1999
).
13.
A. V.
Kiselev
, “
The twelve lectures in the (non)commutative geometry of differential equations
,” preprint IHÉS/M/12/13, (
2012
), p.
140
.
14.
F. A.
Berezin
, “
Introduction to superanalysis
,”
Mathematical Physics and Applied Mathematics
(
Reidel
,
Dordrecht/Boston, MA
,
1987
), Vol. 9.
15.
L. D.
Faddeev
, and
L. A.
Takhtajan
,
Hamiltonian Methods in the Theory of Solitons
(
Springer-Verlag
,
Berlin
,
1987
).
16.
Bäcklund (auto)transformations between PDE appear in the same context. In Ref. 9 we argued that the former, when regarded as the diagrams, are dual to the diagram description of Gardner's deformations.
17.
V. E.
Zakharov
, and
A. B.
Shabat
, “
Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II
,”
Funct. Anal. Appl.
13
(
3
),
166
174
(
1979
).
18.
M.
Marvan
, “
A direct procedure to compute zero-curvature representations. The case
$\mathfrak {sl}_2$
sl2
,” in
Proceedings of a Conference on Secondary Calculus and Cohomological Physics, Moscow, 24–31 August, 1997
(
Diffiety Institute of the Russian Academy of Natural Sciences
,
Moscow
,
1997
), p.
9
; see http://www.emis.de/proceedings/SCCP97 for ELibEMS.
19.
D. A.
Leites
, “
Lie superalgebras
,”
J. Math. Sci.
30
(
6
),
2481
2512
(
1985
).
20.
Yu. I.
Manin
, “
Holomorphic supergeometry and Yang–Mills superfields
,”
J. Math. Sci.
30
(
2
),
1927
1975
(
1985
).
21.
L.
Frappat
,
P.
Sorba
, and
A.
Sciarrino
, “
Dictionary on Lie superalgebras
,” preprint arXiv:hep-th/9607161 (
1996
), p.
146
.
22.
D. A.
Leites
, “
Introduction to the theory of supermanifolds
,”
Russ. Math. Surveys
35
(
1
),
1
64
(
1980
).
23.
P.
Mathieu
, “
Supersymmetric extension of the Korteweg–de Vries equation
,”
J. Math. Phys.
29
(
11
),
2499
2506
(
1988
).
24.
A. V.
Kiselev
and
V.
Hussin
, “
Hirota's virtual multisoliton solutions ofN = 2 supersymmetric Korteweg–de Vries equations
,”
Theor. Math. Phys.
159
(
3
),
832
840
(
2009
);
e-print arXiv:0810.0930 [nlin.SI].
25.
A.
Das
,
W.-J.
Huang
, and
S.
Roy
, “
Zero curvature condition of
$\mathfrak {osp}(2/2)$
osp(2/2)
and the associated supergravity theory
,”
Int. J. Mod. Phys.
18
,
4293
4311
(
1992
).
26.
S.
Andrea
,
A.
Restuccia
, and
A.
Sotomayor
, “
The Gardner category and nonlocal conservation laws for N = 1 super KdV
,”
J. Math. Phys.
46
,
103517
(
2005
).
You do not currently have access to this content.