We use the Gould-Hopper (GH) polynomials to investigate the Novikov-Veselov (NV) equation. The root dynamics of the σ-flow in the NV equation is studied using the GH polynomials and then the Lax pair is found. In particular, when N = 3, 4, 5, one can get the Gold-fish model. The smooth rational solutions of the NV equation are also constructed via the extended Moutard transformation and the GH polynomials. The asymptotic behavior is discussed and then the smooth rational solution of the Liouville equation is obtained.

1.
Athorne
,
C.
and
Nimmo
,
J. J. C.
, “
On the Moutard transformation for integrable partial differential equations
,”
Inverse probl.
7
,
809
826
(
1991
).
2.
Bagdanov
,
L. V.
, “
The Veselov-Novikov equation as a natural two-dimensional generalization of the KdV equation
,”
Theor. Mat. Fiz.
70
,
309
314
(
1997
).
3.
Basalaev
,
M. Y.
,
Dubrovsky
,
V. G.
, and
Topovsky
,
A. V.
, “
New exact multi line soliton and periodic solutions with constant asymptotic values at infinity of the NVN integrable nonlinear evolution equation via dibar-dressing method
,” e-print arXiv:0912.2155.
4.
Bretti
,
G.
and
Ricci
,
P. E.
, “
Multi-dimensional Extensions of the Bernouli and Appell polynomials
,”
Taiwan J. Math.
8
(
3
),
415
428
(
2004
).
5.
Calogero
,
F.
, “
The neatest many-body problem amenable to exact treatments (a “goldfish”?)
,”
Physica D
152–153
,
78
84
(
2001
).
6.
Dubrovin
,
B. A.
,
Krichever
,
I. M.
, and
Novikov
,
S. P.
, “
The Schördinger equation in a periodic field and Riemann surfaces
,”
Dokl. Akad. Nauk SSSR
229
(
1
),
15
18
(
1976
).
7.
Dubrovsky
,
V. G.
and
Formusatik
,
I. B.
, “
The construction of exact rational solutions with constant asymptotic values at infinity of two-dimensional NVN integrable nonlinear evolution equations via dbar-dressing method
,”
J. Phys. A
34A
,
1837
1851
(
2001
).
8.
Dubrovsky
,
V. G.
and
Formusatik
,
I. B.
, “
New lumps of Veselov-Novikov equation and new exact rational potentials of two-dimensional Schrödinger equation via dbar-dressing method
,”
Phys. Lett.
313
(
1–2
),
68
76
,
2003
.
9.
Ferapontov
,
E. V.
, “
Stationary Veselov-Novikov equation and isothermally asymptotic surfaces in projective differential geometry
,” Preprint SFB 288 No. 318, Berlin (
1988
);
10.
Gould
,
H. W.
and
Hopper
,
A. T.
, “
Operational formulas connected with two generalizations of Hermite polynomials
,”
Duke Math. J.
29
,
51
63
(
1962
).
11.
Grinevich
,
P. G.
, “
Rational solitons of the Veselov-Novikov equation are reflectionless two-dimensional potentials at fixed energy
,”
Theoret. Mat. Fiz.
69
(2),
307
310
(
1986
).
12.
Grinevich
,
P. G.
and
Manakov
,
S. V.
, “
Inverse scattering problem for the two-dimensional Schrödinger operator, the dbar-method and nonlinear equations
,”
Funct. Anal. Appl.
20
(
2
),
94
103
(
1986
).
13.
Grinevich
,
P.
,
Mironov
,
A.
, and
Novikov
,
S.
, “
New reductions and nonlinear systems for 2D Schrödinger operators
,” e-print arXiv:1001.4300.
14.
Hu
,
H. C.
,
Lou
,
S. -Y.
, and
Liu
,
Q. -P.
, “
Darboux transformation and variable separation approach: The Nizhnik-Novikov-Veselov equation
,”
Chin. Phys. Lett.
20
,
1413
1415
(
2003
).
15.
Hu
,
H. -C.
and
Lou
,
S. -Y.
, “
Construction of the Darboux transformaiton and solutions to the modified Nizhnik-Novikov-Veselov equation
,”
Chin. Phys. Lett.
21
(
11
),
2073
2076
(
2004
).
16.
Krichever
,
I.
, “
A characterization of Prym varieties
,”
Int. Math. Res. Notices
2006
,
36
.
17.
Konopelchenko
,
B. G.
,
Introduction to Multidimensional Integrable Equations: The Inverse Spectral Transform in 2+1-Dimensions
(
Plenum
,
New York
,
1992
).
18.
Konopelchenko
,
B. G.
and
Landolfi
,
G.
, “
Induced surfaces and their integrable dynamics II: Generalized Weierstrass representations in 4-d spaces and deformations via DS hierarchy
,”
Stud. Appl. Math.
104
,
129
169
(
2000
).
19.
Liu
,
S. -Q.
,
Wu
,
C. -Z.
, and
Zhang
,
Y.
, “
On the Drinfeld-Sokolov hierarchies of D type
,”
Int. Math. Res. Notices
2010
;
20.
Ma
,
W. -X.
and
You
,
Y.
, “
Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions
,”
Trans. Am. Math. Soc.
357
(
5
),
1753
1778
(
2005
).
21.
Manakov
,
S. V.
, “
The inverse scattering method and two-dimensional evolution equations
,”
Usp. Mat. Nauk.
31
(
5
),
245
246
(
1976
).
22.
Matveev
,
V. B.
and
Salle
,
M. A.
,
Darboux Transformations and Solitons
,
Springer series in Nonlinear Dynamics
(
Springer
,
Berlin
,
1991
).
23.
Mironov
,
A. E.
, “
The Novikov-Veselov hierarchy of equations and integrable deformations of minimal Lagrangian tori in CP2
,”
Siberian Electronic Math. Rep.
1
,
38
46
(
2004
).
24.
Mironov
,
A. E.
, “
Relationship between symmetries of the Tzitzeica equation and the Novikov-Veselov hierarchy
,”
Math. Notes
82
(
4
),
569
572
(
2007
).
25.
Moutard
,
Th. F.
, “
Sur les equations differentielles line'ares du second ordre
,”
C.R. Acad. Sci. Paris
80
,
729
(
1875
);
Moutard
,
Th. F.
,
J. Ecole Politechnique
45
,
1
11
(
1878
).
26.
Nimmo
,
J. J. C.
, “
Darboux transformations in (2 + 1) dimensions
,” in
Proceedings of the NATO ARW Applications of Analytic and Geometric Methods to Nonlinear Differential Equations
,
Exeter, U.K.
, edited by
P.
Clarkson
, NATO ASI Series 309 (
Kluwer
,
Netherlands
,
1993
), pp.
183
192
.
27.
Novikov
,
S. P.
, “
Two-dimensional Schrödinger operators in periodic fields
,”
J. Sov. Math.
28
(
1
),
1
20
(
1985
).
28.
Novikov
,
S. P.
and
Veselov
,
A. P.
, “
Two-dimensional Schördinger operator: Inverse scattering transform and evolutional equations
,”
Physica D
18
(
1–3
),
267
273
(
1986
).
29.
Ohta
,
Y.
, “
Pfaffian solution for the Veselov-Novikov equation
,”
J. Phys. Soc. Jpn.
61
(
11
),
3928
3933
(
1992
).
30.
Pashaev
,
O. K.
and
Gurkan
,
Z. N.
, “
Abelian Chern-Simons vortices and holomorphic Burgers' hierarchy
,”
Theor. Math. Phys.
152
,
1017
1029
(
2007
).
31.
Shiota
,
T.
, “
Prym varieties and soliton equations
,” Infinite-dimensional Lie Algebras and Groups, Adv. Ser. Math. Phys. 7 (Luminy-Marseille,
1988
), pp.
407
448
(World Scientific Publishing, Teaneck, NJ, 1989).
32.
Taimanov
,
I. A.
and
Tsarev
,
S. P.
, “
Two-dimensional rational solitons and their blow-up via the Moutard transformation
,”
Theoret. Math. Phys.
157
(
2
),
1525
1541
(
2008
).
33.
Takasaki
,
K.
, “
Dispersionless Hirota equations of two-component BKP hierarchy
,”
Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
2
(
57
),
1
22
(
2006
);
34.
Veselov
,
A. P.
and
Novikov
,
S. P.
, “
Finite-zone, two-dimensional, potential Schördinger operators: Explicit formulas and evolution equations
,”
Dokl. Akad. Nauk SSSR
279
(
1
),
20
24
(
1984
).
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