The complexification of the independent variables of nonlinear integrable evolution partial differential equations (PDEs) in two space dimensions, like the celebrated Kadomtsev-Petviashvili and Davey-Stewartson (DS) equations, yields nonlinear integrable equations in genuine 4 + 2, namely, in four real space dimensions (x1, x2, y1, y2) and two real time dimensions (t1, t2), as opposed to two complex space dimensions and one complex time dimension. The associated initial value problem for such equations, namely, the problem where the dependent variables are specified for all space variables at t1 = t2 = 0, can be solved via a non-local d-bar formalism. Here, the details of this formalism for the 4 + 2 DS system are presented. Furthermore, the linearised version of the 3 + 1 reduction of the 4 + 2 DS system is discussed. The construction of the nonlinear 3 + 1 reduction remains open, in spite of the fact that multi-soliton solutions for the 3 + 1 DS system already exist.
REFERENCES
1.
A. S.
Fokas
, A. R.
Its
, A. A.
Kapaev
, and V. Yu.
Novokshenov
, Painlevé Transcendents: The Riemann-Hilbert Approach
(AMS
, 2006
).2.
A. S.
Fokas
and M. J.
Ablowitz
, “The inverse scattering transform for the Benjamin-Ono equation—A pivot to multidimensional problems
,” Stud. Appl. Math.
68
, 1
–10
(1983
).3.
A. S.
Fokas
and B.
Fuchssteiner
, “The hierarchy of the Benjamin-Ono equation
,” Phys. Lett. A
86
, 341
–345
(1981
).4.
C. S.
Gardner
, J. M.
Greene
, M. D.
Kruskal
, and R. M.
Miura
, “Method for solving the Korteweg-deVries equation
,” Phys. Rev. Lett.
19
, 1095
–1097
(1967
).5.
A. R.
Its
, “The Riemann-Hilbert problem and integrable systems
,” Not. Am. Math. Soc.
50
, 1389
–1400
(2003
).6.
L. D.
Faddeev
, “The inverse problem in the quantum theory of scattering. II
,” Itogi Nauki Tekhniki. Ser. Sovrem. Probl. Mat.
3
, 93
–180
(1974
) (in Russian);L. D.
Faddeev
, [J. Sov. Math.
5
, 334
–396
(1976
) (in English)].7.
V. E.
Zakharov
and S. V.
Manakov
, “Soliton theory
,” Sov. Sci. Rev., Sect. A: Phys. Rev.
1
, 133
–190
(1979
).8.
S. V.
Manakov
, “The inverse scattering transform for the time-dependent Schrödinger equation and Kadomtsev-Petviashvili equation
,” Physica D
3
, 420
–427
(1981
).9.
V. E.
Zakharov
and S. V.
Manakov
, “Construction of higher-dimensional nonlinear integrable systems and of their solutions
,” Funkts. Anal. Ego Prilozh.
19
, 11
–25
(1985
) (in Russian);V. E.
Zakharov
and S. V.
Manakov
, [Funct. Anal. Appl.
19
, 89
–101
(1985
) (in English)].10.
H.
Segur
, “Comments on inverse scattering for the Kadomtsev-Petviashvili equation
,” in Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems
, edited by M.
Tabor
and Y. M.
Treve
(La Jolla Institute
, 1981
);11.
A. S.
Fokas
and M. J.
Ablowitz
, “On the inverse scattering of the time-dependent Schrödinger equation and the associated Kadomtsev-Petviashvili (I) equation
,” Stud. Appl. Math.
69
, 211
–228
(1983
).12.
M. J.
Ablowitz
, D.
Bar Yaacov
, and A. S.
Fokas
, “On the inverse scattering transform for the Kadomtsev-Petviashvili equation
,” Stud. Appl. Math.
69
, 135
–143
(1983
).13.
A. S.
Fokas
, “Inverse scattering of first-order systems in the plane related to nonlinear multidimensional equations
,” Phys. Rev. Lett.
51
, 3
–6
(1983
).14.
A. S.
Fokas
and M. J.
Ablowitz
, “Method of solution for a class of multidimensional nonlinear evolution equations
,” Phys. Rev. Lett.
51
, 7
–10
(1983
).15.
R.
Beals
and R.
Coifman
, “Scattering, transformations spectrales et équations d’évolution non linéaires
,” Sémin. Équations aux Dérivées Partielles (Polytech.)
Exp. No. 22, 1
–9
(1980–1981
);R.
Beals
and R.
Coifman
, “Scattering, transformations spectrales et équations d’évolution non linéaires. II
,” Sémin. Équations aux Dérivées Partielles (Polytech.)
Exp. No. 21, 1
–8
(1981–1982
).16.
R.
Beals
and R.
Coifman
, “Linear spectral problems, non-linear equations and the -method
,” Inverse Probl.
5
, 87
–130
(1989
).17.
A. S.
Fokas
and L. Y.
Sung
, “On the solvability of the N-wave, the Davey-Stewartson and the Kadomtsev-Petviashvili equations
,” Inverse Probl.
8
, 673
–708
(1992
).18.
X.
Zhou
, “Inverse scattering transform for the time dependent Schrödinger equation with applications to the KPI equation
,” Commun. Math. Phys.
128
, 551
–564
(1990
).19.
A. S.
Fokas
, “Nonlinear Fourier transforms, integrability and nonlocality in multidimensions
,” Nonlinearity
20
, 2093
–2113
(2007
).20.
A. S.
Fokas
, “Nonlinear Fourier transforms and integrability in multidimensions
,” Contemp. Math.
458
, 71
–80
(2008
).21.
A. S.
Fokas
, “Integrable nonlinear evolution partial differential equations in 4 + 2 and 3 + 1 dimensions
,” Phys. Rev. Lett.
96
, 190201-1
–190201-4
(2006
).22.
A. S.
Fokas
, “Kadomtsev-Petviashvili equation revisited and integrability in 4 + 2 and 3 + 1
,” Stud. Appl. Math.
122
, 347
–359
(2009
).23.
A. S.
Fokas
, “The D-bar method, inversion of certain integrals and integrability in 4 + 2 and 3 + 1 dimensions
,” J. Phys. A: Math. Theor.
41
, 344006-1
–344006-16
(2008
).24.
M.
Dimakos
and A. S.
Fokas
, “Davey-Stewartson type equations in 4 + 2 and 3 + 1 possessing soliton solutions
,” J. Math. Phys.
54
, 081504-1
–081504-30
(2013
).25.
Z.-Z.
Yang
and Z.-Y.
Yan
, “Symmetry groups and exact solutions of new (4 + 1)-dimensional Fokas equation
,” Commun. Theor. Phys.
51
, 876
–880
(2009
).26.
S.
Zhang
and H.-Q.
Zhang
, “Fractional sub-equation method and its applications to nonlinear fractional PDEs
,” Phys. Lett. A
375
, 1069
–1073
(2011
).27.
Y.
He
, “Exact solutions for (4 + 1)-dimensional nonlinear Fokas equation using extended F-expansion method and its variant
,” Math. Probl. Eng.
2014
, 972519-1
–972519-11
.28.
S.
Zhang
and M.
Chen
, “Painlevé integrability and new exact solutions of the (4 + 1)-dimensional Fokas equation
,” Math. Probl. Eng.
2015
, 367425-1
–367425-7
.29.
S.
Zhang
, C.
Tian
, and W.-Y.
Qian
, “Bilinearization and new multisoliton solutions for the (4 + 1)-dimensional Fokas equation
,” Pramana
86
, 1259
–1267
(2016
).30.
M. O.
Al-Amr
and S.
El-Ganaini
, “New exact traveling wave solutions of the (4 + 1)-dimensional Fokas equation
,” Comput. Math. Appl.
74
, 1274
–1287
(2017
).31.
L.
Cheng
and Y.
Zhang
, “Lump-type solutions for the (4 + 1)-dimensional Fokas equation via symbolic computations
,” Mod. Phys. Lett. B
31
, 1750224-1
–1750224-9
(2017
).32.
C. A.
Gómez
, H.
Garzón G.
, and J. C.
Hernández R.
, “On exact solutions for (4 + 1)-dimensional Fokas equation with variable coefficients
,” Adv. Stud. Theor. Phys.
11
, 765
–771
(2017
).33.
A. S.
Fokas
, “Soliton multidimensional equations and integrable evolutions preserving Laplace’s equation
,” Phys. Lett. A
372
, 1277
–1279
(2008
).34.
A. S.
Fokas
and D. A.
Pinotsis
, “The Dbar formalism for certain linear non-homogeneous elliptic PDEs in two dimensions
,” Eur. J. Appl. Math.
17
, 323
–346
(2006
).35.
M. J.
Ablowitz
, A. S.
Fokas
, and M. D.
Kruskal
, “Note on solutions to a class of nonlinear singular integro-differential equations
,” Phys. Lett. A
120
, 215
–218
(1987
).36.
E. A.
Kuznetsov
, M. D.
Spector
, and V. E.
Zakharov
, “Formation of singularities on the free surface of an ideal fluid
,” Phys. Rev. E
49
, 1283
–1290
(1994
).37.
E. A.
Kuznetsov
, M. D.
Spector
, and V. E.
Zakharov
, “Surface singularities of ideal fluid
,” Phys. Lett. A
182
, 387
–393
(1993
).38.
P.
Constantin
, P. D.
Lax
, and A.
Majda
, “A simple one-dimensional model for the three-dimensional vorticity equation
,” Commun. Pure Appl. Math.
38
, 715
–724
(1985
).39.
M. J.
Ablowitz
and A. S.
Fokas
, Complex Variables, Introduction and Applications
, 2nd ed., Cambridge Texts in Applied Mathematics (Cambridge University Press
, 2003
).© 2018 Author(s).
2018
Author(s)
You do not currently have access to this content.