Integrable nonlinear evolution partial differential equations in \(4+2\) and \(3+1\) dimensions. (English) Zbl 1228.35199

Summary: The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schrödinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.


35Q53 KdV equations (Korteweg-de Vries equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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[1] DOI: 10.1103/PhysRevLett.19.1095
[2] P.D. Lax, Comm. Pure Appl. Math. 21 pp 467– (1968) ISSN: http://id.crossref.org/issn/0079-8185 · Zbl 0162.41103
[3] V.E. Zakharov, Sov. Phys. JETP 34 pp 62– (1972) ISSN: http://id.crossref.org/issn/0038-5646
[4] A.S. Fokas, Stud. Appl. Math. 69 pp 211– (1983) ISSN: http://id.crossref.org/issn/0022-2526 · Zbl 0528.35079
[5] M.J. Ablowitz, Stud. Appl. Math. 69 pp 135– (1983) ISSN: http://id.crossref.org/issn/0022-2526 · Zbl 0527.35080
[6] DOI: 10.1088/0266-5611/8/5/002 · Zbl 0768.35069
[7] DOI: 10.1088/0266-5611/5/2/002 · Zbl 0685.35080
[8] P. Deift, in: Important Developments in Soliton Theory (1993)
[9] DOI: 10.1016/0375-9601(88)90508-7
[10] DOI: 10.1103/PhysRevLett.63.1329
[11] DOI: 10.1007/BF00750662 · Zbl 0807.35138
[12] M.J. Ablowitz, in: Introduction and Applications of Complex Variables (2003) · Zbl 1088.30001
[13] DOI: 10.1007/BF01078388 · Zbl 0597.35115
[14] DOI: 10.1103/PhysRevLett.53.1
[15] DOI: 10.1016/0370-1573(89)90024-0
[16] DOI: 10.1063/1.530377 · Zbl 0767.35087
[17] DOI: 10.1088/0305-4470/29/7/023 · Zbl 0914.35118
[18] P.G. Estevez, J. Nonlinear Math. Phys. 11 pp 164– (2004) ISSN: http://id.crossref.org/issn/1402-9251 · Zbl 1067.35086
[19] DOI: 10.1007/BF01197753 · Zbl 0429.35063
[20] DOI: 10.2307/1971105 · Zbl 0462.35079
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