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On the dispersionless Davey-Stewartson system: Hamiltonian vector field Lax pair and relevant nonlinear Riemann-Hilbert problem for dDS-II system. (English) Zbl 1437.37085

The main novelty of the paper is a new dispersionless system derived from the commutation condition of Lax pair of one-parameter vector fields. It is worth mentioning that the study of these systems, also called heavenly equations, is a new and promising direction of research in the modern theory of integrable dynamical systems.
The paper is well structured. It consists of three parts. The first one is a very detailed review of the Manakov-Santini inverse scattering transform method and its approach to the integrable PDEs of hydrodynamic type, such as the second heavenly Plebański equation, the Dunajski dynamical system, the dispersionless KP system and some others.
The second part contains a derivation of the dispersionless Davey-Stewartson system as a semiclassical limit of the Davey-Stewartson system. An analysis of the Hamiltonian structure of the obtained system is done. The author shows that the dispersionless Davey-Stewartson system is a result of the commutation condition of the one-parameter Hamiltonian vector fields.
The last part contains a construction of the Riemann-Hilbert problem for the obtained dispersionless Davey-Stewartson system.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35Q15 Riemann-Hilbert problems in context of PDEs
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References:

[1] Davey, A.; Stewartson, K., On three dimensional packets of surface waves, Proc. R. Soc. A, 338, 1613, 101-110 (1974) · Zbl 0282.76008 · doi:10.1098/rspa.1974.0076
[2] Ablowitz, M.J., Clarkson, P.A.: Solitons, nonlinear evolution equations and Inverse Scattering. London Math. Society Lecture Note (1991) · Zbl 0762.35001
[3] Anker, D.; Freeman, Nc, On the soliton solutions of Davey-Stewartson equation for long waves, Proc. R. Soc. Lond. A, 360, 529-540 (1978) · Zbl 0384.76016 · doi:10.1098/rspa.1978.0083
[4] Arkadiev, Va; Pogrebkov, Ak; Polivanov, Mc, Closed string-like solutions of the Davey-Stewartson equation, Inverse Probl., 5, L1-L6 (1989) · Zbl 0694.35143 · doi:10.1088/0266-5611/5/1/001
[5] Arkadiev, Va; Pogrebkov, Ak; Polivanov, Mc, Inverse scattering transform method and soliton solutions for Davey-Stewartson II equation, Physica, 36D, 189-197 (1989) · Zbl 0698.35150
[6] Nakanura, A., Explode-decay mode lump solitons of a two-dimensional nonlinear schrödinger equation, Phys. Lett. A, 88, 2, 55-56 (1982) · doi:10.1016/0375-9601(82)90587-4
[7] Nakanura, A., Exact explode-decay soliton solutions of a 2-dimensional nonlinear schrödinger equation, J. Phys. Soc. Jpn., 51, 19-20 (1983) · doi:10.1143/JPSJ.51.19
[8] Satsuma, J.; Ablowitz, Mj, Two-dimensional lumps in nonlinear dispersive systems, J. Math. Phys., 20, 1496-1503 (1979) · Zbl 0422.35014 · doi:10.1063/1.524208
[9] Boiti, M.; Leon, J.; Martina, L.; Pempinelli, F., Scattering of localized solitons in the plane, Phys. Lett., 132A, 432-439 (1988) · doi:10.1016/0375-9601(88)90508-7
[10] Champagne, B.; Winternitz, P., On the infinite-dimensional symmetry group of the Davey-Stewartson equations, J. Math. Phys, 29, 1-8 (1988) · Zbl 0643.35097 · doi:10.1063/1.528173
[11] Omote, M., Infinite-dimensional symmetry algebras and an infinite number of conserved quantities of the (2+1)-dimensional Davey-Stewartson equation, J. Math. Phys., 29, 12, 2599-2603 (1988) · Zbl 0784.35102 · doi:10.1063/1.528102
[12] Tajiri, M., Similarity reductions of the one and two dimensional nonlinear schrödinger equations, J. Phys. Soc. Jpn., 52, 1908-1917 (1983) · doi:10.1143/JPSJ.52.1908
[13] Ganesan, S.; Lakshmanan, M., Singularity-structure analysis and Hirota’s bilinearisation of the Davey-Stewartson equation, J. Phys. A Gen., 103, 20, L1143-L1147 (1987) · Zbl 0656.35141 · doi:10.1088/0305-4470/20/17/003
[14] Fokas, As; Santini, Pm, Recursion operators and bi-Hamiltonian structures in multidimensions. II, Commun. Math. Phys., 116, 3, 449-474 (1988) · Zbl 0706.35129 · doi:10.1007/BF01229203
[15] Santini, Pm; Fokas, As, Recursion operators and bi-Hamiltonian structures in multidimensions. I, Commun. Math. Phys., 115, 375-419 (1988) · Zbl 0674.35074 · doi:10.1007/BF01218017
[16] Gardner, Cs; Greene, Jm; Kruskal, Md; Miura, Rm, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19, 1095-1097 (1967) · Zbl 1061.35520 · doi:10.1103/PhysRevLett.19.1095
[17] Lax, Pd, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math., 21, 467-490 (1968) · Zbl 0162.41103 · doi:10.1002/cpa.3160210503
[18] Zakharov, Ve; Shabat, Ab, Exact theory of two-dimensional self- focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP, 34, 62-69 (1972)
[19] Ablowitz, Mj; Kaup, Dj; Newell, Ac; Segur, H., Method for solving the sine-Gordon equation, Phys. Rev. Lett., 30, 1262-1264 (1973) · doi:10.1103/PhysRevLett.30.1262
[20] Ablowitz, Mj; Kaup, Dj; Newell, Ac; Segur, H., The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math., 53, 249-315 (1974) · Zbl 0408.35068 · doi:10.1002/sapm1974534249
[21] Zakharov, Ve; Manakov, Sv; Novikov, Sp; Pitaevsky, Lp, Theory of Solitons (1984), New York: Plenum, New York · Zbl 0598.35002
[22] Zakharov, Ve; Shabat, Ab, Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II, Func. Anal. Appl., 13, 3, 166-174 (1979) · Zbl 0448.35090 · doi:10.1007/BF01077483
[23] Manakov, Sv; Santini, Pm, Inverse scattering problem for vector fields and the Cauchy problem for the heavenly equation, Phys. Lett. A, 359, 613-619 (2006) · Zbl 1236.37042 · doi:10.1016/j.physleta.2006.07.011
[24] Manakov, Sv; Santini, Pm, The Cauchy problem on the plane for the dispersionless Kadomtsev-Petviashvili equation, JETP Lett., 83, 462-466 (2006) · doi:10.1134/S0021364006100080
[25] Timman, R.: Unsteady motion in transonic flow. Symposium Transsonicum, Aachen 1962 ed K. Oswatitsch. Springer, Berlin, pp. 394-401 (1962) · Zbl 0116.17502
[26] Zobolotskaya, Ea; Kokhlov, Rv, Quasi-plane waves in the nonlinear acoustics of confined beams, Sov. Phys. Acoust., 15, 35-40 (1969)
[27] Kodama, Y., Gibbons, J.: Integrability of the Dispersionless KP Hierarchy. In: Proceedings of the 4th Workshop on Nonlinear and Turbulent Processes in Physics (1990) · Zbl 0751.58037
[28] Manakov, Sv; Santini, Pm, On the solution of the dKP equation: the nonlinear Riemann-Hilbert problem, longtime behaviour, implicit solutions and wave breaking, J. Phys. A Math. Theor., 41, 055204 (2008) · Zbl 1136.35083 · doi:10.1088/1751-8113/41/5/055204
[29] Manakov, Sv; Santini, Pm, On the solutions of the second heavenly and Pavlov equations, J. Phys. A Math. Theor., 42, 404013 (2009) · Zbl 1184.35279 · doi:10.1088/1751-8113/42/40/404013
[30] Plebanski, F., Some solutions of complex Einstein equations, J. Math. Phys., 16, 2395-2402 (1975) · doi:10.1063/1.522505
[31] Manakov, Sv; Santini, Pm, The dispersionless 2D Toda equation: dressing, Cauchy problem, longtime behaviour, implicit solutions and wave breaking, J. Phys. A Math. Theor., 42, 095203 (2009) · Zbl 1165.37039 · doi:10.1088/1751-8113/42/9/095203
[32] Boyer, C.; Finley, Jd, Killing vectors in self-dual, Euclidean Einstein spaces, J. Math. Phys., 23, 1126-1128 (1982) · Zbl 0484.53051 · doi:10.1063/1.525479
[33] Gegenberg, Jd; Das, A., Stationary Riemaniann space-times with self-dual curvature, Gen. Rel. Grav., 16, 817-829 (1984) · Zbl 0545.53039 · doi:10.1007/BF00762935
[34] Hitchin, Nj; Doebner, Hd; Weber, T., Complex manifolds and Einstein’s equations, Twistor Geometry and Nonlinear Systems. Lecture Notes in Mathematics (1982), Berlin: Springer, Berlin
[35] Jones, Pe; Tod, Kp, Minitwistor spaces and Einstein-Weyl spaces, Class. Quantum Grav., 2, 565-577 (1985) · Zbl 0575.53042 · doi:10.1088/0264-9381/2/4/021
[36] Ward, Rs, Einstein-Weyl spaces and SU \(( \infty )\) Toda fields, Class. Quantum Grav., 7, L95-L98 (1990) · Zbl 0687.53044 · doi:10.1088/0264-9381/7/4/003
[37] Mineev-Weinstein, M.; Wigmann, P.; Zabrodin, A., Integrable structure of interface dynamics, Phys. Rev. Lett., 84, 22, 5106-5109 (2000) · doi:10.1103/PhysRevLett.84.5106
[38] Wigmann, P.; Zabrodin, A., Conformal maps and integrable hierarchies, Comm. Math. Phys., 213, 3, 523-538 (2000) · Zbl 0973.37042 · doi:10.1007/s002200000249
[39] Krichever, I.; Marshakov, A.; Zabrodin, A., Integrable structure of the Dirichlet boundary problem in multiply-connected domains, Commun. Math. Phys., 259, 1, 1-44 (2005) · Zbl 1091.37019 · doi:10.1007/s00220-005-1387-5
[40] Yi, G.; Santini, Pm, The inverse spectral transform for the Dunajski hierarchy and some of its reductions, I: Cauchy problem and longtime behavior of solutions, J. Phys. A Math. Theor., 48, 1533, 215203-457 (2015) · Zbl 1331.35313 · doi:10.1088/1751-8113/48/21/215203
[41] Dunajski, M., Anti-self-dual four manifolds with a parallel real spinor, Proc. R. Soc. A, 458, 1205-1222 (2002) · Zbl 1006.53040 · doi:10.1098/rspa.2001.0918
[42] Dubrovin, B.; Grava, T.; Klein, C., On universality of critical behavior in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the tritronquee solution to the Painleve-I equation, J. Nonlinear Sci., 19, 1, 57-94 (2009) · Zbl 1220.37048 · doi:10.1007/s00332-008-9025-y
[43] Klein, C.; Roidot, K., Numerical Study of the semiclassical limit of the Davey-Stewartson II equations, Nonlinearity, 27, 9, 739-740 (2014) · Zbl 1301.35153 · doi:10.1088/0951-7715/27/9/2177
[44] Assainova, O., Klein, C., Mclaughlin, K., Miller, P.: A study of the direct spectral transform for the defocusing Davey-Stewartson ii equation in the semiclassical limit, arXiv:1710.03429v1 (2017) · Zbl 1420.35339
[45] Konopelchenko, Bg, Quasiclassical generalized Weierstrass representation and dispersionless DS equation, J. Phys. A Math. Theor., 40, F995 (2007) · Zbl 1132.53009 · doi:10.1088/1751-8113/40/46/F03
[46] Madelung, E., Quantum theory in hydrodynamic form, Zeitschr. Phys., 40, 322-326 (1926) · JFM 52.0969.06 · doi:10.1007/BF01400372
[47] Yi, G.: On the dispersionless Davey-Stewartson hierarchy: Zakharov-Shabat equations, twistor structure and Lax-Sato formalism, arXiv:1812.10220v2 (2018)
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