Integrable dispersive chains and their multi-phase solutions. (English) Zbl 1420.37100

Marvan, Michal; Pavlov, Maxim V.

Lett. Math. Phys. 109, No. 5, 1219-1245 (2019).

The authors describe the connection of global solutions of the Mikhalev system
\[ a_{1,t} = a_{2,x} ,\,\,\,a_1 a_{2,x} + a_{1,y} = a_2 a_{1,x} + a_{2,t} \]
and multi-phase solutions for integrable two-dimensional dispersive systems associated with the energy-dependent Schrödinger operator.
The presented approach is universal, i.e., applicable to any integrable system whose Lax pair contains coefficients which depend polynomially on the spectral parameter.
The authors’ approach allows reconsidering multi-phase solutions of multi-component dispersive reductions from a unified point of view.

Reviewer: Rakib Efendiev (Baku)

MSC:

37K15	Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
37K05	Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Keywords:

integrable dispersive chains; three-dimensional quasilinear systems; multi-phase solutions

References:

[1]	Adler, VE; Shabat, AB, Model equation of the theory of solitons, Theor. Math. Phys., 153, 1373-1387, (2007) · Zbl 1138.37044
[2]	Alber, S.J.: Hamiltonian systems on the Jacobi varieties. In: Ratiu, T. (ed.) The Geometry of Hamiltonian Systems (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., 22, pp. 23-32. Springer, New York (1991) · Zbl 0753.58013
[3]	Alber, SJ, Associated integrable systems, J. Math. Phys., 32, 916-922, (1991) · Zbl 0850.70175
[4]	Alber, MS; Fedorov, YN, Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians, Inverse Probl., 17, 1-26, (2001) · Zbl 0988.35139
[5]	Antonowicz, M.; Fordy, A., A family of completely integrable multi-Hamiltonian systems, Phys. Lett. A, 122, 95-99, (1987)
[6]	Antonowicz, M.; Fordy, A., Coupled KdV equations with multi-Hamiltonian structures, Physica, 28D, 345-57, (1987) · Zbl 0638.35079
[7]	Antonowicz, M.; Fordy, A., Coupled Harry Dym equations with multi-Hamiltonian structures, J. Phys. A, 21, l269-75, (1988) · Zbl 0673.35088
[8]	Antonowicz, M.; Fordy, A., Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems, Commun. Math. Phys., 124, 465-486, (1989) · Zbl 0696.35172
[9]	Błaszak, M.; Marciniak, K., Stäckel systems generating coupled KdV hierarchies and their finite-gap and rational solutions, J. Phys. A, 41, 485202, (2008) · Zbl 1158.37025
[10]	Brezhnev, Yu.: Historical remarks to finite-gap integration theory: elementary treatment of the theory. arXiv:nlin/0504051
[11]	Chudnovsky, D.V., Chudnovsky, G.V.: Travaux de J. Drach (1919), in Classical Quantum Models and Arithmetic Problems, Lecture Notes in Pure and Applied Mathematics, vol. 92, pp. 445-458. CRC Press (Dekker, New York, 1984)
[12]	Dobrokhotov, SY; Maslov, VP; Mosolov, PP; Gurevich, BM; Sukhov, YM; Zlatoustov, VA; Matveev, VB; Markeev, AP; Zel’dovich, YB; Marchenko, VA; Khruslov, EYa, Sessions of the I.G. Petrovskii seminar on differential equations and mathematical problems of physics, Surv. Math. Sci., 30:6, 197-206, (1975)
[13]	Drach, J., Determination des cas de reduction de l’equation differentialle \(\frac{d^{2}y}{dx^{2}}=[\varphi (x)+h]y\), C. R. Acad. Sci., 168, 47-50, (1919) · JFM 47.0411.03
[14]	Drach, J., Sur l’integration par quadratures de l’equation differentialle \( \frac{d^{2}y}{dx^{2}}=[\varphi (x)+h]y\), C. R. Acad. Sci., 168, 337-340, (1919) · JFM 47.0412.01
[15]	Dubrovin, BA, Inverse problem for periodic finite-zoned potentials in the theory of scattering, Funct. Anal. Appl., 9, 61-62, (1975) · Zbl 0315.35072
[16]	Dubrovin, BA, Periodic problems for the Korteweg-de Vries equation in the class of finite band potentials, Funct. Anal. Appl., 9, 215-223, (1975) · Zbl 0358.35022
[17]	Dubrovin, BA; Novikov, SP, Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg-de Vries equation, Soviet Phys. JETP, 40, 1058-1063, (1974)
[18]	Dubrovin, BA; Novikov, SP, A periodic problem for the Korteweg-de Vries and Sturm-Liouville equations. Their connection with algebraic geometry, Dokl. Akad. Nauk SSSR, 219, 531-534, (1974) · Zbl 0312.35015
[19]	Dubrovin, BA; Matveev, VB; Novikov, SP, Non-linear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian varieties, Russian Math. Surv., 31, 59-146, (1976) · Zbl 0346.35025
[20]	Ferapontov, EV, Integration of weakly-nonlinear hydrodynamic systems in Riemann invariants, Phys. Lett. A, 158, 112-118, (1991)
[21]	Ferapontov, EV; Khusnutdinova, KR, On integrability of (2+1)-dimensional quasilinear systems, Commun. Math. Phys., 248, 187-206, (2004) · Zbl 1070.37047
[22]	Ferapontov, EV; Khusnutdinova, KR, The characterization of 2-component (2+1)-dimensional integrable systems of hydrodynamic type, J. Phys. A Math. Gen., 37, 2949-2963, (2004) · Zbl 1040.35042
[23]	Ferapontov, EV; Khusnutdinova, KR; Klein, C., On linear degeneracy of integrable quasilinear systems in higher dimensions, Lett. Math. Phys., 96, 5-35, (2011) · Zbl 1217.35114
[24]	Gibbons, J.; Tsarev, SP, Reductions of Benney’s equations, Phys. Lett. A, 211, 19-24, (1996) · Zbl 1072.35588
[25]	Gibbons, J.; Tsarev, SP, Conformal maps and reductions of the Benney equations, Phys. Lett. A, 258, 263-270, (1999) · Zbl 0936.35184
[26]	Its, AR; Matveev, VB, Hill’s operator with finitely many gaps, Funct. Anal. Appl., 9, 65-66, (1975) · Zbl 0318.34038
[27]	Its, AR; Matveev, VB, Schroödinger operators with finite-gap spectrum and \(N\)-soliton solutions of the Korteweg-de Vries equation, Theor. Math. Phys., 23, 343-355, (1975)
[28]	Its, A.R., Matveev, V.B.: A class of solutions of the Korteweg-de Vries equation, Problems in mathematical physics, No. 8 (Russian), pp. 70-92, 173. Izdat. Leningrad. Univ., Leningrad (1976) (in Russian)
[29]	Martinez Alonso, L., Schrödinger spectral problems with energy dependent potentials as sources of nonlinear Hamiltonian evolution equations, J. Math. Phys., 21, 2342-2349, (1980) · Zbl 0455.35111
[30]	Martinez Alonso, L.; Shabat, AB, Energy dependent potentials revisited: a universal hierarchy of hydrodynamic type, Phys. Lett. A, 299, 359-365, (2002) · Zbl 0996.37072
[31]	Martinez Alonso, L.; Shabat, AB, Towards a theory of differential constraints of a hydrodynamic hierarchy, J. Nonlinear Math. Phys., 10, 229-242, (2003) · Zbl 1055.35092
[32]	Martinez Alonso, L.; Shabat, AB, Hydrodynamic reductions and solutions of a universal hierarchy, Theor. Math. Phys., 140, 1073-1085, (2004) · Zbl 1178.37067
[33]	Matveev, VB, 30 years of finite-gap integration theory, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 366, 837, (2008) · Zbl 1153.37419
[34]	Mikhalëv, VG, On the Hamiltonian formalism for Korteweg-de Vries type hierarchies, Func. Anal. Appl., 26, 140-142, (1992) · Zbl 0790.58022
[35]	Novikov, SP, The periodic problem for the Korteweg-de Vries equation, Funct. Anal. Appl., 8, 236-246, (1974) · Zbl 0299.35017
[36]	Pavlov, MV, Integrable hydrodynamic chains, J. Math. Phys., 44, 4134-4156, (2003) · Zbl 1062.37078
[37]	Pavlov, MV, Hamiltonian formalism of weakly nonlinear systems in hydrodynamics, Theor. Math. Phys., 73, 1242-1245, (1987) · Zbl 0653.76005
[38]	Pavlov, MV, Integrable dispersive chains and energy dependent Schrödinger operator, J. Phys. A, 47, 295204, (2014) · Zbl 1304.35470
[39]	Shabat, AB, Universal solitonic hierarchy, J. Nonlinear Math. Phys., 12, 614-624, (2005) · Zbl 1362.35083
[40]	Shabat, AB, Symmetric polynomials and conservation laws, Vladikavkazskij matematicheskij zhurnal, 14, 83-94, (2012) · Zbl 1326.37039
[41]	Tsarev, SP, On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type, Soviet Math. Dokl., 31, 488-491, (1985) · Zbl 0605.35075
[42]	Tsarev, SP, The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method, Math. USSR Izvestiya, 37, 397-419, (1991) · Zbl 0796.76014