zbMATH — the first resource for mathematics

Algebro-geometric solutions of the coupled modified Korteweg-de Vries hierarchy. (English) Zbl 1304.37046
Based on the stationary zero-curvature equation and the Lenard recursion equations, the authors derive the coupled modified Korteweg-de Vries (cmKdV) hierarchy associated with a \(3 \times 3\) matrix spectral problem. Resorting to the Baker-Akhiezer function and the characteristic polynomial of the Lax matrix for the cmKdV hierarchy, the authors introduce a trigonal curve with three infinite points and two algebraic functions carrying the data of the divisor. The asymptotic properties of the Baker-Akhiezer function and the two algebraic functions are studied near three infinite points on the trigonal curve. Algebro-geometric solutions of the cmKdV hierarchy are obtained in terms of the Riemann theta function.

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
14H70 Relationships between algebraic curves and integrable systems
35Q53 KdV equations (Korteweg-de Vries equations)
Full Text: DOI
[1] Ablowitz, M. J.; Clarkson, P. A., Solitons, nonlinear evolution equation and inverse scattering, (1991), Cambridge University Press Cambridge · Zbl 0762.35001
[2] Airault, H.; McKean, H. P.; Moser, J., Rational and elliptic solutions of the Korteweg-de Vries equation a related many-body problem, Comm. Pure Appl. Math., 30, 95-148, (1977) · Zbl 0338.35024
[3] Alber, M. S.; Fedorov, Y. N., Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians, Inverse Problems, 17, 1017-1042, (2001) · Zbl 0988.35139
[4] Athorne, C.; Fordy, A., Generalised KdV and mkdv equations associated with symmetric spaces, J. Phys. A, 20, 1377-1386, (1987) · Zbl 0642.58034
[5] Baldwin, S.; Eilbeck, J. C.; Gibbons, J.; Ônishi, Y., Abelian functions for cyclic trigonal curves of genus 4, J. Geom. Phys., 58, 450-467, (2008) · Zbl 1211.37082
[6] Belokolos, E. D.; Bobenko, A. I.; Enol’skii, V. Z.; Its, A. R.; Mateveev, V. B., Algebro-geometric approach to nonlinear integrable equations, (1994), Springer Berlin · Zbl 0809.35001
[7] Brezhnev, Y. V., Finite-band potentials with trigonal curves, Theoret. Math. Phys., 133, 1657-1662, (2002) · Zbl 1087.34561
[8] Buchstaber, V. M.; Enolskii, V. Z.; Leykin, D. V., Uniformization of Jacobi varieties of trigonal curves and nonlinear differential equations, Funct. Anal. Appl., 34, 159-171, (2000) · Zbl 0978.58012
[9] Cao, C. W.; Wu, Y. T.; Geng, X. G., Relation between the Kadomtsev-Petviashvili equation and the confocal involutive system, J. Math. Phys., 40, 3948-3970, (1999) · Zbl 0947.35138
[10] Date, E.; Tanaka, S., Periodic multi-soliton solutions of Korteweg-de Vries equation and Toda lattice, Progr. Theoret. Phys. Suppl., 59, 107-125, (1976)
[11] Dickson, R.; Gesztesy, F.; Unterkofler, K., Algebro-geometric solutions of the Boussinesq hierarchy, Rev. Math. Phys., 11, 823-879, (1999) · Zbl 0971.35065
[12] Dubrovin, B. A., Theta functions and nonlinear equations, Russian Math. Surveys, 36, 11-92, (1981) · Zbl 0549.58038
[13] Dubrovin, B. A., Matrix finite-gap operators, Rev. Sci. Tech., 23, 33-78, (1983)
[14] Eilbeck, J. C.; Enolski, V. Z.; Matsutani, S.; Ônishi, Y.; Previato, E., Abelian functions for purely trigonal curves of genus three, Int. Math. Res. Not. IMRN, (2007), Art. ID rnm 140, 38 pp · Zbl 1210.14032
[15] Farkas, H. M.; Kra, I., Riemann surfaces, (1992), Springer New York · Zbl 0764.30001
[16] Geng, X. G.; Cao, C. W., Decomposition of the \((2 + 1)\)-dimensional gardner equation and its quasi-periodic solutions, Nonlinearity, 14, 1433-1452, (2001) · Zbl 1160.37405
[17] Geng, X. G.; Dai, H. H.; Zhu, J. Y., Decomposition of the discrete Ablowitz-Ladik hierarchy, Stud. Appl. Math., 118, 281-312, (2007)
[18] Geng, X. G.; Wu, L. H.; He, G. L., Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions, Phys. D, 240, 1262-1288, (2011) · Zbl 1223.37093
[19] Geng, X. G.; Wu, L. H.; He, G. L., Quasi-periodic solutions of the Kaup-Kupershmidt hierarchy, J. Nonlinear Sci., 23, 527-555, (2013) · Zbl 1309.37070
[20] Geng, X. G.; Xue, B., A three-component generalization of Camassa-Holm equation with N-peakon solutions, Adv. Math., 226, 827-839, (2011) · Zbl 1207.35254
[21] Gesztesy, F.; Ratneseelan, R., An alternative approach to algebro-geometric solutions of the AKNS hierarchy, Rev. Math. Phys., 10, 345-391, (1998) · Zbl 0974.35107
[22] Griffiths, P.; Harris, J., Principles of algebraic geometry, (1994), Wiley New York · Zbl 0836.14001
[23] Hirota, R., “molecule solution” of coupled modified KdV equation, J. Phys. Soc. Jpn., 66, 2530-2532, (1997) · Zbl 0944.35074
[24] Iwao, M.; Hirota, R., Soliton solutions of a coupled modified KdV equation, J. Phys. Soc. Jpn., 66, 577-588, (1997) · Zbl 0946.35078
[25] Krichever, I. M., Algebraic-geometric construction of the zaharov-sabat equations and their periodic solutions, Soviet Math. Dokl., 17, 394-397, (1976) · Zbl 0361.35007
[26] Krichever, I. M., Integration of nonlinear equations by the methods of algebraic geometry, Funct. Anal. Appl., 11, 12-26, (1977) · Zbl 0368.35022
[27] Li, L. C.; Nenciu, I., The periodic defocusing Ablowitz-Ladik equation and the geometry of Floquet CMV matrices, Adv. Math., 231, 3330-3388, (2012) · Zbl 1300.37047
[28] Ma, Y. C.; Ablowitz, M. J., The periodic cubic Schrödinger equation, Stud. Appl. Math., 65, 113-158, (1981) · Zbl 0493.35032
[29] Matveev, V. B.; Smirnov, A. O., On the Riemann theta function of a trigonal curve and solutions of the Boussinesq and KP equations, Lett. Math. Phys., 14, 25-31, (1987) · Zbl 0641.35061
[30] Matveev, V. B.; Smirnov, A. O., Symmetric reductions of the Riemann-function and some of their applications to the Schrödinger and Boussinesq equations, Amer. Math. Soc. Transl., 157, 227-237, (1993) · Zbl 0807.35137
[31] McKean, H. P., Integrable systems and algebraic curves, (Grmela, M.; Marsden, J. E., Global Analysis, Lecture Notes in Math., vol. 755, (1979), Springer Berlin), 83-200 · Zbl 0449.35080
[32] Miller, P. D.; Ercolani, N. N.; Krichever, I. M.; Levermore, C. D., Finite genus solutions to the Ablowitz-Ladik equations, Comm. Pure Appl. Math., 48, 1369-1440, (1995) · Zbl 0869.34065
[33] Miura, R. M., Korteweg-de Vries equation and generalizations I. A remarkable explicit nonlinear transformation, J. Math. Phys., 9, 1202-1204, (1968) · Zbl 0283.35018
[34] Mumford, D., Tata lectures on theta I, II, (1984), Birkhäuser Boston
[35] Ônishi, Y., Abelian functions for trigonal curves of degree four and determinantal formulae in purely trigonal case, Int. J. Math., 20, 427-441, (2009) · Zbl 1222.14066
[36] Previato, E., The Calogero-Moser-krichever system and elliptic Boussinesq solitons, (Harnad, J.; Marsden, J. E., Hamiltonian Systems, Transformation Groups and Spectral Transform Methods, (1990), CRM Montreal), 57-67 · Zbl 0749.35047
[37] Previato, E., Monodromy of Boussinesq elliptic operators, Acta Appl. Math., 36, 49-55, (1994) · Zbl 0837.33012
[38] Previato, E.; Verdier, J. L., Boussinesq elliptic solitons: the cyclic case, (Ramanan, S.; Beauville, A., Proc. Indo-French Conf. on Geometry, Dehli, (1993), Hindustan Book Agency Delhi), 173-185 · Zbl 0844.14016
[39] Smirnov, A. O., A matrix analogue of Appell’s theorem and reductions of multidimensional Riemann theta-functions, Math. USSR Sb., 61, 379-388, (1988) · Zbl 0677.33002
[40] Tsuchida, T.; Wadati, M., The coupled modified Korteweg-de Vries equations, J. Phys. Soc. Jpn., 67, 1175-1187, (1998) · Zbl 0973.35170
[41] Wadati, M., The modified Korteweg-de Vries equation, J. Phys. Soc. Jpn., 34, 1289-1296, (1973) · Zbl 1334.35299
[42] Yajima, N.; Oikawa, M., A class of exactly solvable nonlinear evolution equations, Progr. Theoret. Phys., 54, 1576-1577, (1975) · Zbl 1079.37512
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.