The Whitham equations revisited. (English) Zbl 0887.14011
The paper reviews the theory of the Whitham equations being developed in the last decade for the completely integrable partial differential equations and their solutions in terms of abelian functions of algebraic curves. The authors “attempt to organize a subset of information in a coherent and more or less rigorous manner, while clarifying various formulas and assertions in the literature”. The paper contains necessary facts from the theory of Riemann surfaces. The hyperelliptic case (KdV equation) and its extension to the \(KP\)-equation are discussed. The principal part of the paper devoted to the square eigenfunctions approach to the averaging \((\psi \psi^*, \psi\) is the Baker-Akhiezer function). The connections to symplectic geometry, Landau-Ginzburg theory, Seiberg-Witten theory are pointed out. The paper presents a valuable contribution to the subject.
Reviewer: V.Z.Enol’skij (Kyïv)
MSC:
| 14H05 | Algebraic functions and function fields in algebraic geometry |
| 30F30 | Differentials on Riemann surfaces |
| 14H55 | Riemann surfaces; Weierstrass points; gap sequences |
| 14H40 | Jacobians, Prym varieties |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
Keywords:
\(KP\)-equation; KdV equation; Whitham equations; completely integrable partial differential equations; abelian functions of algebraic curves; Riemann surfaces; Baker-Akhiezer functionReferences:
| [1] | DOI: 10.1007/BF02099527 · Zbl 0756.35074 · doi:10.1007/BF02099527 |
| [2] | DOI: 10.1016/0960-0779(94)E0096-8 · Zbl 1080.37591 · doi:10.1016/0960-0779(94)E0096-8 |
| [3] | DOI: 10.1142/S0217732394002355 · Zbl 1021.81782 · doi:10.1142/S0217732394002355 |
| [4] | Baker H., Abelian functions (1897) |
| [5] | Belokolos E., Algebro-geometric approach to nonlinear integrable equations (1994) · Zbl 0809.35001 |
| [6] | Bilal A., hep-th 9601007 |
| [7] | DOI: 10.1137/0152052 · Zbl 0757.34014 · doi:10.1137/0152052 |
| [8] | Bonelli G., hep-th 9605090 |
| [9] | Carroll R., Topics in soliton theory (1991) · Zbl 0777.35072 |
| [10] | DOI: 10.1080/00036819308840162 · Zbl 0812.35119 · doi:10.1080/00036819308840162 |
| [11] | Carroll, R. 1995.Proc. First World Cong. Nonlinear Analysts, Vol. 1, 241–252. Berlin: deGruyter. |
| [12] | Carroll R., Jour. Phys. A 28 pp 6373– (1995) |
| [13] | DOI: 10.1007/BF02430644 · Zbl 0814.35117 · doi:10.1007/BF02430644 |
| [14] | Carroll R., World Scientific 4 pp 24– (1995) |
| [15] | Carroll R., solv-int 9511009 |
| [16] | Carroll R., solv-int 9606005, Proc. Second World Congress Nonlinear Analysts (1996) |
| [17] | Carroll R., hep-th 9607219, Mod. Phys. Lett. (1996) |
| [18] | Carroll R., Some kernels on a Riemannn surface (1996) |
| [19] | Carroll R., Lecture notes on Kdv, KP, Toda, TFT, susy, integrability, Riemann surfaces, matrix models,LG and SW theories, etc., 542 pages (1995) |
| [20] | Carroll R., ep-th 9610216 |
| [21] | Cherednik I., Basic methods of solition theory (1996) · Zbl 0909.35002 |
| [22] | D’Hoker E., hep-th 9609041, 9609145, and 9610156 |
| [23] | DOI: 10.1016/0550-3213(91)90129-L · doi:10.1016/0550-3213(91)90129-L |
| [24] | DOI: 10.1016/0550-3213(95)00609-5 · Zbl 0996.37507 · doi:10.1016/0550-3213(95)00609-5 |
| [25] | DOI: 10.1070/RM1989v044n06ABEH002300 · Zbl 0712.58032 · doi:10.1070/RM1989v044n06ABEH002300 |
| [26] | Dubrovin B., Dokl. Akad. Nauk. SSSR 27 pp 665– (1983) |
| [27] | DOI: 10.1070/RM1981v036n02ABEH002596 · Zbl 0549.58038 · doi:10.1070/RM1981v036n02ABEH002596 |
| [28] | Dubrovin B., Math. Phys. Rev. 3 pp 1– (1982) |
| [29] | Dubrovin B., Math. Phys. Rev. 9 pp 3– (1991) |
| [30] | DOI: 10.1070/IM1982v019n02ABEH001418 · Zbl 0501.14016 · doi:10.1070/IM1982v019n02ABEH001418 |
| [31] | Dubrovin B., Lecture notes in math. 1620 pp 120– (1995) |
| [32] | DOI: 10.1016/0550-3213(92)90137-Z · doi:10.1016/0550-3213(92)90137-Z |
| [33] | DOI: 10.1007/BF02099286 · Zbl 0753.58039 · doi:10.1007/BF02099286 |
| [34] | DOI: 10.1142/S0217732396000151 · Zbl 1022.81764 · doi:10.1142/S0217732396000151 |
| [35] | DOI: 10.1007/978-1-4612-2034-3 · doi:10.1007/978-1-4612-2034-3 |
| [36] | Fay J., Springer Lect. Notes Math. 352 (1973) · Zbl 0281.30013 · doi:10.1007/BFb0060090 |
| [37] | DOI: 10.1002/cpa.3160330605 · Zbl 0454.35080 · doi:10.1002/cpa.3160330605 |
| [38] | Forster O., Lectures on Riemann surfaces (1992) |
| [39] | Forest M., Stud. Appl. Math. 68 pp 11– (1983) · Zbl 0541.35071 · doi:10.1002/sapm198368111 |
| [40] | DOI: 10.1142/S0217979292001067 · Zbl 0801.35120 · doi:10.1142/S0217979292001067 |
| [41] | Gibbons, J. and Kodama, Y. 1994.Singular limits of dispersive waves, 61–66. Plenum. |
| [42] | Griffiths P., Introduction to algebraic curves (1989) · Zbl 0696.14012 |
| [43] | Grinevich, P., Orlov, A. and Schulman, E. 1993.Important developments in soliton theory, 283–301. Springer. |
| [44] | Itoyama H., hep-th 9511126, 9512161, and 9601168 |
| [45] | DOI: 10.1007/BF01225258 · Zbl 0648.35080 · doi:10.1007/BF01225258 |
| [46] | DOI: 10.1142/S0217751X96001000 · Zbl 1044.81739 · doi:10.1142/S0217751X96001000 |
| [47] | Kodama Y., World Scientific 11 pp 166– (1990) |
| [48] | Kodama Y., Lecture Lyon workshop (1991) |
| [49] | DOI: 10.1016/0375-9601(89)90255-7 · doi:10.1016/0375-9601(89)90255-7 |
| [50] | DOI: 10.1016/0375-9601(88)90354-4 · doi:10.1016/0375-9601(88)90354-4 |
| [51] | DOI: 10.1007/BF01077626 · Zbl 0688.35088 · doi:10.1007/BF01077626 |
| [52] | DOI: 10.1070/RM1989v044n02ABEH002044 · Zbl 0699.35188 · doi:10.1070/RM1989v044n02ABEH002044 |
| [53] | DOI: 10.1007/BF00994629 · Zbl 0840.35095 · doi:10.1007/BF00994629 |
| [54] | Krichever I., Math. Phys. Rev. 9 pp 1– (1991) |
| [55] | DOI: 10.1007/BF01078472 · Zbl 0637.35060 · doi:10.1007/BF01078472 |
| [56] | DOI: 10.1002/cpa.3160470403 · Zbl 0811.58064 · doi:10.1002/cpa.3160470403 |
| [57] | DOI: 10.1007/BF02099016 · Zbl 0753.35095 · doi:10.1007/BF02099016 |
| [58] | Krichever I., hep-th 9604199 |
| [59] | Krichever I., hep-th 9611158 |
| [60] | Lax, P., Levermore, C. and Venakides, S. 1993.Important developments in soliton theory, 205–241. Springer. · Zbl 0819.35122 |
| [61] | Maltsev A., solv-int 9611008 |
| [62] | Marshakov A., hep-th 9607109 |
| [63] | Martinec E. Warner N. 1996 hep-th 9509161 |
| [64] | DOI: 10.1016/0370-2693(95)00920-G · doi:10.1016/0370-2693(95)00920-G |
| [65] | DOI: 10.1063/1.528732 · Zbl 0711.35114 · doi:10.1063/1.528732 |
| [66] | McLaughlin D., Physica 3D 31 pp 335– (1981) |
| [67] | Mumford D., Lectures on theta, 1 and 2, Birkhäuser, (1983) · Zbl 0509.14049 · doi:10.1007/978-1-4899-2843-6 |
| [68] | Nakatsu T., hep-th 9509162 |
| [69] | Novikov S., Theory of Solitons (1984) |
| [70] | DOI: 10.1016/0550-3213(94)90124-4 · Zbl 0996.81510 · doi:10.1016/0550-3213(94)90124-4 |
| [71] | Sonnenschein J., hep-th 9510129 |
| [72] | Springer G., Introduction to Riemann surfaces (1981) |
| [73] | Takasaki K., hep-th 9603069 |
| [74] | DOI: 10.1142/S0129055X9500030X · Zbl 0838.35117 · doi:10.1142/S0129055X9500030X |
| [75] | Takasaki K., hep-th 9603069 |
| [76] | DOI: 10.1002/cpa.3160380202 · Zbl 0571.35095 · doi:10.1002/cpa.3160380202 |
| [77] | DOI: 10.1016/0550-3213(87)90219-7 · doi:10.1016/0550-3213(87)90219-7 |
| [78] | Whitham G., Linear and nonlinear waves (1974) · Zbl 0373.76001 |
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