Levi, Decio Toward a unification of the various techniques used to integrate nonlinear partial differential equations: Bäcklund and Darboux transformations vs. dressing method. (English) Zbl 0621.35095 Rep. Math. Phys. 23, 41-56 (1986). After a historical survey on integrable nonlinear partial differential equations, we briefly review three methods often used to obtain their exact solutions, i.e. the Bäcklund transformation, the Darboux transformation and the dressing method introduced by Zakharov and Shabat in 1979. Then we elucidate their interrelations. Cited in 4 Documents MSC: 35Q99 Partial differential equations of mathematical physics and other areas of application 35G20 Nonlinear higher-order PDEs 35A30 Geometric theory, characteristics, transformations in context of PDEs 35C05 Solutions to PDEs in closed form Keywords:survey; integrable nonlinear partial differential equations; exact solutions; Bäcklund transformation; Darboux transformation; dressing method PDFBibTeX XMLCite \textit{D. Levi}, Rep. Math. Phys. 23, 41--56 (1986; Zbl 0621.35095) Full Text: DOI References: [1] Korteweg, D. J.; Vries, G. De, Philos. Magazine, 39, 422-443 (1895), Ser. \(5 N^O\) [2] Boussinesq, J., J. Math. Pures Appl., 17, 55-108 (1872), Ser. \(2 N^O\) [3] Bianchi, L.; Spoerri, Lezioni di geometria differenziale (1894), Pisa [4] Darboux, G., Leçons sur la theorie generale des surfaces et les applications geometriques du calcul infinitesimal, Vol. III (1894), Gauthier-Villars: Gauthier-Villars Paris · JFM 25.1159.02 [5] Bäcklund, A. V., Om ytor med konstant negativ krökning, t. XIX (1883), Lund Univ. Arsskrift [6] Rubinstein, J., J. Math. Phys., 11, 258-266 (1970) [7] Bianchi, L., Ann. R. Scuola Normale Superiore Pisa, 2, 285-340 (1879) [8] Fermi, E.; Pasta, J. R.; Ulam, S. M., Studies of non-linear problems, (Collected papers of E. Fermi, Vol. II (1965), Chicago Univ. Press: Chicago Univ. Press Chicago), 977-988 [9] Zabusky, N. J.; Kruskal, M. D., Phys. Rev. Lett., 15, 240-243 (1965) [10] Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M., Phys. Rev. Lett., 19, 1095-1097 (1967) [11] Calogero, F.; Degasperis, A., Spectral transform and solitons: tools to solve and investigate non-linear evolution equations, Vol. 1 (1982), North-Holland Press Co: North-Holland Press Co Amsterdam · Zbl 0501.35072 [12] Zakharov, V. E.; Faddeev, L. D., Funk. Anal. Pril., 5, 18-27 (1971), in Russian [13] Lax, P. D., Comm. Pure Appl. Math., 21, 467-490 (1968) [14] Adler, M., Inv. Math., 50, 219-248 (1979) [15] Zakharov, V. E.; Shabat, A. B., Funk. Anal. Pril., 13, 13-22 (1979), in Russian [16] Muskelishvili, N. I., Singular integral equations (1953), Noordhoff: Noordhoff Groningen [17] Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segur, H., Stud. Appl. Math., 53, 249-315 (1974) [18] Levi, D.; Sym, A.; Wojciechowski, S., J. Phys. A: Math. & Gen., 16, 2423-2432 (1983) · Zbl 0548.35098 [19] Choodnovsky, D. V.; Choodnovsky, G. V., Il Nuovo Cimento, 40B, 339-353 (1977) [20] Bruschi, M.; Ragnisco, O., Il Nuovo Cimento, 88B, 119-139 (1985) [21] Ichikawa, Y. H.; Konno, K.; Wadati, M., New integrable non-linear evolution equations leading to exotic solitons, (Horton, C. W.; Reichl, L. E.; Szebehely, A. C., Long-time prediction in dynamics (1983), J. Wiley & Sons: J. Wiley & Sons New York), 345-365 [22] Kruskal, M. D., Non-linear wave equations, (Moser, J., Dynamical systems, theory and applications. Dynamical systems, theory and applications, Lect. Notes in Phys., Vol. 38 (1975), Springer-Verlag: Springer-Verlag Berlin), 310-354 [23] Zakharov, V. E.; Mikhailov, A. V., Zh. Eksp. Theor. Fiz., 74, 1953-1973 (1978), in Russian [24] Takhtajan, L. A., Phys. Lett., 64A, 235-237 (1977) [25] Kadomtsev, B. B.; Petviashvili, V. I., Dokl. Akad. Nauk SSSR, 192, 753-756 (1970), in Russian [26] Levi, D.; Ragnisco, O.; Sym, A., Lett. Nuovo Cimento, 33, 401-406 (1982) [27] Darboux, G., C.R. Acad. Sci. Paris, 94, 1456-1460 (1882) [28] Levi, D.; Ragnisco, O.; Sym, A., Il Nuovo Cimento, 83B, 36-42 (1984) [29] Hirota, R., Direct method of finding exact solutions of non-linear evolution equations, (Miura, R. M., Bäcklund transformations. Bäcklund transformations, Lect. Notes Math., Vol. 515 (1976), Springer-Verlag: Springer-Verlag Berlin), 40-68 [30] Pöppe, C., Physica, 9D, 103-139 (1983) [31] Fokas, A. S.; Ablowitz, M. J., Phys. Rev. Lett., 47, 1096-1100 (1981), Santini, P.M., Ablowitz, M.J. and Fokas, A.S.: The direct linearization of a class of non-linear evolution equations, INS≠35 Clarkson College of Technology, submitted to J. Math. Phys. [32] Boiti, M.; Pempinelli, F.; Tu, G. Z., Phys. Lett., 93A, 107-110 (1983) [33] Fokas, A. S.; Ablowitz, M. J., Lecture on the inverse scattering transform for multidimensional (2+1) models, (Wolf, K. B., Non-linear phenomena. Non-linear phenomena, Lect. Notes in Phys. (1983), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0505.76031 [34] Levi, D.; Ragnisco, O.; Bruschi, M., Il Nuovo Cimento, 58A, 56-66 (1980) [35] Sym, A.; Corones, J., Phys. Lett., 68A, 305-307 (1978) [36] Mikhailov, A. V., Physica, 3D, 73-117 (1981) [37] Neugebauer, G.; Meinel, R., Determinant representation of the n-fold AKNS Bäcklund transformations and a separation of the Gel’fand-Levitan-Marchenko equation (10/10/83), Friedrich-Schiller University: Friedrich-Schiller University Jena, submitted to Physica D on October 1982 [38] Wadati, M.; Sanuki, H.; Konno, K., Progr. Theor. Phys., 53, 419-436 (1975) 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.