×

zbMATH — the first resource for mathematics

On a class of physically important integrable equations. (English) Zbl 1194.35363
Summary: A methodology introduced by Fuchssteiner and the author is used to derive a class of physically important integrable evolution equations. Among these equations are integrable generalizations of the Korteweg-deVries (KdV), of the modified KdV, of the nonlinear Schrödinger (NLS), and of the sine-Gordon equations. The modeling of water waves, as well as general asymptotic considerations, are used to illustrate the occurrence of the generalized modified KdV and NLS equations, respectively.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Taniuti, T.; Wei, C.C.; Taniuti, T.; Kodama, Y.; Taniuti, T., J. phys. soc. Japan, Prog. theor. phys., J. phys. soc. Japan, 45, 298, (1978)
[2] Calogero, F., Why are certain nonlinear PDEs both widely applicable and integrable, () · Zbl 0808.35001
[3] Gardner, C.S.; Green, J.M.; Kruskal, M.D.; Miura, R.M., Important developments in soliton theory, (), 19, 1095, (1967), for recent developments, see
[4] Gel’fand, I.M.; Dorfman, I.; Gel’fand, I.M.; Dorfman, I., Funct. anal. appl., Funct. anal. appl., 14, 71, (1980)
[5] Fokas, A.S.; Fuchssteiner, B., Lett. nuovo cimento, 28, 299, (1980)
[6] Fuchssteiner, B.; Fokas, A.S., Physica D, 4, 47, (1981)
[7] Fuchssteiner, B., Prog. theor. phys., 65, 861, (1981)
[8] Holm, D.; Camassa, R., Phys. rev. lett., 71, 1671, (1993)
[9] Fokas, A.S.; Santini, P.M., An inverse acoustic problem and linearization of moderate amplitude dispersive waves, (1994), (preprint)
[10] Fokas, A.S.; Liu, Q.M., Asymptotic integrability of water waves, (1994), preprint · Zbl 0973.35502
[11] Rosenau, P., Phys. rev. lett., (1994)
[12] Whitham, G.B., Linear and nonlinear waves, (1974), Wiley-Interscience NY · Zbl 0373.76001
[13] Benjamin, T.B.; Bona, J.L.; Mahony, J.J., Phil. trans. roy. soc. A, 272, 47, (1972)
[14] Kodama, Y., Phys. lett. A, 112, 193, (1985)
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.