Solitons from sine waves: Analytical and numerical methods for non- integrable solitary and cnoidal waves. (English) Zbl 0611.35080

The ”FKDV” equation, \(u_ t+uu_ x-u_{xxxxx}=0\), is used as a testbed for a variety of analytical and numerical methods that can be applied to solitary waves and cnoidal waves of ”non-integrable” differential equations, that is to say, to equations which cannot be solved by the inverse scattering transform. The basic tools are (i) Padé approximants formed from power series in the amplitude; (ii) a Newton- Kantorovich/pseudospectral Fourier/continuation numerical method; (iii) singular perturbation theory for two interacting solitons of almost identical phase speed; (iv) bifurcation and branch-switching methods; (v) the imbricate-soliton series. A number of new results for the FKDV equation are obtained including extensive numerical calculations of the spatially periodic solutions with one peak (”cnoidal wave”) and two peaks (”bicnoidal wave”) per period, an analytical expression for the double- peaked soliton (”bion”), calculation of both the limit and bifurcation points for the bicnoidal wave, and finally the computation of accurate analytical approximations to the cnoidal wave for all amplitudes. More important, all of these analytical and numerical tools are highly effective for this equation in spite of the fact that it cannot be solved by the inverse scattering transform. Work now in progress will apply these methods to non-integrable equations in two space dimensions.


35Q99 Partial differential equations of mathematical physics and other areas of application
76B25 Solitary waves for incompressible inviscid fluids
35B10 Periodic solutions to PDEs
Full Text: DOI Link


[1] Gorshkov, K. A.; Ostrovskii, L. A.; Papko, V. V., Sov. Phys. JETP, 44, 306 (1976)
[2] Gorshkov, K. A.; Ostrovskii, L. A.; Papko, V. V., Sov. Phys. Dokl., 22, 378 (1977)
[3] Gorshov, K. A.; Papko, V. V., Sov. Phys. JETP, 46, 92 (1977)
[4] Gorshkov, K. A.; Ostrovskii, L. A.; Papko, V. V.; Pikovsky, A. S., Phys. Lett., 74A, 177 (1979)
[5] Gorshkov, K. A.; Ostrovskii, L. A., Physica, 3D, 428 (1981) · Zbl 1194.37119
[6] Imada, M., J. Phys. Soc. Japan, 52, 1946 (1983)
[7] Kano, K.; Nakayama, T., J. Phys. Soc. Japan, 50, 361 (1981)
[8] Kawahara, T., J. Phys. Soc. Japan, 33, 260 (1972)
[9] Nagashima, H.; Kuwahara, M., J. Phys. Soc. Japan, 50, 3792 (1981)
[10] Nayfeh, A., Perturbation Methods, ((1973), Wiley: Wiley New York), 58 · Zbl 0265.35002
[11] Ostrovskii, L. A.; Pelinovskii, E. N., Sov. Phys. Dokl., 15, 1097 (1971)
[12] Yamamoto, Y.; Takizawa, E. I., J. Phys. Soc. Japan, 1421 (1981)
[13] Yoshimura, K.; Watanabe, S., J. Phys. Soc. Japan, 51, 3028 (1982)
[14] Bender, C. M.; Orszag, S. A., Advanced Mathematical Methods for Scientists and Engineers, ((1978), McGraw-Hill: McGraw-Hill New York), 383 · Zbl 0417.34001
[15] Boyd, J. P., Mon. Wea. Rev., 106, 1192 (1978)
[16] Gottlieb, D.; Orszag, S. A., Numerical Analysis of Spectral Methods (1977), SIAM: SIAM Philadelphia · Zbl 0412.65058
[17] J.P. Boyd, J. Comp. Phys., to be published.; J.P. Boyd, J. Comp. Phys., to be published.
[18] Chan, T. F.; Keller, H. B., SIAM J. Sci. Stat. Comput., 3, 173 (1982) · Zbl 0497.65028
[19] Toda, M., Phys. Rept., 18, 1 (1975)
[20] Orszag, S., J. Comp. Phys., 37, 70 (1980) · Zbl 0476.65078
[21] Boyd, J. P., J. Math. Phys., 23, 375 (1982) · Zbl 0502.35009
[22] McLaughlin, D. W.; Scott, A. C., Phys. Rev., A18, 1652 (1978)
[23] Keener, J. P.; McLaughlin, D. W., Phys. Rev., A16, 777 (1977)
[24] Norton, H. J., Comp. J., 7, 76 (1964) · Zbl 0133.08704
[25] Kantorovich, L. V., Dokl. Akad. Nauk SSSR, 59, 1237 (1948) · Zbl 0031.05701
[26] Lax, P., Commun. Pure Appl. Math., 21, 467 (1968) · Zbl 0162.41103
[27] Young, D. M.; Gregory, R. T., (A Survey of Numerical Mathematics, vol. 2 (1973), Addison-Wesley: Addison-Wesley Reading, MA), 918 · Zbl 0262.65002
[28] Finlayson, B. A., The Method of Weighted Residuals and Variational Principles (1972), Academic Press: Academic Press New York · Zbl 0319.49020
[29] Boyd, J. P., SIAM J. Appl. Math., 44, 952 (1984) · Zbl 0559.76019
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.