The double cnoidal wave of the Korteweg–de Vries equation: An overview
dc.contributor.author | Boyd, John P. | en_US |
dc.date.accessioned | 2010-05-06T22:37:42Z | |
dc.date.available | 2010-05-06T22:37:42Z | |
dc.date.issued | 1984-12 | en_US |
dc.identifier.citation | Boyd, John P. (1984). "The double cnoidal wave of the Korteweg–de Vries equation: An overview." Journal of Mathematical Physics 25(12): 3390-3401. <http://hdl.handle.net/2027.42/70734> | en_US |
dc.identifier.uri | https://hdl.handle.net/2027.42/70734 | |
dc.description.abstract | Earlier work of the author on the spatially periodic solutions of the Korteweg–de Vries equation is here extended via an in‐depth treatment of a special case. The double cnoidal wave is the simplest generalization of the ordinary cnoidal wave discovered by Korteweg and de Vries in 1895. In the limit of small amplitude, the double cnoidal wave is the sum of two noninteracting linear sine waves. In the oppositie limit of large amplitude, it is the sum of solitary waves of two different heights repeated periodically over all space. Although special, the double cnoidal wave is important because it is but the particular case N=2 of a broad family of solutions known variously as ‘‘N‐polycnoidal waves,’’ ‘‘finite gap,’’ ‘‘finite zone’’ solutions, ‘‘waves on a circle,’’ or ‘‘N‐phase wave trains.’’ It has been shown by others that the set of N‐polycnoidal waves gives the general initial value solution to the Korteweg–de Vries equation. This present work is the core of a three‐part treatment of the double cnoidal wave. This part, the overview, presents graphic examples in all the important parameter regimes, explains how collision phase shifts alter the average speed of the two wave phases from the ‘‘free’’ velocities of the two solitary waves, describes the different branches or modes of the double cnoidal wave (it is possible to have many solitary waves on each spatial period provided they are of only two distinct sizes), and contrasts the results of this work with the very limited numerical calculations of previous authors. The second part describes how the problem of numerically calculating the double cnoidal wave can be reduced down to solving four algebraic equations by perturbation theory. The third part explains how the so‐called ‘‘modular transformation’’ of the Riemann theta functions is important in interpreting N‐polycnoidal waves. | en_US |
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dc.format.extent | 1224202 bytes | |
dc.format.mimetype | text/plain | |
dc.format.mimetype | application/pdf | |
dc.publisher | The American Institute of Physics | en_US |
dc.rights | © The American Institute of Physics | en_US |
dc.title | The double cnoidal wave of the Korteweg–de Vries equation: An overview | en_US |
dc.type | Article | en_US |
dc.subject.hlbsecondlevel | Physics | en_US |
dc.subject.hlbtoplevel | Science | en_US |
dc.description.peerreviewed | Peer Reviewed | en_US |
dc.contributor.affiliationum | Department of Atmospheric and Oceanic Science, University of Michigan, Ann Arbor, Michigan 48109 | en_US |
dc.description.bitstreamurl | http://deepblue.lib.umich.edu/bitstream/2027.42/70734/2/JMAPAQ-25-12-3390-1.pdf | |
dc.identifier.doi | 10.1063/1.526109 | en_US |
dc.identifier.source | Journal of Mathematical Physics | en_US |
dc.identifier.citedreference | J. P. Boyd, J. Math. Phys. 23, 375 (1982). | en_US |
dc.identifier.citedreference | W. E. Ferguson, Jr., H. Flaschka, and D. W. McLaughlin, J. Comput. Phys. 45, 157 (1982). | en_US |
dc.identifier.citedreference | R. Hirota, Phys. Rev. Lett. 27, 1192 (1971). | en_US |
dc.identifier.citedreference | R. Hirota and J. Satsuma, Prog. Theor. Phys. Suppl. 59, 64 (1976). | en_US |
dc.identifier.citedreference | A. Nakamura, J. Phys. Soc. Jpn. 47, 1701 (1949). | en_US |
dc.identifier.citedreference | J. P. Boyd, J. Math. Phys. 25, 3402 (1984). | en_US |
dc.identifier.citedreference | J. P. Boyd, J. Math. Phys. 25, 3415 (1984). | en_US |
dc.identifier.citedreference | R. Hirota and M. Ito, J. Phys. Soc. Jpn. 50, 338 (1981). | en_US |
dc.identifier.citedreference | A. Nakamura and Y. Matsuno, J. Phys. Soc. Jpn. 48, 653 (1980). | en_US |
dc.identifier.citedreference | A. Nakamura, J. Phys. Soc. Jpn. 48, 1365 (1980). | en_US |
dc.identifier.citedreference | M. G. Forest and D. W. McLaughlin, J. Math. Phys. 23, 1248 (1982); H. Flaschka, M. G. Forest, and D. W. McLaughlin, Commun. Pure. Appl. Math. 33, 739 (1980). | en_US |
dc.identifier.citedreference | J. Zagrodzinski, Lett. Nuovo Cimento 30, 266 (1981); J. Zagrodziński and M. Jaworski, Phys. Lett. A 92, 427 (1982); J. Zagrodziński and M. Jaworski, Z. Phys. B 49, 75 (1982); J. Zagrodziński, J. Math. Phys. 24, 46–52 (1983). | en_US |
dc.identifier.citedreference | S. P. Novikov, in Solitons, edited by R. K. Bullough and P. J. Caudrey (Springer‐Verlag, New York, 1980), p. 325. | en_US |
dc.identifier.citedreference | Forest and McLaughlin11 have stressed the usefulness of polycnoidal wave theory in studying a “high density of solitons,” which does not necessarily imply periodic boundary conditions. The theory of baroclinic instability in the atmosphere, which has been studied via the sine‐Gordon equation [J. D. Gibbon, I. N. James, and I. M. Moroz, Proc. R. Soc. London. Ser. A 367, 219 (1979)], most emphatically does involve periodic boundary conditions, but the unstable waves are wavenumbers 4, 5, and 6 for typical values of the parameters, so that this is a problem where the components of a polycnoidal wave have ratios k2/k1k2∕k1 and so on which are fractions rather than integers. | en_US |
dc.identifier.citedreference | R. Grimshaw, Proc. R. Soc. London. Ser. A 368, 359 (1979). | en_US |
dc.identifier.citedreference | The constant is 12a where a ≡ R11k12+2R12k1k2+R22k22+R22k22.a≡R11k12+2R12k1k2+R22k22+R22k22. | en_US |
dc.identifier.citedreference | Some care is necessary in interpreting Fig. 3 correctly in terms of sine waves. First, the fheta function θ01(X,Y;T) ≡ θ00(X+½Y+½;T)θ01(X,Y;T)≡θ00(X+12Y+12;T) is used to generate the graphs; as explained in Appendix A, θ01θ01 is more convenient in the solitary wave regime, but it differs by phase factors from the θ0θ0 which is used everywhere when discussing theta Fourier series. Second, expanding the logarithm of the theta function and then taking the second derivative multiplies all Fourier components by a minus sign. Thus, the “fundamental” referred to in the text is −cos[2π(X+½)] = cos(2πX).−cos[2π(X+12)]=cos(2πX). Third, the graphs were made in the [1, 1] representation, i.e., k1 = k2 = 1,k1=k2=1, which is unnatural for Fourier series as explained in Sec. VI of this work, in Appendix A of Ref. 6, and Sec. VI of Ref. 7. The second harmonic is proportional to −cos[2π(X+Y+½+½)] = −[2π(X+Y)].−cos[2π(X+Y+12+12)]=−[2π(X+Y)]. Thus, the fundamental and second harmonic are out of phase at t = 0t=0 at x = 0x=0 and the fundamental has a peak there, even though naive use of (2.3) would seem to imply both should be negative for x = t = 0.x=t=0. I ask the reader’s indulgence for this long‐winded explanation, but as is the theme of Ref. 7, the need to use different sets of wavenumbers to interpret the double cnoidal wave as sine waves or as solitons sometimes even left the author confused!. | en_US |
dc.identifier.citedreference | P. D. Lax, Commun. Pure. Appl. Math. 21, 467 (1968). He showed that if δ1δ1 and δ2δ2 areas denned in Sec. V of Ref. 6 and one defines r = δ12/δ22,r=δ12∕δ22, then the nonoverlapping collision (Fig. 5) occurs if r>2.618.r>2.618. When e>3.0,e>3.0, the large peak simply decreases to a certain lower bound and then begins to increase again, but no local minimum appears at x = 0x=0 (Fig. 4). For intermediate r. there is an interval in time when there is but a single maximum (as true also for larger r), but there are two peaks with a shallow minimum between them when the phase factors ϕ1ϕ1 and ϕ2ϕ2 both = 0,=0, as true for smaller r. | en_US |
dc.identifier.citedreference | B. Fornberg and G. B. Whitham, Philos. Trans. R. Soc. London. Ser. A 289, 373 (1978). | en_US |
dc.identifier.citedreference | Although the author was not aware of it at the time Boyd1 was written, M. Toda, Phys. Rep. 18, 1 (1975), has shown that the series of displaced single solitons, (3.8) of Ref. 1, in fact is an exact solution of the KdV equation! The implied statement in Ref. 1 that it is only an approximate solution is therefore incorrect. See Appendix B for further discussion. | en_US |
dc.identifier.citedreference | G. B. Whitham, Nonlinear Waves (Wiley, New York, 1974), p. 584. | en_US |
dc.identifier.citedreference | J. M. Hyman, Rocky Mount. J. Math. 8, 95 (1978). | en_US |
dc.identifier.citedreference | H. P. McKean and P. van Moerbeke, Invent. Math. 30, 217 (1975). | en_US |
dc.identifier.citedreference | V. Makhankov, Comput. Phys. Comm. 21, 1 (1980). | en_US |
dc.identifier.citedreference | H. E. Rauch and H. M. Farkas, Theta Functions with Applications to Riemann Surfaces (Williams and Wilkins, Baltimore, 1974), p. 229. | en_US |
dc.identifier.citedreference | J. P. Boyd, SIAM J. Appl. Math, (in press). | en_US |
dc.identifier.citedreference | I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, 4th ed. (Academic, New York, 1965), pp. 911–912. | en_US |
dc.owningcollname | Physics, Department of |
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