Abstract
The history of this subject is that GARDNER, GREENE, KRUSKAL and MIURA [34] suggested the inverse scattering scheme of solution for the KdV (22) in 1967, ZAKHAROV and SHABAT (ZS) [35] solved the NLS (24) in 1971, and WADATI and TODA [36] used the method of [34] to find the explicit solution of the KdV whilst WADATI [37] used the method of ZS [35] to solve the modified KdV (23) in 1972. Then HIROTA [38] and CAUDREY, GIBBON, EILBECK and BULLOUGH [39] found the multisoliton solutions of the s-G in 1972–73 and these were rapidly followed by the inverse scattering solution by ABLOWITZ, KAUP, NEWELL and SEGUR (AKNS) [40]. AKNS then gave a generalised ZS inverse scattering scheme [16]. Most of the work of this paper II is concerned with the AKNS-ZS scheme.
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References
D.J. Korteweg and G. de Vries, Phil. Mag. 39, 422 (1895).
J.S. Russell, Report on Waves, British Association Reports (1844).
G.B. Whitham, Linear and Non-linear Waves (John Wiley & Sons, New York, 1974) pp. 580–585.
J.D. Gibbon and J.C. Eilbeck, J. Phys. A: Gen. Phys. 5, L132 (1972).
R.K. Dodd and R.K. Bullough, Proc. Roy. Soc. A 351, 499 (1976).
R.K. Bullough, Solitons. Lectures at the Winter College on the Interaction of Radiation and Condensed Matter. I.C.T.P., Trieste, March 1976. To be published by the I.A.E.A., Vienna (1977).
J.M. Burgers, Adv. Appl. Mech. 1, 171 (1948).
S. Coleman, Classical lumps and their quantum descendants. Lectures at the 1975 International School of Subnuclear Physics “Ettore Majorana” (1975).
H. Haken, in Synergetics in Cooperative Phenomena, Edited by H. Haken (North Holland, Amsterdam, 1974).
J.A. Krumhansl and J.R. Schrieffer, Phys. Rev. B 11, 3535 (1975).
R.F. Dashen, B. Hasslacher, and A. Neveu, Phys. Rev. D 3424 (1975).
P.J. Caudrey, J.C. Eilbeck and J.D. Gibbon, Il Nuovo Cimento 25, 497 (1975).
J.D. Gibbon and J.C. Eilbeck, J. Phys. A: Gen. Phys. 5, L122 (1972).
P.J. Caudrey, J.C. Eilbeck, and J.D. Gibbon, J. Inst. Math. Appiics. 14, 375 (1975).
R. Hirota, Phys. Rev. Lett. 27, 1192 (1971).
M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Phys. Rev. Lett. 31, 125 (1973).
M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Studies in Appl. Math. 53, 249 (1974).
E. Fermi, J.R. Pasta and S.M. Ulam, Studies of nonlinear problems I, Los Alamos Rept. LA-1940 (May 1955) and Collected Works of E. Fermi Vol. II (Univ. of Chicago Press, 1965) pp. 978–88.
J. Gibbons, S.G. Thornhill, M.J. Wardrop, and D. ter Haar, On the theory of Langmuir Solitons, preprint Univ. of Oxford, Dept. of Theoretical Phys. Ref. 36776 (1976).
V.E. Zakharov, Zh. Eksp. Teor. Fiz. 62, 1745 (1972) (Soviet Phys. J.E.T.P. 35, 908 (1975).
G.L. Lamb, Rev. Mod. Phys. 43, 99 (1971).
J.C. Eilbeck, P.J. Caudrey, J.D. Gibbon, and R.K. Bullough, 6, 1337 (1973).
B.B. Kadomtsev and V.I. Petviashvili, Dokl. Akad. Nauk. SSR, 192, 1337 (1973).
V.E. Zakharov and A.B. Shabat, Funkt. Anal. i Ego Prilozh. 8, 43 (1974).
H.C. Yuen and B.M. Lake, Phys. Fluids 18, 956 (1975).
R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures on Physics Vol. III (Addison-Wesley, Reading, Mass., 1965) pp. 21.14–21.18.
J.D. Gibbon, P.J. Caudrey, R.K. Bullough and J.C. Eilbeck, Lett. al Nuovo Cimento 8, 775 (1973).
H.M. Gibbs and R.E. Slusher, Phys. Rev. A 6, 2326 (1972).
T.A. Fulton and R.C. Dynes, Solid State Comm. 12, 57 (1973).
S.P. Novikov, Funkt. Anal. i Ego Prilozh. 8, 54 (1974).
P.D. Lax, Comm. Pure and Appl. Maths. 28, 141 (1975).
R.K. Bullough, P.J. Caudrey, J.D. Gibbon, S. Duckworth, H.M. Gibbs, B. Bölger, and L. Baede, Optics Communications 18, 200 (1976).
R.K. Bullough and P.J. Caudrey, Bumping spin waves in the B-phase of liquid 3He. Preprint (April, 1977).
C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Phys. Rev. Lett. 19, 1095 (1967).
V.E. Zakharov and A.B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) (Soviet Phys. J.E.T.P. 34, 62 (1972)).
M. Wadati and M. Toda, J. Phys. Soc. Japan 32, 1403 (1972).
M. Wadati, J. Phys. Soc. Japan 32, 1681 (1972).
R. Hirota, J. Phys. Soc. Japan 33, 1459 (1972).
P.J. Caudrey, J.D. Gibbon, J.C. Eilbeck and R.K. Bullough, Phys. Rev. Lett. 30, 237 (1973).
M. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Phys. Rev. Lett. 30, 1262 (1973).
P.P. Eisenhart, A treatise on the differential geometry of curves and surfaces (Dover, New York, 1960).
R.K. Dodd and R.K. Bullough, Proc. Roy. Soc. A 352, 481 (1977).
P.J. Caudrey, R.K. Dodd and J.D. Gibbon, Proc. Roy. Soc. A 351, 407 (1976).
P.D. Lax, Comm. Pure and Appl. Math. 21, 467 (1968).
R.K. Dodd and R.K. Bullough. To be published (April 1977).
M.A. Naimak, Linear differential operators Pt. II (Harrap, London, 1968).
V.E. Zakharov and L.D. Fadeev, Funkt. Anal. i Ego Prilozh. 18 (1971).
H. Flaschka and A.C. Newell, “Integrable systems of non linear evolution equations” in Dynamical Systems Theory and Applications. Springer Lecture Notes in Physics. Edited by J. Moser (Springer Verlag, Heidelberg, 1975).
For example V.E. Korepin and L.D. Fadeev, Teoreticheskaya i Matematicheskaya Fiz. 25, 147 (1975).
A. Luther, Phys. Rev. B 14, 2153 (1976).
R.J. Baxter, Phys. Rev. Lett. 26, 832 (1971).
J.D. Johnson, S. Krinsky and B.M. McCoy, Phys. Rev. A 8, 2526 (1973).
R.F. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D 10, 4130 (1974).
D.J. Kaup, J. Math. Phys. 16, 2036 (1975).
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Bullough, R.K., Dodd, R.K. (1977). II. Mathematical Structures. In: Haken, H. (eds) Synergetics. Springer Series in Synergetics, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-66784-8_11
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