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References
N. J. Zabusky and M. D. Kruskal, Phys. Rev. Lett. 15, 240–243 (1965).
The best general introduction to solitons remains the classic review by A. C. Scott; F. Y. F. Chu, and D. W. McLaughlin, in Proc. IEEE, 61 1443–1483 (1973).
D. J. Korteweg and G. de Vries, Phil. Mag. 39, 422–443, (1895).
V. I. Talanov, Zh. Eksp. Teor. Fiz. Pis. Red. 2, 218–222 (1965); trans JETP Lett. 2, 138–141 (1965); P. L. Kelley, Phys. Rev. Lett. 15, 1005–1008 (1965).
V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118–134 (1971); trans. Soviet Physics JETP 34, 62–69 (1972).
For a good review see R. Rajaraman, Phys. Lett. 21C, 229–313 (1975).
C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Phys. Rev. Lett. 19 1095–97 (1967).
M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Stud. In App. Math. 53 249–315 (1974).
E. Fermi, J. Pasta, and S. Ulam, “Studies of Nonlinear Problems. I”, Los Alamos Scientific Laboratory Report, LA-1940 (1955).
A notable exception to this rule is the sine-Gordon equation, where the analytic two soliton solution was found prior to the numerical experiments which showed the soliton nature of the solution in (6.b). Compare A. Seeger, M. Donth, and A. Kochendörfer, Z. Phys. 134, 173–193 (1953) to J. K. Perring and T. H. R. Skyrme, Nuc. Phys. 31, 550–555 (1962).
For a recent review of the status of this “experimental mathematics,” see V. G. Makhankov, Phys. Lett. 35C, 1–128 (197 ).
P. Lax, Comm. Pure and App. Math., 21, 467–490 (1968).
J. Corones and D. McLaughlin, Phys. Rev. A10, 2051–2062 (1974).
H. Wahlquist and F. Estabrook, J. Math. Phys. 16, 1–7 (1975).
G. Lamb, Phys. Rev. Lett. 37, 235–237 (1976).
J. Corones, B. Markovsky, and V. Rizov, J. Math. Phys. 18, 2207–2213 (1977).
F. Lund and T. Regge, Phys. Rev. D 14, 1524–1535 (1976); F. Lund,Phys. Rev. Lett. 38, 1175–1178 (1977).
More precisely, we shall consider only the time independence of the point spectrum of L; for the continuum, it is easiest to work directly with R(λ).
Studying the asymptotic behavior as × →-∞ is equally simple but yields no new information; see ref. (2).
I. M. Gelfand and B. M. Levitan, Izvest. Akad. Nauk. SSSR Ser. Mat. 15 309–360 (1951) [Am. Math. Soc. Trams. 1, 253–304 (1953)].
V. A. Marchenko, Dokl. Akad. Nauk. SSR 72, 457–460 (1950); 104, 695–698 (1955); Z. S. Agranovich and V. A. Marchenko, The Inverse Problem of Scattering Theory (Gordon and Breach, New York, 1963).
V. Bargman, Rev. Mod. Phys. 21, 488–493 (1949); R. Jost and W. Kahn; Det. Kgl. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 27, No. 9 (1953).
See, for example, P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, pp. 1649–1659 (McGraw-Hill, New York, 1953).
The absence of a factor of “i” here is purely conventional and is for notational convenience only.
R. Hermann, Phys. Rev. Lett. 36, 835–836 (1976).
J. Corones, J. Math. Phys. 18, 163–164 (1977).
M. Crampin, F. Pirani, P. Robinson, Lett. Math. Phys. 2, 15–19 (1977).
M. Crampin, Phys. Lett. 66A, 170–172 (1978).
J. Corones, “An Illustration of the Lie Group Framework for Soliton Equations: Generalizations of the Lund-Regge Model”, J. Math. Phys., to appear.
Bäcklund Transformations, The Inverse Scattering Method, Solitons, and Their Applications, Robert M. Miura, ed., Lecture Notes in Mathematics 515, A. Dold and B. Eckmann, eds. (Springer Verlag, Berlin, 1976).
M. Wadati, H. Sanuki, K. Konno, Prog. Theor. Phys. 53, 419–436 (1975).
See, for example, R. Dashen, B. Hasslacher, and A. Neveu, Phys. Rev. D 10, 4114–4129, 4130-4138 (1975); J. Goldstone and R. Jackiw, Phys. Rev. D 11, 1486–1498 (1975); A. Neveu and J. Gervais, eds, Phys. Lett. 23C, 237–374 (1976).
R. Dashen, B. Hasslacher, and A. Neveu, Phys. Rev. D 11, 3424–3450 (1975).
A. Luther, Phys. Rev. B. 14, 2153–2159 (1976).
V. Zakharov and L. Faddeev, Funkt. Anal. i Ego. Pril. 5, 18–27 (1971); (Transl.) Funkt. Anal. and its Applic. 5, 280–287 (1972).
For a pedagogical presentation, see the article by H. Flaschka and D. McLaughlin, in ref. [30].
V. Zakharov, Zh. Eksp. Teor. Fiz. 65, 219–225 (1973); (Transl.) Sov. Phys. JETP 38, 108–110 (1974).
F. Tappert, private communication see also J. C. Goldstein, Los Alamos Informal Report, LA 6833-MS (1977).
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Campbell, D. (1979). Solitons. In: Beiglböck, W., Böhm, A., Takasugi, E. (eds) Group Theoretical Methods in Physics. Lecture Notes in Physics, vol 94. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-09238-2_5
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