Abstract
A general approach is given to obtain the system of ordinary differential equations which determines the pure soliton solutions for the class of generalized Korteweg-de Vries equations (cf. [6]). This approach also leads to a system of ordinary differential equations for the pure soliton solutions of the sine-Gordon equation.
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References
Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: Nonlinear-evolution equations of physical significance. Phys. Rev. Letters31, 125–127 (1973)
Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: The inverse scattering transform-Fourier analysis for nonlinear problems. Studies Appl. Math.53, 249–315 (1974)
Caudrey, P.J., Gibbon, J.D., Eilbeck, J.C., Bullough, R.K.: Exact multisoliton solutions of the self induced transparency and sine-Gordon equations. Phys. Rev. Letters30, 237–238 (1973)
Flaschka, H., Newell, A.C.: Integrable systems of nonlinear evolution equations. Lecture notes in physics, Vol. 38, pp. 355–440. Berlin-Heidelberg-New York: Springer 1975
Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg-de Vries equation. Phys. Rev. Letters19, 1095–1097 (1967)
Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Korteweg-de Vries equation and generalizations. VI. Methods for exact solution. Comm. Pure Appl. Math.27, 97–133 (1974)
Hirota, R.: Exact solution of the sine-Gordon equation for multiple collisions of solitons. J. Phys. Soc. Japan33, 1459–1463 (1972)
Kato, T.: Quasi-linear equations of evolution with applications to partial differential equations. Lecture notes in Mathematics, Vol. 448, pp. 25–70. Berlin-Heidelberg-New York: Springer 1974
Kruskal, M.D.: The Korteweg-de Vries equation and related evolution equations. Lectures in applied mathematics, Vol. 15, pp. 61–84. Providence: American Mathematical Society 1974
Kruskal, M.D., Zabusky, N.J.: Interaction of solutions in a collisionless plasma and the recurrence of initial states. Phys. Rev. Letters15, 240–243 (1965)
Kruskal, M.D.: Nonlinear wave equations. Lecture notes in physics, Vol. 38, pp. 310–354. Berlin-Heidelberg-New York: Springer 1975
Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math.21, 467–490 (1968)
Lax, P.D.: Invariant functionals of nonlinear equations of evolution. Proc. internat. conf. functional analysis and related topics (Tokyo 1969), pp. 240–251. Tokyo: Univ. of Tokyo Press 1970
Lax, P.D.: Almost periodic behaviour of nonlinear waves. Adv. Math.16, 368–379 (1975)
Miura, R.M.: Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Math. Phys.9, 1202–1204 (1968)
Miura, R.M., Gardner, C.S., Kruskal, M.D.: Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion. J. Math. Phys.9, 1204–1209 (1968)
Wadati, M., Toda, M.: The exactN-soliton solution of the Korteweg-de Vries equation. J. Phys. Soc. Japan32, 1403–1411 (1972)
Zakharov, V.E.: Kinetic equation for solitons. Soviet Phys. JETP33, 538–541 (1971)
Zakharov, V.E., Shabat, B.A.: Exact theory of two-dimensional self-focusing and one-dimensional self modulation of waves in nonlinear media. Zh. Eksp. Teor. Fiz.61, 118 (1972); (Sov. Phys. JETP34, 62–69 (1972))
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Fuchssteiner, B. Pure soliton solutions of some nonlinear partial differential equations. Commun.Math. Phys. 55, 187–194 (1977). https://doi.org/10.1007/BF01614547
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DOI: https://doi.org/10.1007/BF01614547