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Some features of one-soliton solutions of the Korteweg - de Vries equation

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Abstract

Using the method of inverse scattering problem [1, 2], we study solutions of the Korteweg - de Vries equation under initial conditions in the form of two nonsoliton pulses with not very large amplitudes. It is shown that if the distance between these pulses is not large, then they evolve to one soliton and an oscillating nonlinear tail for t → ∞. As the distance between the pulses or the pulse amplitudes increase, two solitons and an oscillating nonlinear tail are formed. Similar behavior is observed for solutions of the nonlinear Schrödinger equation. The only difference is that three, but not two, solitons are formed if the distance between two initial inphase pulses increases. The results of analytical consideration are illustrated by the numerical solution of the Korteweg - de Vries equation.

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Authors and Affiliations

  1. Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia

    S. N. Vlasov & E. V. Koposova

Authors
  1. S. N. Vlasov
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  2. E. V. Koposova
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Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 47, Nos. 10–11, pp. 1008– 1012, October–November, 2004.

This work is supported by the Council on the Grants of the President of the Russian Federation for Support of Leading Scientific Schools (grant NSh-1637.2003.2) and the Program “Mathematical Methods in the Nonlinear Dynamics” of the Presidium of the Russian Academy of Sciences.

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Vlasov, S.N., Koposova, E.V. Some features of one-soliton solutions of the Korteweg - de Vries equation. Radiophys Quantum Electron 47, 905–909 (2004). https://doi.org/10.1007/s11141-005-0030-9

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  • DOI: https://doi.org/10.1007/s11141-005-0030-9

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