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On the Korteweg-de Vries equation: an associated equation

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  • Eugene P. Schlereth
  • Ervin Y. Rodin

Abstract

The purpose of this paper is to describe a relationship between the Korteweg-de Vries (KdV) equation u t − 6 u u x + u x x x = 0 and another nonlinear partial differential equation of the form z t + z x x x − 3 z x z x x z = H ( t ) z . The second equation will be called the Associated Equation (AE) and the connection between the two will be explained. By considering AE, explicit solutions to KdV will be obtained. These solutions include the solitary wave and the cnoidal wave solutions. In addition, similarity solutions in terms of Airy functions and Painlevé transcendents are found. The approach here is different from the Inverse Scattering Transform and the results are not in the form of solutions to specific initial value problems, but rather in terms of solutions containing arbitrary constants.

Suggested Citation

  • Eugene P. Schlereth & Ervin Y. Rodin, 1984. "On the Korteweg-de Vries equation: an associated equation," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 7, pages 1-15, January.
    • Handle: RePEc:hin:jijmms:237357
    DOI: 10.1155/S0161171284000272
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