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Soliton interaction for the extended Korteweg-de Vries equation

Journal Article


Abstract


  • Soliton interactions for the extended Korteweg-de Vries (KdV) equation are examined. It is shown that the extended KdV equation can be transformed (to its order of approximation) to a higher-order member of the KdV hierarchy of integrable equations. This transformation is used to derive the higher-order, two-soliton solution for the extended KdV equation. Hence it follows that the higher-order solitary-wave collisions are elastic, to the order of approximation of the extended KdV equation. In addition, the higher-order corrections to the phase shifts are found. To examine the exact nature of higher-order, solitary-wave collisions, numerical results for various special cases (including surface waves on shallow water) of the extended KdV equation are presented. The numerical results show evidence of inelastic behaviour well beyond the order of approximation of the extended KdV equation; after collision, a dispersive wavetrain of extremely small amplitude is found behind the smaller, higher-order solitary wave.

Publication Date


  • 1996

Citation


  • Marchant, T. R., & Smyth, N. F. (1996). Soliton interaction for the extended Korteweg-de Vries equation. IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications), 56(2), 157-176. doi:10.1093/imamat/56.2.157

Scopus Eid


  • 2-s2.0-0030122439

Start Page


  • 157

End Page


  • 176

Volume


  • 56

Issue


  • 2

Abstract


  • Soliton interactions for the extended Korteweg-de Vries (KdV) equation are examined. It is shown that the extended KdV equation can be transformed (to its order of approximation) to a higher-order member of the KdV hierarchy of integrable equations. This transformation is used to derive the higher-order, two-soliton solution for the extended KdV equation. Hence it follows that the higher-order solitary-wave collisions are elastic, to the order of approximation of the extended KdV equation. In addition, the higher-order corrections to the phase shifts are found. To examine the exact nature of higher-order, solitary-wave collisions, numerical results for various special cases (including surface waves on shallow water) of the extended KdV equation are presented. The numerical results show evidence of inelastic behaviour well beyond the order of approximation of the extended KdV equation; after collision, a dispersive wavetrain of extremely small amplitude is found behind the smaller, higher-order solitary wave.

Publication Date


  • 1996

Citation


  • Marchant, T. R., & Smyth, N. F. (1996). Soliton interaction for the extended Korteweg-de Vries equation. IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications), 56(2), 157-176. doi:10.1093/imamat/56.2.157

Scopus Eid


  • 2-s2.0-0030122439

Start Page


  • 157

End Page


  • 176

Volume


  • 56

Issue


  • 2