Opuscula Mathematica (Jan 2017)

The inverse scattering transform in the form of a Riemann-Hilbert problem for the Dullin-Gottwald-Holm equation

  • Dmitry Shepelsky,
  • Lech Zielinski

DOI
https://doi.org/10.7494/OpMath.2017.37.1.167
Journal volume & issue
Vol. 37, no. 1
pp. 167 – 187

Abstract

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The Cauchy problem for the Dullin-Gottwald-Holm (DGH) equation \[u_t-\alpha^2 u_{xxt}+2\omega u_x +3uu_x+\gamma u_{xxx}=\alpha^2 (2u_x u_{xx} + uu_{xxx})\] with zero boundary conditions (as \(|x|\to\infty\)) is treated by the Riemann-Hilbert approach to the inverse scattering transform method. The approach allows us to give a representation of the solution to the Cauchy problem, which can be efficiently used for further studying the properties of the solution, particularly, in studying its long-time behavior. Using the proposed formalism, smooth solitons as well as non-smooth cuspon solutions are presented.

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