@Article{Shepelsky2017,
author = "Shepelsky, Dmitry and Zielinski, Lech",
title = "The inverse scattering transform in the form of a Riemann-Hilbert problem for the Dullin-Gottwald-Holm equation",
journal = "Opuscula Math.",
year = "2017",
volume = "37",
number = "1",
pages = "167--187",
doi = "http://dx.doi.org/10.7494/OpMath.2017.37.1.167",
url = "http://dx.doi.org/10.7494/OpMath.2017.37.1.167",
abstract = "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.",
keywords = "Dullin-Gottwald-Holm equation, Camassa-Holm equation, inverse scattering transform, Riemann-Hilbert problem",
issn = "1232-9274"
}