Opuscula Math. 37, no. 1 (2017), 167-187
http://dx.doi.org/10.7494/OpMath.2017.37.1.167

 
Opuscula Mathematica

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

Dmitry Shepelsky
Lech Zielinski

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.

Mathematics Subject Classification: 35Q53, 37K15, 35Q15, 35B40, 35Q51, 37K40.

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  • Dmitry Shepelsky
  • Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkiv, Ukraine
  • V. N. Karazin Kharkiv National University, 4 Svobody Square, 61022 Kharkiv, Ukraine
  • Lech Zielinski
  • LMPA, Université du Littoral Côte d'Opale, 50 rue F. Buisson, CS 80699, 62228 Calais, France
  • Communicated by Alexander Gomilko.
  • Received: 2016-04-19.
  • Accepted: 2016-09-24.
  • Published online: 2016-12-14.
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Cite this article as:
Dmitry Shepelsky, Lech Zielinski, The inverse scattering transform in the form of a Riemann-Hilbert problem for the Dullin-Gottwald-Holm equation, Opuscula Math. 37, no. 1 (2017), 167-187, http://dx.doi.org/10.7494/OpMath.2017.37.1.167

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