Opuscula Mathematica
Opuscula Math. 37, no. 1 (), 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



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.
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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ISSN 1232−9274, e-ISSN 2300−6919, DOI http://dx.doi.org/10.7494/OpMath
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