TY  - JOUR
ID  - Shepelsky2017
LB  - Shepelsky2017
AU  - Shepelsky, Dmitry
AU  - Zielinski, Lech
TI  - The inverse scattering transform in the form of a Riemann-Hilbert problem for the Dullin-Gottwald-Holm equation
ST  - The inverse scattering transform in the form of a Riemann-Hilbert problem for the Dullin-Gottwald-Holm equation
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2017
VL  - 37
IS  - 1
SP  - 167
EP  - 187
DO  - http://dx.doi.org/10.7494/OpMath.2017.37.1.167
UR  - http://dx.doi.org/10.7494/OpMath.2017.37.1.167
AB  - 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.
KW  - Dullin-Gottwald-Holm equation
KW  - Camassa-Holm equation
KW  - inverse scattering transform
KW  - Riemann-Hilbert problem
SN  - 1232-9274
ER  -