Global weak solutions and breaking waves to the Degasperis-Procesi equation with linear dispersion
The author makes a new study of an initial value problem for the Degasperis-Procesi (DP) equation with strong linear dispersion in the form \[ u_{t}-u_{txx}+4u\, u_{x} + \gamma \left(u-u_{xx}\right)_{x} = 3u_{x}u_{xx}+u\, u_{xxx}, \] where \(\gamma (\in \mathbb{R})\) is a dispersive parameter. He then considers the both global strong and weak solutions of the above equation. This is followed by a study of precise breaking and permanent wave solution of the above equation. It is shown that, when the wave breaking occurs, there is at least one point where the slope of the wave profile is infinite at the breaking time. It is noted that the breaking point depends on the dispersive parameter \(\gamma\). Included is also the proof of existence and uniqueness of global weak solution of the above equation with strong dispersion. This is a new and interesting work on global weak solutions and breaking waves to the DP equation with linear dispersion.
scientific article; zbMATH DE number 5626911
scientific article