Dynamical behaviors and new exact bounded travelling wave solutions for the modified Degasperis-Procesi equation (Q2907776)

Consider the partial differential equation NEWLINE\[NEWLINEu_t- u_{xxt}+ 4uu_x= 3u_x u_{xx}+ uu_{xxx}.\tag{\(*\)}NEWLINE\]NEWLINE The authors look for solutions of \((*)\) in the form NEWLINE\[NEWLINEu(x,t)= \mu+ \psi(\xi),\quad \xi= x-ct,NEWLINE\]NEWLINE where \(c>0\) is the wave speed, \(\mu\) is some constant. This problem can be reduced to the study of a planar autonomous system with respect to the existence of heteroclinic, homoclinic and periodic trajectories. The authors establish under special relations between \(c\) and \(\mu\) such solutions.