Periodic wave solutions and solitary wave solutions for the new soliton equation (Q2856122)

Consider the soliton equation NEWLINE\[NEWLINEu_t= a\Biggl({1\over u}\Biggr)_{xxx}+ b\Biggl({1\over u^2}\Biggr)_x,\tag{\(*\)}NEWLINE\]NEWLINE where \(a\) and \(b\) are real constants, and look for solutions of the form NEWLINE\[NEWLINEu(x,t)= \phi(x- ct)\equiv \phi(\xi).\tag{\(**\)}NEWLINE\]NEWLINE Substituting \((**)\) into \((*)\) and integrating the corresponding ordinary differential equation, the author arrives at the autonomous system NEWLINE\[NEWLINE{d\phi\over d\xi}= y,\quad {dy\over d\xi}= {6ay^2+ b\phi^2+ c\phi^3\over 2a\phi}NEWLINE\]NEWLINE having the first integral NEWLINE\[NEWLINEH(\phi,y)\equiv {y^2\over \phi^6}+ {b\over 4a\phi^4}+ {c\over a\phi}.NEWLINE\]NEWLINE Under certain assumptions on the parameters \(a\), \(b\), \(c\) the author derives explicit parametric representations for three types of traveling wave solutions of \((*)\).