Exact solutions to a nonlinearly dispersive Schrödinger equation (Q2470148)

In the present study the authors deal with the following nonlinearly dispersive Schrödinger equation (shortly \(\text{NLS}(m,n)\)): \[ iu_t+ (u\cdot|u|^{n-1})_{xx}+\mu u|u|^{m-1}= 0,\tag{1} \] where \(n\) is a positive integer, \(m\) is an integer, \(\mu= \pm1\), and \(i^2= -1\). The authors use the method of dynamical systems to find exact solutions of (1). More precisely, they obtain the existence of solitary paterns, smooth and non-smooth periodic patterns and breather solutions of (1). The authors also discuss the bifurcations of phase portrait of (1).