Envelope compact and solitary pattern structures for the GNLS(m,n,p,q) equations

In this Letter, to further understand the role of nonlinear dispersion in the generalized nonlinear Schrödinger equation, we introduce and study the generalized nonlinear Schrödinger equation with nonlinear dispersion (called GNLS(m,n,p,q) equation): iut+a(u|u|n−1)xx+bu|u|m−1+ic(u|u|p−1)xxx+id(u|u|q−1)x=0. Some new envelope compacton solutions and solitary pattern solutions of GNLS(m,n,p,q) equation are obtained via the gauge transformation and some direct ansatze. In particular, it is shown that GNLS(m,n,p,q) equation with linear dispersion gives rise to envelope compactons and solitary patterns, which implies that nonlinear dispersion is not necessary condition for GNLS(m,n,p,q) equation to admit envelope compactons and solitary patterns. Moreover, some unusually local conservation laws are presented for GNLS+(n,n,n,n) equation and GNLS−(n,n,n,n) equation, respectively.