Envelope compactons and solitary patterns

In this Letter, to understand the role of nonlinear dispersion in complex nonlinear wave equations, we introduce and study nonlinear Schrödinger equation with nonlinear dispersion (called NLS(m,n) equation): iut+(u|u|n−1)xx+μu|u|m−1=0. As a consequence, we obtain some envelope compactons for NLS+(n,n) equation and envelope solitary patterns for NLS−(n,n) equation. Moreover, we also show that NLS(m,1) equation, which is nonlinear wave equation with linear dispersion possess both envelope compactons and solitary patterns. Finally, some unusually local conservation laws are given for NLS+(n,n) equation and NLS−(n,n) equation, respectively.