Diversity of solitons in a generalized nonlinear Schrödinger equation with self-steepening and higher-order dispersive and nonlinear terms

In this article, we show that if the nonlinear Schrödinger (NLS) equation is generalized by simultaneously taking into account higher-order dispersion, a quintic nonlinearity, and selfsteepening terms, the resulting equation is interesting as it has exact soliton solutions which may be (depending on the values of the coefficients) stable or unstable, standard or "embedded," fixed or "moving" (i.e., solitons which advance along the retarded-time axis). We investigate the stability of these solitons by means of a modified version of the Vakhitov-Kolokolov criterion, and numerical tests are carried out to corroborate that these solitons respond differently to perturbations. It is shown that this generalized NLS equation can be derived from a Lagrangian density which contains an auxiliary variable, and Noether's theorem is then used to show that the invariance of the action integral under infinitesimal gauge transformations generates a whole family of conserved quantities. Finally, we study if this equation has the Painlevé property.