We construct certain higher order smooth positon and breather positon solutions of an extended nonlinear Schrödinger equation with the cubic and quartic nonlinearity. We utilize the generalized Darboux transformation method to construct the aforementioned solutions. The three well-known equations, namely nonlinear Schrödinger equation, Hirota equation, and generalized nonlinear Schrödinger equation, are sub-cases of the considered extended nonlinear Schrödinger equation. The solutions which we construct are more general. We analyze how the positon and breather positon solutions of the constituent equations get modified by the higher order nonlinear and dispersion terms. Our results show that the width and direction of the smooth positon and breather-positon solutions are highly sensitive to higher-order effects. Further, we carryout an asymptotic analysis to predict the behaviour of positons. We observe that during collision positons exhibit a time-dependent phase shift. We also present the exact expression of time-dependent phase shift of positons. Finally, we show that this time-dependent phase shift is directly proportional to the higher order nonlinear and dispersion parameters.

In this paper, we investigate smooth positon and breather-positon solutions of a generalized nonlinear Schrödinger (GNLS) equation which contains higher order nonlinear effects. With the help of generalized Darboux transformation (GDT) method we construct N th order smooth positon solutions of GNLS equation. We study the effect of higher order nonlinear terms on these solutions.Our investigations show that the positon solutions are highly compressed by higher order nonlinear effects. The direction of positons are also get changed. We also derive N th order breather-positon (B-P) solution with the help of GDT. We show that these B-Ps are well compressed by the effect of higher order nonlinear terms but the period of B-P solution is not affected as in the breather solution case.