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We construct certain higher order smooth positon and breather positonsolutions of an extended nonlinear Schr\"odinger equation with the cubic andquartic nonlinearity. We utilize the generalized Darboux transformation methodto construct the aforementioned solutions. The three well-known equations,namely nonlinear Schr\"odinger equation, Hirota equation, and generalizednonlinear Schr\"odinger equation, are sub-cases of the considered extendednonlinear Schr\"odinger equation. The solutions which we construct are moregeneral. We analyze how the positon and breather positon solutions of theconstituent equations get modified by the higher order nonlinear and dispersionterms. Our results show that the width and direction of the smooth positon andbreather-positon solutions are highly sensitive to higher-order effects.Further, we carryout an asymptotic analysis to predict the behaviour ofpositons. We observe that during collision positons exhibit a time-dependentphase shift. We also present the exact expression of time-dependent phase shiftof positons. Finally, we show that this time-dependent phase shift is directlyproportional to the higher order nonlinear and dispersion parameters.

Higher order smooth positon and breather positon solutions of an extended nonlinear Schr\"odinger equation with the cubic and quartic nonlinearity