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We study various properties of solutions of an extended nonlinearSchr\"{o}dinger (ENLS) equation, which arises in the context of geometricevolution problems -- including vortex filament dynamics -- and governspropagation of short pulses in optical fibers and nonlinear metamaterials. Forthe periodic initial-boundary value problem, we derive conservation lawssatisfied by local in time, weak $H^2$ (distributional) solutions, andestablish global existence of such weak solutions. The derivation is obtainedby a regularization scheme under a balance condition on the coefficients of thelinear and nonlinear terms -- namely, the Hirota limit of the considered ENLSmodel. Next, we investigate conditions for the existence of traveling wavesolutions, focusing on the case of bright and dark solitons. The balancecondition on the coefficients is found to be essential for the existence ofexact analytical soliton solutions; furthermore, we obtain conditions whichdefine parameter regimes for the existence of traveling solitons for variouslinear dispersion strengths. Finally, we study the modulational instability ofplane waves of the ENLS equation, and identify important differences betweenthe ENLS case and the corresponding NLS counterpart. The analytical results arecorroborated by numerical simulations, which reveal notable differences betweenthe bright and the dark soliton propagation dynamics, and are in excellentagreement with the analytical predictions of the modulation instabilityanalysis.

Conservation laws, exact travelling waves and modulation instability for an extended nonlinear Schr\"odinger equation