Abstract
ThispaperdealswiththenonlinearSchr¨ odingerequation(NSE)withpolynomial(powertype)andpowerlogarithmicnonlinearities.Athirdordergeneralizationof NSE-thesocalledRKLEwithapplications intelecommunications isconsideredtoo. In somespecial casesthesolitonsolutionsortheblowingupsolutionscanbeexpressedby elementaryorspecial functions(elliptic,hyperbolic).Weare lookingfor thewavesolutionsansatzhavingrealnon-negativeamplitudeandlinearphase.Theamplitudesatisfies secondordernonlinearODEandit isconstructedbyusingthemethodoffirst integral (inmechanicprocesseswithonedegreeof freedomaredescribedbysuchtypeODEs). Sothesolutionsarefoundintointegral formandone-parametricfamilyofperiodicsolutionsisconstructed. Several illustrativeexamplesofperiodic,blowingupandsolitontype solutionsareproposed.
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