The present paper is aimed at a consistent analytical description of a combined interplay of linear frequency modulation of the high-frequency carrier, nonlinearity, and inhomo- geneity of an optical fiber on the propagation of an arbitrary mode feasible in the fiber, with vortex modes included. Formal asymptotic procedure is applied to the problem of wave propa- gation and equations for the mode structure and the envelope of the chirped pulse are derived. Lagrangian is proposed enabling to reformulate the envelope equation as a variational problem which is then solved for the case of travelling pulse of an arbitrary shape. Explicit expressions for the pulse parameters are derived via characteristics of the gradient optical fiber and the structure of propagating mode.