Theoretical investigation on self-focusing of an elliptical \(q\) -Gaussian laser beam carrying an intensity ripple over its cross section in plasma with axial density ramp has been presented. The optical nonlinearity of plasma has been modeled by the relativistic mass nonlinearity of plasma electrons under the effect on intense field of laser beam. Variational theory has been invoked to obtain semi analytical solutions of the wave equations for the fields of main beam and that of ripple. Emphasis is put on the evolutions of the beam widths of main beam and that of ripple.
