We analyze the nonlinear coupled mode equations describing the $\check{C}erenkov$ second harmonic generation in a nonlinear dielectric waveguide. An approximate analytic solution in a closed form is newly obtained for the pump depletion and nonlinear phase-shift of the fundamental field. Our solution includes as particular cases the previous approximate solutions derived by others. The validity of our solution is also checked by the nonlinear beam propagation method. The higher order analysis for the nonlinear coupled mode equations is also performed. The solution can be also expressed in terms of a leaky mode phase-mismatch and it shows explicitly that a resonant pump depletion and nonlinear phase-shift of the fundamental field occurs when the phase-matching to the lowest order leaky mode is accomplished.
