Stopping of generalized solitary and periodic waves in optical waveguide with varying and constant parameters

We demonstrate the possibility of stopping the soliton pulses in optical waveguide with varying and constant parameters exhibiting a Kerr nonlinear response. By using the similarity transformation, we have found the constraint condition for varying waveguide parameters and derive the exact analytical solutions for self-similar bright, kink, dark and rectangular solitary and periodic waves for nonlinear Schr\"{o}dinger equation with variable coefficients. All these generalized wave solutions depend on five arbitrary parameters and two free integration constants. It is found that the velocity of solitons is related to a free parameter $q$, which play an important role in the dynamic behavior of soliton's evolution. The precise expression of soliton's velocity shows that the solitons can be nearly stopped for appropriate values of free parameter $q$. The possibility for stopping of soliton pulses can also be realized when all parameters of nonlinear Schr\"{o}dinger equation are constant. The numerical simulations show that the stopping behavior of solitons and rectangular solitary waves can be achieved for appropriate values of free parameters characterizing these wave solutions.