Localization of the sine-Gordon equation solutions

•Distributed control algorithms for the wave localization at arbitrary initial conditions for the sine-Gordon equation.•Localization of the waves moving in both directions by means of the feedforward (nonfeedback) control.•Two localized wave analytical solutions are provided by the feedback distributed control algorithm. Localization of the waves of the sine-Gordon equation depends on the shape of the initial condition. It is shown how initially motionless Gaussian distribution may be modified to obtain propagation of localized waves in both directions. However, the resulting localized wave profile is described neither by an asymptotic envelope- wave solution to the sine-Gordon equation nor by its exact traveling breather solution. The distributed control algorithms are developed to achieve wave localization independent of the shape of the initial condition. It is shown that localization of the waves in both directions is achieved by means of a feedforward (nonfeedback) control. The waves are similar to the envelope wave solution. The feedback distributed algorithm is shown to provide both localized waves according to analytical solutions and their unidirectional propagation.