The dressing chain of discrete symmetries and proliferation of nonlinear equations

In the examples of sine-Gordon and Korteweg–de Vries (KdV) equations, we propose a direct method for using dressing chains (discrete symmetries) to proliferate integrable equations. We give a recurrent procedure (with a finite number of steps in general) that allows the step-by-step production of an integrable system and its $L$–$A$ pair from the known $L$–$A$ pair of an integrable equation. Using this algorithm, we reproduce a number of known results for integrable systems of the KdV type. We also find a new integrable equation of the sine-Gordon series and investigate its simplest soliton solution of the double $\pi$-kink type.