Application of the inverse scattering method to the equation $\sigma_{xt}=e^\sigma$

It is shown that the equation $\sigma_{xt}=e^\sigma$, which arises in many branches of physics and mathematics, can be exactly solved by means of the inverse scattering problem. Here, $N$-soliton solutions are constructed. These solutions describe one movIng soliton and $N-1$ fixed solitons. The phase shift of the moving soliton resulting from scattering on fixed solitons is found. Conservation laws are constructed. The method used in the paper differs somewhat from the ordinary method of the inverse scattering problem.