On the possibility of exact reciprocal transformations for one-soliton solutions to equations of the Lobachevsky class

Problems on reciprocal transformation of solutions to equations of $\Lambda^2$-class (equations related with special coordinate nets on the Lobachevsky plane $\Lambda^2$) are discussed. A method of the construction of solutions to one analytic differential equation of $\Lambda^2$-class by a given solution of another analytic differential equation of this class is proposed. The reciprocal transformation of one-soliton solutions of the sine-Gordon equation and one-soliton solutions of the modified Korteweg–de Vries equation is obtained. This result confirms the possibility of the construction of such transition.