Integration of the equation formed by the combination of the equations AKNS(+1) and AKNS(-1) using the inverse problem for periodic Dirac operator

Abstract In this paper, the equation formed by a combination of the equations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>AKNS</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{AKNS}(+1) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>AKNS</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\operatorname{AKNS}(-1)} was integrated using the inver problem method for the self-adjoint periodic Dirac operator in the class of periodic infinite-gap functions. In addition, an infinite system of Dubrovin differential equations that represents evolution of spectral data of the Dirac operator is derived and it is proved that the Cauchy problem for the system of Dubrovin differential equations has a unique solution in the class of three times continuously differentiable periodic infinite-gap functions.