The Cauchy Problem for the Modified Korteweg-De Vries-Liouville (Mkdv-L) Equation With an Additional Term in the Class of Periodic Infinite-Gap Functions

In this paper, the inverse spectral problem method is used to integrate a modified Korteweg-de Vries-Liouville (mKdV-L) equation with an additional term in the class of periodic infinite-gap functions.The evolution of the spectral data of the periodic Dirac operator is introduced, and the coefficient of the Dirac operator is a solution for a modified Korteweg-de Vries-Liouville equation with an additional term.A simple algorithm for deriving the Dubrovin system of differential equations is proposed.The solvability of the Cauchy problem for a Dubrovin infinite system of differential equations in the class of six times continuously differentiable periodic infinite-gap functions is proven.It is proven that there is a global solution of the Cauchy problem for a modified Korteweg-de Vries-Liouville equation with an additional term for sufficiently smooth initial conditions.