Negative Order Modified Korteweg–de Vries–Liouville (nmKdV-L) Equation in the Class of Periodic Infinite-gap Functions

In this paper, the inverse spectral problem method is used to integrate the nonlinear negative order modified Korteweg–de Vries–Liouville (nmKdV–L) equation in the class of periodic infinite-gap functions. The evolution of the spectral data of the periodic Dirac operator is introduced whose coefficient is a solution of the nonlinear negative order modified Korteweg–de Vries–Liouville equation. A simple algorithm for deriving Dubrovin system of differential equations is proposed. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of twice continuously differentiable periodic infinite gap functions is proved. It is shown that there is a global solution of the mixed problem for the nonlinear nmKdV–L equation for sufficiently smooth initial data.