MIXED PROBLEM FOR THE NEGATIVE ORDER MODIFIEDKORTEWEG-DE VRIES–COSINE GORDON EQUATION

In this paper, the negative order modified Korteweg-de Vriescosine Gordon equation (nmKdV-coshG) in the class of periodic infinitegap functions is integrated using the inverse spectral problem method.It is demonstrated that the Cauchy problem for the infinite Dubrovin differential equation system is solvable in the class of twice continuously differentiable periodic infinite-gap functions.It is demonstrated that the mKdV-coshG equation is satisfied by the sum of the uniformly convergent functional series constructed by solving the Dubrovin differential equation system.Furthermore, it is demonstrated that a global solution to the mixed problem for the negative order modified Korteweg-de Vries-cosine Gordon equation exists for sufficiently smooth initial data.