A Mixed Problem for the Negative Order Nonlinear Modified Korteweg–de Vries Equation with a Special Term

In this paper, a mixed problem for the negative order nonlinear modified Korteweg–de Vries (mKdV) equation with a special free term is considered. It is shown that the mixed problem for the negative order nonlinear mKdV equation with a special free term can be integrated using the inverse spectral problem method. The evolution of the spectral data of the Dirac operator with a periodic potential associated with the solution to the negative order mKdV equation with a special free term is determined. The obtained results allow us to apply the inverse problem method to solving the mixed problem for the negative order nonlinear mKdV equation with a special free term. The solvability of the Cauchy problem for the infinite Dubrovin–Trubowits system of differential equations in the class of triply continuously differentiable periodic functions is proved. It is shown that the solution constructed using the Dubrovin–Trubowits system of equations and the trace formula satisfies the mixed problem for the negative order nonlinear mKdV equation with a special free term.

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