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- Title
On the Modified Korteweg–De-Vries Equation with Loaded Term.
- Authors
Khasanov, A. B.; Allanazarova, T. Zh.
- Abstract
We apply the method of inverse spectral problem to find the solution to the Cauchy problem for a modified Korteweg–de-Vries equation (mKdV) in the class of periodic infinite-gap functions. A simple procedure is proposed for the derivation of the Dubrovin system of differential equations. We prove the solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of five times continuously differentiable periodic infinite-gap functions. It is shown that the sum of a uniformly convergent function series constructed from the solutions of the infinite system of Dubrovin equations and the formulas for the first trace satisfy the mKdV equation. Moreover, it is proved that: (i) if the initial function is a real π-periodic analytic function, then the solution of the Cauchy problem for the mKdV equation with loaded term is also a real analytic function with respect to the variable x; (ii) if the number π 2 is a period (antiperiod) of the original function, then π 2 is also a period (antiperiod) in the variable x of the solution to the Cauchy problem for the mKdV equation with loaded term.
- Subjects
DIFFERENTIAL equations; TRACE formulas; ANALYTIC functions; EQUATIONS; INVERSE problems; CAUCHY problem
- Publication
Ukrainian Mathematical Journal, 2022, Vol 73, Issue 11, p1783
- ISSN
0041-5995
- Publication type
Article
- DOI
10.1007/s11253-022-02030-4