The paper is devoted to the integration of the loaded modified Kortewegde Vries equation in the class of rapidly decreasing functions. It is well known that loaded differential equations in the literature are usually called equations containing in the coefficients or in the right-hand side any functionals of the solution, in particular, the values of the solution or its derivatives on manifolds of lower dimension. In this paper, we consider the Cauchy problem for the loaded modified Korteweg-de Vries equation. The problem is solved using the inverse scattering method, i.e. the evolution of the scattering data of a non-self-adjoint Dirac operator is derived, the potential of which is a solution to the loaded modified Korteweg-de Vries equation in the class of rapidly decreasing functions. A specific example is given to illustrate the application of the results obtained.

Aknazar Khasanov, Dr. Sci. (Phys.–Math.), Prof., Samarkand State University, 15, University Boulevard, Samarkand, 140104, Republic of Uzbekistan, tel.: (99866)239-14-36, email: ahasanov2002@mail.ru

Umid Hoitmetov, Cand. Sci. (Phys.–Math.), Khorezm Branch of the V. I. Romanovskiy Institute of Mathematics, Urgench State University, 14, Kh. Alimjan st., Urgench, 220100, Republic of Uzbekistan, (99862) 224-67-00, email: x.umid@urdu.uz

Khasanov A.B., Hoitmetov U.A. On Integration of the Loaded mKdV Equation in the Class of Rapidly Decreasing Functions. The Bulletin of Irkutsk State University. Series Mathematics, 2021, vol. 38, pp. 19-35. https://doi.org/10.26516/1997-7670.2021.38.19

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