Abstract
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.
Highlights
The inverse scattering method (ISM) traces its origins to the work of Gardner, Greene, Kruskal and Miura [19]. They managed to find a global solution to the Cauchy problem for the Korteweg-de Vries (KdV) equation by reducing it to the inverse scattering problem for the Sturm-Liouville operator
The application of the ISM to the nonlinear Schrodinger equation (NLS) equation, modified Korteweg-de Vries equation (mKdV) and sine-Gordon equations is based on the scattering problem for the Dirac operator on the entire axis
We study the loaded mKdV equation, namely, consider the following equation ut + 6u2ux + uxxx + γ(t)u(0, t)ux(x, t) = 0, (1.1)
Summary
The inverse scattering method (ISM) traces its origins to the work of Gardner, Greene, Kruskal and Miura [19] They managed to find a global solution to the Cauchy problem for the Korteweg-de Vries (KdV) equation by reducing it to the inverse scattering problem for the Sturm-Liouville operator. In this direction, the following important result was obtained. The application of the ISM to the NLS equation, mKdV and sine-Gordon equations is based on the scattering problem for the Dirac operator on the entire axis. We study the loaded mKdV equation, namely, consider the following equation ut + 6u2ux + uxxx + γ(t)u(0, t)ux(x, t) = 0,.
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