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Inverse scattering problem for the operator representing sum of the operator of the third derivative on semi-axis and of the operator of multiplication by a real function is studied in this paper. Properties of Jost solutions of such an operator are studied and it is shown that these Jost solutions are solutions of the Riemann boundary value problem on a system of rays. The main system of linear singular integral equations is derived. This system is equivalent to the solution of inverse scattering problem. [ABSTRACT FROM AUTHOR]

Direct and inverse scattering problem for an operator with non-local potential is solved in the paper. The method is based on the Riemann boundary value problem on a bundle of three straight lines. Description of scattering problem data is given. [ABSTRACT FROM AUTHOR]

The work is devoted to solving the Cauchy problem for the modified Korteweg-de Vries–sine-Gordon (mKdV–sG) equation with an additional term in the class of rapidly decreasing functions. The problem is solved by the inverse scattering method. Several examples are given illustrating the application of the obtained results. [ABSTRACT FROM AUTHOR]

The work is devoted to solving the Cauchy problem for the sine-Gordon equation with an additional term and a self-consistent source in the class of rapidly decreasing functions. The problem is solved by the inverse scattering method. Several special cases of the sine-Gordon equation with an additional term are given, which can be integrated using the inverse scattering method, for example, the loaded sine-Gordon equation. Several examples are given to illustrate the application of the obtained results. [ABSTRACT FROM AUTHOR]