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In this paper we study the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a real, positive, multi-humped, fairly smooth but not necessarily analytic potential decaying at infinity. We provide the rigorous semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues, the norming constants, and the reflection coefficient. [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]