Inverse scattering and soliton solutions of nonlocal complex reverse-spacetime mKdV equations
The paper is devoted to a nonlocal modification of the Korteweg-de Vries equation. The nonlocality of the model manifests in the presence, in the corresponding nonlinear differential equation, of a term depending on inverted space-time coordinates. In addition, the author considers even larger, multicomponent class of mKdV equations. It turns out that such nonlolcal generalization possesses a Lax pair formulation which results in the complete integrability of the model. Then the author applies the inverse scattering method by means of the corresponding Riemann-Hilbert problem, which is one of the equivalent approaches to finding the explicit solutions. The investigation is done along the lines of classical works on the inverse scattering method. The author routinely formulates the related spectral problem, finds the time evolution of the scattering data, and finds the corresponding Gelfand-Levitan-Marchenko integral equations, which encode the soliton solutions. The presentation of the paper is quite complete, and contains many technical details and explicit calculations.
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