Abstract
The dressing method [1] has been a powerful tool for obtaining new integrable equations as well as characterizing large classes of solutions of these equations. This method is applicable to both equations in 1 + 1, i. e. one spatial and one temporal dimensions [1], [2], as well as to equations in 2 + 1, i. e. in two spatial and one temporal dimensions [2]–[4]. The usual. dressing method for equations in 1 + 1 yields solutions which can be thought of as perturbations of the zero solution. It is based on a local Riemann-Hilbert (RH) problem which is in general inadequate for capturing solutions which are perturbations of an arbitrary exact solution. It has been recently shown by Fokas and Zakharov [5] that for equations in 2 +1 solvable by a nonlocal RH problem, it is possible to use the same analytic structure associated with decaying solutions to capture bounded but non-decaying solutions. In particular this method is capable of capturing dromions as well as perturbations of line-solitons. This extended dressing method for nonlocal RH problems is illustrated in §2.
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Fokas, A.S., Sung, LY., Zakharov, V.E. (1991). Recent Developments in Multidimensional Inverse Scattering. In: Makhankov, V.G., Pashaev, O.K. (eds) Nonlinear Evolution Equations and Dynamical Systems. Research Reports in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-76172-0_9
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DOI: https://doi.org/10.1007/978-3-642-76172-0_9
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