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Dressing via nonlocal Riemann–Hilbert problem In the previous chapter we illustrated the eﬃciency of the dressing approach using the local Riemann–Hilbert (RH) problem for solution of the Cauchy problem for a number of (1+1)-dimensional nonlinear integrable equations. The essential progress in the development of the inverse spectral transform (IST) formalism has been achieved owing to the perception that the nonlocal RH problem can serve as a natural frame for solving nonlinear equations in 2+1 dimensions. Manakov [305] was the ﬁrst to apply the nonlocal RH problem to treat the Kadomtsev–Petviashvili (KP) equation by means of the IST method. Besides, there exists an important class of (1+1)-dimensional nonlinear integrodiﬀerential equations which cannot be solved by the methods discussed in Chap. 8. This chapter contains an exposition of basic points related to the appli- cation of the nonlocal RH problem. We consider three featured examples. In Sect. 9.1 we consider the (1+1)-dimensional integrodiﬀerential Benjamin–Ono (BO) equation . At the very beginning we work with real function u(x, t)tak- ing into account important constraints imposed on the spectral data by the reality condition, due to Kaup, Lakoba, and Matsuno [231]. Basic steps in ap- plication of the nonlocal RH problem developed for
Published: Jan 1, 2007
Keywords: Spectral Problem; Hilbert Problem; Spectral Equation; Lump Solution; Wronskian Relation
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