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Beginning with this chapter, we proceed to a description of the second (mostly analytic) aspect of the dressing method. In this chapter we will show how to dress a seed solution of a (1+1)-dimensional nonlinear equation making use of the local Riemann–Hilbert (RH) problem. First we formulate in Sect. 8.1 a general approach to the RH problem based dressing method [354] in terms of the Lax representation associated with a given nonlinear equation. Then in the subsequent sections we will illustrate with examples of speciﬁc nonlinear equations the power of the RH problem method. Throughout this chapter we stress two basic facts concerning the applicability of the RH problem to solve nonlinear equations: (1) the RH problem naturally arises in the context of non- linear equations and (2) this approach is substantially universal. In Sect. 8.2 we concretize the main ideas by means of the classic example of the nonlinear Schrod ¨ inger (NLS) equation. Sections 8.3 and 8.4 are devoted to mathemat- ically more complicated equations: the modiﬁed NLS (MNLS) equation and the Ablowitz–Ladik (AL) equation. These two examples are particularly in- teresting from the point of view of the RH problem. Indeed, the reader will see that
Published: Jan 1, 2007
Keywords: Soliton Solution; Spectral Problem; Homoclinic Orbit; Hilbert Problem; Homoclinic Solution
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