Abstract
This chapter focuses on two asymptotic models for internalwaves, the Benjamin–Ono (BO) and Intermediate LongWave (ILW) equations, which are integrable by inverse scattering techniques (IST). After briefly recalling their (rigorous) derivations, we will review old and recent results on the Cauchy problem, comparing those obtained by IST and PDE techniques and also results more closely connected to the physical origin of the equations. We will consider mainly the Cauchy problem on the whole real line, but comment in some detail on important new results obtained by IST techniques for the periodic BO equation. We will also briefly discuss some relevant closely related problems, for instance the higher-order extensions, but the twodimensional (KP-like) versions of the BO and ILWequations will be studied in more detail in the chapter devoted to KP-type equations.
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