Benjamin-Ono and Intermediate Long Wave Equations: Modeling, IST and PDE

Abstract

This survey article is focused on two asymptotic models for internal waves, the Benjamin-Ono (BO) and Intermediate Long Wave (ILW) equations that are integrable by inverse scattering techniques (IST). After recalling briefly 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 connected to the physical origin of the equations. We will consider mainly the Cauchy problem on the whole real line with only a few comments on the periodic case. We will also briefly discuss some close relevant problems in particular the higher order extensions and the two-dimensional (KP like) versions of the BO and ILW equations.Remark 3. The above derivation was performed for purely gravity waves. Surface tension effects result in adding a third order dispersive term in the asymptotic models. One gets for instance the so-called Benjamin equation (see [30,31]):where δ > 0 measures the capillary effects. This equation, which is in some sense close to the KdV equation, is not known to be integrable. Its solitary waves, the existence of which was proven in [30,31] by the degree-theoretic approach, present oscillatory tails. We refer to [145] for the Cauchy problem in L 2 and to [12,21] for further results on the existence and stability of solitary wave solutions and to [46] for numerical simulations.In presence of surface tension, the ILW equation has to be modified in the same way. We are not aware of mathematical results on the resulting equation.Remark 4. The BO equation was fully justified in [219] as a model of long internal waves in a two-fluid system by taking into account the influence of the surface

Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono Equation

“…We state this result in Theorem [1.3.1] below. For recent mathematical surveys of the classical and quantum periodic Benjamin-Ono equations, see §5.2 in Saut [51] and §1.1.6 in Okounkov [46], respectively.…”

“…From the perspective of dispersive equations, it is a coincidence that the symplectic space M (a) of (1.1) corresponds to its critical regularity s c = −1/2 as in Proposition [3.1.2]. For discussion of criticality in (1.1), see Saut [51].…”

Finite gap conditions and small dispersion asymptotics for the classical periodic Benjamin–Ono equation

“…We discuss the notation ε in § [9.2] and write v = v(x, t; ε) for solutions of (1.1). For a comprehensive survey of research on (1.1) see Saut [68].…”

On the Integrability of theBenjamin‐OnoEquation on the Torus

“…Using that u is real valued, one verifies that Q + u is sesquilinear. The claimed estimates are obtained from (16) as follows: since n ≤ 1 + |n| one has f 2 1/2 ≤ −i∂ x f |f + f 2 , and hence by (16),…”

“…[12], [14]). For an excellent survey, including the derivation of (1), we refer to the recent article by J.C. Saut [16].…”