Direct Scattering for the Benjamin-Ono Equation with Rational Initial Data

The scattering transform for the Benjamin–Ono equation in the small-dispersion limit

“…Here, the potential u 0 : R → R is the initial condition u(x, 0) = u 0 (x) for (1), and the operator C + : L 2 (R) → H + (R) is the self-adjoint orthogonal projection onto the Hardy space. As first noted in [11] and generalized in [19], the direct scattering problem for the BO equation can be effectively solved for rational potentials of the form…”

“…, φ P (λ)) T is the unique nullvector of A(λ j ) normalized by the condition (12). The scattering data associated with the rational potential u 0 are as follows [19].…”

“…In [19] a different formula for γ j was shown to be valid for rational potentials u 0 . While that formula is especially useful for exact calculations (see Section 6), the general formula (15) turns out to be better suited to small-dispersion asymptotics; see Section 5.2.…”

“…In this paper we use recently obtained exact formulae [19] to study the asymptotic behavior of the scattering data for the BO equation (1) in the limit → 0. These formulae hold for rational initial data with simple poles, and for convenience we adopt here the further condition of a single local extremum (Definition 3.1).…”

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Using exact formulae for the scattering data of the Benjamin-Ono equation valid for general rational potentials recently obtained in [19], we rigorously analyze the scattering data in the small-dispersion limit. In particular, we deduce precise asymptotic formulae for the reflection coefficient, the location of the eigenvalues and their density, and the asymptotic dependence of the phase constant (associated with each eigenvalue) on the eigenvalue itself.

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