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This research investigates the limiting behavior of solutions to the Benjamin–Ono Cauchy problem under specific scaling conditions.

February 25, 2026Open Access

Universality in the Small-Dispersion Limit of the Benjamin–Ono Equation

Key Points

This research investigates the limiting behavior of solutions to the Benjamin–Ono Cauchy problem under specific scaling conditions.
Examined double-scaling limits where dispersion approaches zero.
Analyzed solutions for rational initial data at critical points.
Utilized contour integrals for explicit representations of solutions.
Identified universal limiting profiles independent of initial data details.
Established comparisons with known results for the Korteweg-de Vries equation.
Demonstrated consistent behaviors in multiple analytical scenarios.

Abstract

Abstract We examine the solution of the Benjamin–Ono Cauchy problem for rational initial data in three types of double-scaling limits in which the dispersion tends to zero while simultaneously the independent variables either approach a point on one of the two branches of the caustic curve of the inviscid Burgers equation, or approach the critical point where the branches meet. The results reveal universal limiting profiles in each case that are independent of details of the initial data. We compare the results obtained with corresponding results for the Korteweg-de Vries equation found by Claeys–Grava in three papers (Claeys and Grava in Commun Math Phys 286:979–1009, 2009, Commun Pure Appl Math 63:203–232, 2010, SIAM J Math Anal 42:2132–2154, 2010). Our method is to analyze contour integrals appearing in an explicit representation of the solution of the Cauchy problem, in various limits involving coalescing saddle points.

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