The zero-dispersion limit for the Benjamin--Ono equation on the circle

2025/09/17 by Mæhlen, Ola
#35B40 #35Q53 #37K15 #47B35 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.14134

Abstract

Using the explicit formula of P. Gérard, we characterize the zero-dispersion limit for solutions of the Benjamin--Ono equation on the circle \mathbbT= ℝ/2πℤ with bounded initial data u₀∈ L^∞(\mathbbT,ℝ). The result generalizes the work of L. Gassot, who focused on periodic bell-shaped data, and complements the work of Gérard and X. Chen who identified the zero-dispersion limit on the line with u₀∈ L²∩ L^∞(ℝ). Here, as well as in the mentioned cases, the characterization agrees with the one first obtained by Miller--Xu for bell-shaped data on the line: The zero-dispersion limit is given as an alternating sum of the characteristics appearing in the (multivalued) solution of Burgers' equation. From this characterization, we compute regularity properties of the zero-dispersion limit, including maximum principles and an Oleinik estimate.

Citations

• Universality in the Small-Dispersion Limit of the Benjamin-Ono Equation
• The Benjamin-Ono Initial-Value Problem for Rational Data with Application to Long-Time Asymptotics and Scattering
• Justification of the Benjamin-Ono equation as an internal water waves model
• The zero dispersion limit for the Benjamin--Ono equation on the line
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• On the integrability of the Benjamin-Ono equation on the torus
• On critical behaviour in systems of Hamiltonian partial differential equations
• Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations
• On the zero-dispersion limit of the Benjamin-Ono Cauchy problem for positive initial data
• The Zero-Dispersion Limit for the Odd Flows in the Focusing Zakharov-Shabat Hierarchy
• Semiclassical Soliton Ensembles for the Focusing Nonlinear Schroedinger Equation

Related

• The zero dispersion limit for the Benjamin--Ono equation on the line
• The Benjamin-Ono equation in the zero-dispersion limit for rational initial data: generation of dispersive shock waves
• Universality in the Small-Dispersion Limit of the Benjamin-Ono Equation
• On the zero-dispersion limit of the Benjamin-Ono Cauchy problem for positive initial data
• The Benjamin-Ono Hierarchy with Asymptotically Reflectionless Initial Data in the Zero-Dispersion Limit