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Explicit formula for the Benjamin--Ono equation with square integrable and real valued initial data and applications to the zero dispersion limit

Xi Chen

TL;DR

This work extends the explicit solution formula for the Benjamin–Ono equation on the real line to initial data in $L_r^2(\mathbb{R})$, enabling a direct representation of the solution via a perturbed Lax-pair framework. It also broadens the zero-dispersion limit to more singular initial data by proving a weak $L^2$-convergence and deriving a holomorphic characterization of the limit through a generalized resolvent, culminating in a principal-value logarithmic formula and a piecewise algebraic description for the limit. Central tools include the Hardy space $L_+^2(\mathbb{R})$, Toeplitz operators $T_u$, the operator $G$, and the Kato–Rellich theorem to control perturbations and ensure maximal dissipativity. The paper also derives a notable integral identity that aids the second equality in the zero-dispersion expression and discusses short-time versus long-time limits, highlighting open questions for higher-regularity initial data and the torus setting. Overall, the results deepen the understanding of explicit BO dynamics and asymptotics under reduced regularity, with implications for nonlocal dispersive equations and spectral methods.

Abstract

In this paper, we extend G{é}rard's formula for the solution of the Benjamin--Ono equation on the line to square integrable and real valued initial data. Combined with this formula, we also extend the G{é}rard's formula for the zero dispersion limit of the Benjamin--Ono equation on the line to more singular initial data. In the derivation of the extension of the formula for the zero dispersion limit, we also find an interesting integral equality, which might be useful in other contexts.

Explicit formula for the Benjamin--Ono equation with square integrable and real valued initial data and applications to the zero dispersion limit

TL;DR

This work extends the explicit solution formula for the Benjamin–Ono equation on the real line to initial data in , enabling a direct representation of the solution via a perturbed Lax-pair framework. It also broadens the zero-dispersion limit to more singular initial data by proving a weak -convergence and deriving a holomorphic characterization of the limit through a generalized resolvent, culminating in a principal-value logarithmic formula and a piecewise algebraic description for the limit. Central tools include the Hardy space , Toeplitz operators , the operator , and the Kato–Rellich theorem to control perturbations and ensure maximal dissipativity. The paper also derives a notable integral identity that aids the second equality in the zero-dispersion expression and discusses short-time versus long-time limits, highlighting open questions for higher-regularity initial data and the torus setting. Overall, the results deepen the understanding of explicit BO dynamics and asymptotics under reduced regularity, with implications for nonlocal dispersive equations and spectral methods.

Abstract

In this paper, we extend G{é}rard's formula for the solution of the Benjamin--Ono equation on the line to square integrable and real valued initial data. Combined with this formula, we also extend the G{é}rard's formula for the zero dispersion limit of the Benjamin--Ono equation on the line to more singular initial data. In the derivation of the extension of the formula for the zero dispersion limit, we also find an interesting integral equality, which might be useful in other contexts.
Paper Structure (11 sections, 15 theorems, 173 equations)

This paper contains 11 sections, 15 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. The Benjamin–Ono equation
  3. The Lax pair
  4. The explicit formula
  5. Zero dispersion limit
  6. Maximally dissipative operators and the Kato-Rellich theorem
  7. Structure of the paper
  8. Proof of the extension of the explicit formula
  9. Proof of the extension of the formula for the zero dispersion limit
  10. Final comments and open problems
  11. Appendix

Key Result

Theorem 1.1

For every $u_{0} \in H_r^{s}(\mathbb{R})$ with $s > 3-\sqrt{33 / 4}$, there exists a unique solution $u \in C\left(\mathbb{R}, H_{r}^{s}(\mathbb{R})\right)$ of (0.1) with $u(0) = u_0$. Also, for every $T > 0$, the flow map $u_{0} \in H^{s}(\mathbb{R}) \mapsto u \in C\left([-T, T], H^{s}(\mathbb{R})\

Theorems & Definitions (39)

  • Theorem 1.1: 5, 12, 19 13, 7
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 29 more