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The Benjamin-Ono Initial-Value Problem for Rational Data with Application to Long-Time Asymptotics and Scattering

Elliot Blackstone, Louise Gassot, Patrick Gérard, Peter D. Miller

TL;DR

This work solves the Benjamin-Ono initial-value problem on $\mathbb{R}$ for real, $L^2$ rational data with simple poles by yielding a closed-form, determinant-based solution that depends explicitly on $(t,x)$ and the dispersion parameter $\varepsilon$. The solution is expressed either as a ratio of contour-integral determinants $\det(\mathbf{A})/\det(\mathbf{B})$ or as a logarithmic derivative $i\varepsilon\partial_x\log\det(\overline{\mathbf{B}})$, with rigorous conditions ensuring the determinants never vanish for $t>0$. The paper develops a resolvent-based formulation for the rational data, derives Hardy-space inclusion criteria for the resolvent function, and provides a detailed analysis of exceptional and non-exceptional pole configurations. It also establishes long-time asymptotics for the special initial data equal to minus a soliton, proving $L^2$-scattering to a linear dispersive flow and linking the limit to a distorted Fourier transform $\alpha(\lambda)$. The combination of exact determinant formulas, Hardy-space analysis, and spectral theory of the Lax operator yields concrete insights into soliton resolution and scattering in the Benjamin-Ono equation, with explicit results for both general rational data and important special cases.

Abstract

We show that the initial-value problem for the Benjamin-Ono equation on $\mathbb{R}$ with $L^2(\mathbb{R})$ rational initial data with only simple poles can be solved in closed form via a determinant formula involving contour integrals. The dimension of the determinant depends on the number of simple poles of the rational initial data only and the matrix elements depend explicitly on the independent variables $(t,x)$ and the dispersion coefficient $ε$. This allows for various interesting asymptotic limits to be resolved quite efficiently. As an example, and as a first step towards establishing the soliton resolution conjecture, we prove that the solution with initial datum equal to minus a soliton exhibits scattering.

The Benjamin-Ono Initial-Value Problem for Rational Data with Application to Long-Time Asymptotics and Scattering

TL;DR

This work solves the Benjamin-Ono initial-value problem on for real, rational data with simple poles by yielding a closed-form, determinant-based solution that depends explicitly on and the dispersion parameter . The solution is expressed either as a ratio of contour-integral determinants or as a logarithmic derivative , with rigorous conditions ensuring the determinants never vanish for . The paper develops a resolvent-based formulation for the rational data, derives Hardy-space inclusion criteria for the resolvent function, and provides a detailed analysis of exceptional and non-exceptional pole configurations. It also establishes long-time asymptotics for the special initial data equal to minus a soliton, proving -scattering to a linear dispersive flow and linking the limit to a distorted Fourier transform . The combination of exact determinant formulas, Hardy-space analysis, and spectral theory of the Lax operator yields concrete insights into soliton resolution and scattering in the Benjamin-Ono equation, with explicit results for both general rational data and important special cases.

Abstract

We show that the initial-value problem for the Benjamin-Ono equation on with rational initial data with only simple poles can be solved in closed form via a determinant formula involving contour integrals. The dimension of the determinant depends on the number of simple poles of the rational initial data only and the matrix elements depend explicitly on the independent variables and the dispersion coefficient . This allows for various interesting asymptotic limits to be resolved quite efficiently. As an example, and as a first step towards establishing the soliton resolution conjecture, we prove that the solution with initial datum equal to minus a soliton exhibits scattering.

Paper Structure

This paper contains 19 sections, 16 theorems, 180 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General solution formula
  3. The action of the resolvent for rational u0
  4. Conditions for f to lie in the Hardy space
  5. The exceptional case: $\mathrm{i} c_1/\epsilon\in \mathbb{N}_{<0}$
  6. The non-exceptional case with $\operatorname{Re}(\mathrm{i} c_1/\epsilon)<0$
  7. The non-exceptional case with $\mathrm{i} c_1/\epsilon=M\in\mathbb{N}$
  8. The non-exceptional case with $\operatorname{Re}(\mathrm{i} c_1/\epsilon)\geq 0$ and $\mathrm{i} c_1/\epsilon\not\in\mathbb{N}$
  9. Derivation of the solution formulæ
  10. Derivation of formula
  11. Derivation of formula
  12. Nonvanishing of the denominators
  13. Long time asymptotics for specific initial data
  14. Spectral theory of the Lax operator
  15. A distorted Fourier transform
  16. ...and 4 more sections

Key Result

Theorem 1.3

\newlabelthm:inversion-formula0 Let $\epsilon>0$. The solution of the Cauchy initial-value problem for the Benjamin-Ono equation eq:BO with rational initial condition $u(x,0)=u_0(x)$ of the form eq:rationalIC is An equivalent formula is Also, we have $\det(\mathbf{B}(t,x))\neq 0$ and $\det(\overline{\mathbf{B}}(t,x))\neq 0$ for all $(t,x)\in\mathbb{R}^2$ with $t>0$ and all $\epsilon>0$.

Figures (2)

  • Figure 1: Left: admissible branch cuts of $h(z)$ in the $z$-plane for a rational initial condition with $N=5$. Right: corresponding contours $C_1,C_2,C_3,C_4,C_5$ and $C_0$ for a situation where $2$ is the only exceptional index.
  • Figure 2: A demonstration of the convergence guaranteed by Proposition \ref{['prop:weak-asymptotics']}. In the left/right panel, the red curve is the real/imaginary part of $\alpha(-y)$, respectively, see \ref{['eq:minus-soliton-alpha']}.

Theorems & Definitions (43)

  • Definition 1.1: Branch cuts of $h$
  • Definition 1.2: Contours
  • Theorem 1.3: Solution of Benjamin-Ono for rational initial data
  • Remark 1.4: The limit $t\downarrow 0$ and negative times
  • Remark 1.5: Choice of branch cuts
  • Remark 1.6: Two equivalent formulæ
  • Remark 1.7: Comparison with the Calogero-Moser equation
  • Remark 1.8: Comparison with the $N$-soliton formula
  • Remark 1.9: An explicit invariant space for $u(t,x)$
  • Remark 1.10: Implementation
  • ...and 33 more