Asymptotic stability of Benjamin--Ono multisolitons in $L^2(\mathbb R)$

Technical Summary: Asymptotic Stability of Benjamin–Ono Multisolitons in $L^2(\mathbb{R})$

Problem Statement
The paper investigates the long-time asymptotic behavior of solutions to the Benjamin–Ono (BO) equation, a model for interfacial waves in stratified fluids:
$$\partial_t u = H\partial_x^2 u - 2u\partial_x u, \quad u: \mathbb{R}_t \times \mathbb{R}_x \to \mathbb{R},$$
where $H$ is the Hilbert transform. While the existence and orbital stability of multisoliton solutions (nonlinear structures that behave like particles) have been established in higher regularity spaces (specifically $H^s$ for $s > -1/2$), the question of asymptotic stability in the critical $L^2(\mathbb{R})$ setting remained open. Specifically, it was unknown whether solutions emanating from small $L^2$ perturbations of an $N$-soliton profile converge to a new multisoliton configuration as $t \to \pm \infty$, or if other coherent structures could emerge.

A major challenge in the $L^2$ setting is the lack of spatial decay hypotheses required by traditional inverse scattering methods. Without such decay, the spectral theory of the associated Lax operator becomes delicate, particularly regarding the potential presence of embedded singular continuous spectra, which could theoretically obstruct soliton resolution.

Methodology
The authors employ a dynamical approach based on the complete integrability of the BO equation, specifically utilizing the Lax pair formulation and an explicit solution formula derived by Gérard. The methodology proceeds through several key technical stages:

1. Lax Pair and Spectral Analysis: The authors analyze the Lax operator $L_u = -i\partial_x - \Pi(uf)$ acting on the Hardy space $L^2_+(\mathbb{R})$. They establish that for $u \in L^2(\mathbb{R})$, $L_u$ is a self-adjoint operator with essential spectrum $[0, \infty)$. Crucially, they prove that the negative spectrum consists of simple eigenvalues and that the corresponding eigenfunctions possess sufficient regularity (belonging to the domain of the generator of Fourier translations, $X$).
2. The Peter Operator and Unitary Flow: A central technical innovation is the rigorous construction of the unitary flow $U(t)$ associated with the Peter operator $P_u$ (defined as $P_u = -i\partial_x^2 - 2\partial_x\Pi(uf) + 2u'_+f$). Unlike previous works that required higher regularity ($H^3$), the authors extend the definition of $U(t)$ to $L^2$ solutions via approximation. They prove that this flow encapsulates the BO dynamics, satisfying the intertwining relation $L_{u(t)}U(t) = U(t)L_{u(0)}$ and the evolution equation $\partial_t U(t) = P_u(t)U(t)$.
3. Explicit Formula and Asymptotic Profiles: Leveraging the explicit formula $u_+(t, z) = \frac{1}{2\pi i} I_+((X - 2tL_{u_0} - z)^{-1}\Pi u_0)$, the authors analyze the solution's behavior along spacetime rays $x = 2t\lambda$. They decompose the solution into contributions from the discrete spectrum (eigenvalues) and the continuous spectrum.
   • They prove that for any speed $\lambda$ not corresponding to an eigenvalue of $L_{u_0}$, the solution decays to zero in local $L^2$ windows.
   • For speeds corresponding to eigenvalues, the solution converges to a single soliton profile dictated by the associated eigenfunction.
4. Spectral Stability and Equicontinuity: To handle perturbations of multisolitons, the authors establish spectral stability bounds, showing that the $N$ largest eigenvalues of the perturbed Lax operator remain close to the original parameters, while any "spectral dust" (additional eigenvalues generated by the perturbation) is controlled by the size of the $L^2$ perturbation. They further utilize $L^2$-equicontinuity of the BO flow to upgrade weak convergence (in the upper half-plane) to strong local $L^2$ convergence on the real line.

Key Contributions and Results

• Theorem 1.5 (Long-time Asymptotics for General Data): The paper establishes a dichotomy for any $L^2$ solution $u(t)$. On windows traveling at speed $2\lambda$, the solution either converges to zero (if $\lambda$ is not an eigenvalue of the initial Lax operator) or converges to a single soliton profile determined by the eigenfunction associated with $\lambda$. This result rules out the emergence of new coherent structures in the $L^2$ regime, suggesting that any embedded singular continuous spectrum leads only to decaying radiation.
• Theorem 1.4 / Theorem 6.1 (Asymptotic Stability of Multisolitons): The main result proves that $N$-multisolitons are asymptotically stable in $L^2(\mathbb{R})$. Specifically, if the initial data $u_0$ is a small $L^2$ perturbation of an $N$-soliton profile $Q_{\Lambda, \vec{c}}$, the solution $u(t)$ evolves such that, in windows traveling with the constituent solitons, it converges to a new multisoliton profile $Q_{\tilde{\Lambda}, \tilde{c}}(t)$.
  • The new spectral parameters $\tilde{\Lambda}$ and spatial centers $\tilde{c}$ are determined by the initial data and remain close to the original parameters.
  • The multisoliton decouples asymptotically into a sum of well-separated single solitons.
  • The convergence holds in the local $L^2$ topology, and the global $L^2$ distance between the solution and the limiting multisoliton remains bounded by the size of the initial perturbation.

Significance
The paper claims to resolve the asymptotic stability of Benjamin–Ono multisolitons under the weakest possible regularity assumptions ($L^2$), matching the threshold for global well-posedness. By avoiding the stringent spatial decay hypotheses required by inverse scattering techniques, the authors demonstrate that the soliton resolution phenomenon holds even when the initial data lacks rapid decay. The work clarifies the role of the Lax operator's spectral properties in determining long-time dynamics, showing that the discrete spectrum fully characterizes the coherent structures in the asymptotic limit, while the continuous spectrum contributes only to radiation that disperses. This provides a complete description of the long-time behavior of $L^2$ solutions without relying on the existence of embedded singular continuous spectra to generate new structures.