Infinite-order multisoliton solutions to the Benjamin--Ono equation and soliton resolution

1. Problem Statement

The paper investigates the long-time asymptotic behavior of solutions to the Benjamin–Ono (BO) equation on the real line:
$$\partial_t u - \partial_x |D_x| u + \partial_x (u^2) = 0, \quad u(0, x) = u_0(x)$$
where $|D_x|$ is the Fourier multiplier with symbol $|\xi|$.

Context and Gap:

• Finite Soliton Resolution: Previous work (specifically [GGM26]) established that for initial data $u_0$ with sufficient spatial decay (specifically $x u_0 \in L^2(\mathbb{R})$), the solution decomposes as $t \to \infty$ into a finite superposition of solitons plus a radiative remainder that vanishes in Sobolev spaces.
• The Challenge: This paper addresses the case where the initial data has slow spatial decay such that $u_0 \in L^2(\mathbb{R})$ but $x u_0 \notin L^2(\mathbb{R})$. In this regime, the number of solitons involved in the solution is infinite.
• Objective: To construct a class of "infinite-order multisoliton" solutions and prove that they still exhibit a "soliton resolution" phenomenon, decoupling into an infinite superposition of independent solitons with distinct velocities, without a radiative term.

2. Methodology

The authors employ the integrability of the Benjamin–Ono equation, specifically utilizing the Lax pair structure and a recent explicit formula for the solution derived by Gérard [Gé23].

A. The Explicit Formula

The proof relies on the explicit representation of the solution $u(t, x)$ via the Lax operator $L_{u_0}$ and the adjoint of the multiplication operator $X^*$.

• Lax Operator: $L_{u_0} = -i \frac{d}{dx} - T_{u_0}$, acting on the Hardy space $L^2_+(\mathbb{R})$.
• Explicit Solution: The holomorphic extension of the projection $\Pi u(t, z)$ is given by:
  $$\Pi u(t, z) = \frac{1}{2i\pi} I_+ \left( (X^* - 2t L_{u_0} - z)^{-1} \Pi u_0 \right)$$
  where $I_+$ denotes the limit of the Fourier transform at $0^+$.

B. Spectral Analysis of the Lax Operator

A crucial step is analyzing the spectral properties of $L_{u_0}$ for the specific class of initial data defined by an infinite sum of soliton profiles.

• Eigenfunctions: The authors prove that the eigenfunctions $\phi_j$ of $L_{u_0}$ associated with negative eigenvalues $\lambda_j$ belong to the domain of $X^*$ ($D(X^*)$). This is non-trivial because the standard assumption $x u_0 \in L^2$ is violated.
• Key Identity: They establish a "distorted Plancherel formula" linking the $L^2$ norm of the initial data to the eigenvalues:
  $$\|\Pi u_0\|_{L^2}^2 = \sum_{k=1}^\infty 2\pi |\lambda_k|$$
  This identity confirms that the infinite sum of solitons corresponds to an infinite discrete spectrum of the Lax operator.

C. Asymptotic Analysis

The long-time behavior is analyzed by studying the operator $\Omega_t$ defined in the explicit formula.

• Decoupling: By decomposing the initial data into components parallel and orthogonal to the eigenfunctions of $L_{u_0}$, the authors show that as $t \to \infty$, the term associated with the $j$-th eigenvalue concentrates around the velocity $c_j = 1/\text{Im}(p_j^\infty) = 2|\lambda_j|$.
• Remainder Control: They prove that the interaction terms between different solitons and the radiative part vanish in the limit, provided the number of solitons $N$ is taken to infinity after the time limit $t \to \infty$.

3. Key Contributions

1. Construction of Infinite-Order Multisolitons:
   The authors define a class of initial data $u_0$ as an infinite sum of soliton profiles:
   $$u_0(x) = \sum_{j=1}^\infty \frac{2\text{Im}(p_j)}{|x + p_j|^2}$$
   subject to a convergence condition on the parameters $(p_j)$:
   $$\sum_{j,k} \frac{\text{Im}(p_j)}{|p_j - p_k|^2} < \infty$$
   They prove that despite $x u_0 \notin L^2$, this data belongs to $H^s(\mathbb{R})$ for all $s \ge 0$.

2. Extension of Soliton Resolution:
   They generalize the soliton resolution conjecture to the infinite-order case. The solution $u(t, x)$ asymptotically behaves as an infinite superposition of solitons:
   $$u(t, x) \approx \sum_{j=1}^\infty R_{p_j^\infty}(x - c_{p_j^\infty} t)$$
   where $R_p$ is the standard soliton profile.

3. Refinement of Spectral Theory:
   The paper provides a rigorous proof that eigenfunctions of the Lax operator for $L^2$ potentials (without $x u_0 \in L^2$) lie in the domain of $X^*$. This allows the application of the explicit formula without the standard decay assumptions.

4. Inverse Spectral Problem Insight:
   The authors identify a map from the initial parameters $(p_j)$ to the asymptotic parameters $(p_j^\infty)$. They note that characterizing the range of this map is equivalent to solving an inverse spectral problem for the Lax operator, which remains an open problem.

4. Main Results

Theorem 1.2 (Soliton Resolution for Infinite-Order Multisolitons):
Let $u_0$ be an infinite-order multisoliton satisfying the condition (1.4).

1. Regularity: $u_0 \in H^s(\mathbb{R})$ for all $s \in \mathbb{R}$.
2. Asymptotic Decomposition: There exists a sequence of parameters $(p_j^\infty)$ with strictly increasing imaginary parts such that:
   $$\lim_{N \to \infty} \limsup_{t \to \infty} \left\| u(t, \cdot) - \sum_{j=1}^N R_{p_j^\infty}(\cdot - c_{p_j^\infty} t) \right\|_{L^2} = 0$$
3. Global Convergence: The infinite sum $u_{sol}(t, x) = \sum_{j=1}^\infty R_{p_j^\infty}(x - c_{p_j^\infty} t)$ defines a continuous function. Furthermore, for any $s \ge 1/2$:
   $$\| u(t, \cdot) - u_{sol}(t, \cdot) \|_{L^\infty} + \| u(t, \cdot) - u_{sol}(t, \cdot) \|_{\dot{H}^s} \xrightarrow{t \to \infty} 0$$
   Crucially, the radiative remainder term vanishes completely in the limit; the solution is purely a superposition of solitons.

5. Significance

• Theoretical Advancement: This work bridges the gap between finite and infinite soliton interactions in integrable systems. It demonstrates that the "soliton resolution" phenomenon is robust even when the initial data lacks the standard moment conditions ($x u_0 \in L^2$) required for finite soliton counts.
• Methodological Innovation: The ability to handle the domain of the operator $X^*$ for potentials with slow decay is a significant technical achievement in spectral theory for the Benjamin–Ono equation.
• Open Problems: The paper highlights the complexity of the inverse spectral problem for these specific potentials, suggesting that while the forward evolution is understood, the precise characterization of the asymptotic parameters from the initial data remains a challenging open question.

In summary, Gassot and Gérard successfully extend the soliton resolution theory to infinite-order solutions of the Benjamin–Ono equation, proving that even with slow-decaying initial data, the long-time dynamics are governed entirely by a countable family of non-interacting solitons.