On the convergence of explicit formulas for $L^2$ solutions to the Benjamin-Ono and continuum Calogero-Moser equations

1. Problem Statement

The paper addresses the numerical approximation of two nonlocal, integrable dispersive partial differential equations (PDEs) on the torus $\mathbb{T}$:

1. The Benjamin–Ono (BO) Equation:
   $$\partial_t u = \partial_x (|\partial_x|u - u^2)$$
   This models internal waves in deep stratified fluids.
2. The Continuum Calogero–Moser (CCM) Equation:
   $$i\partial_t u + \partial_{xx} u \pm \frac{2}{i} u \partial_x \Pi(|u|^2) = 0$$
   This is a derivative nonlinear Schrödinger (DNLS) type equation, related to the Calogero–Moser particle system.

The Core Challenge:
Existing numerical convergence results for these equations typically require initial data with high regularity (e.g., $u_0 \in H^s$ with $s > 1$). However, the equations are globally well-posed for rough initial data in $L^2(\mathbb{T})$ (for BO) and $L^2_+(\mathbb{T})$ (for CCM, the Hardy space). The authors aim to construct and analyze numerical schemes that converge for these low-regularity $L^2$ solutions, a regime where standard spectral or finite-difference methods often fail to provide rigorous error bounds or structure preservation.

2. Methodology

Instead of discretizing the PDEs directly (e.g., via finite differences or standard spectral methods), the authors exploit the complete integrability of the systems.

A. Explicit Formulas

The method relies on recent discoveries of explicit formulas for the solutions, which express the Fourier coefficients of the solution $u(t)$ in terms of the initial data $u_0$ and specific operators:

• Riesz–Szegő Projector ($\Pi$): Projects onto non-negative frequencies.
• Shift Operator ($S^*$): A left-shift in frequency space.
• Lax Operators ($L_{u_0}$): Semi-bounded self-adjoint operators defined on the Hardy space.
  • For BO: $L_{u_0} f = -i\partial_x f - \Pi(u_0 f)$.
  • For CCM: $L_{u_0} f = -i\partial_x f \mp u_0 \Pi(u_0 f)$.

The explicit formulas for the $k$-th Fourier coefficient $\hat{u}(t, k)$ are given by:
$$\hat{u}(t, k) = \langle (e^{it(I+2L_{u_0})}S^*)^k \Pi u_0, 1 \rangle \quad (\text{BO})$$
$$\hat{u}(t, k) = \langle (e^{-it(I+2L_{u_0})}S^*)^k u_0, 1 \rangle \quad (\text{CCM})$$

B. Discretization Strategy

The authors design a fully-discrete scheme by truncating the operators in the explicit formulas:

1. Spatial Discretization: Replace the infinite-dimensional operators with finite-dimensional truncations using frequency cut-offs $\Pi_n$.
2. Iterative Truncation: Unlike previous work [3] which used a fixed truncation $K$ for all iterations, this paper introduces a sequence of truncation parameters $n_K(k)$ for each iteration step $k$.
   • The scheme computes $\hat{u}_K(t, k)$ using a truncated Lax operator $L_{n(k)}$.
   • This flexibility allows for the construction of schemes that preserve specific physical quantities (mass, momentum, $L^2$ norm).

C. Key Analytical Tools

To prove convergence for $L^2$ data, the authors avoid standard energy estimates used for smooth solutions. Instead, they utilize:

• Operator Theory: Treating the truncated Lax operators $L_n$ as perturbations of the free operator $L_0 = -i\partial_x$.
• Resolvent Convergence: Proving that the resolvents $(L_n + \kappa)^{-1}$ converge in operator norm to $(L + \kappa)^{-1}$ as $n \to \infty$.
• Unitary Group Convergence: Using the norm resolvent convergence to establish that the unitary groups $e^{itL_n}$ converge strongly to $e^{itL}$ uniformly on compact time intervals and compact subsets of $L^2$.
• Compactness Arguments: Combining weak convergence (derived from the operator limits) with the boundedness of the $L^2$ norm to upgrade the result to strong $L^2$ convergence.

3. Key Contributions

1. Generalization of Numerical Schemes

The authors generalize the scheme from [3] by introducing variable truncation parameters $n_K(k)$. This allows for the design of structure-preserving schemes:

• Mass Conservation: All schemes in this class preserve the average (mass) of the solution.
• $L^2$ Norm Preservation: By choosing $n(k) \leq K - k$, the authors construct schemes that exactly preserve the $L^2$ norm of the truncated initial data on the discrete level. This is a significant improvement over previous methods where the discrete $L^2$ norm might drift.

2. Convergence for Rough Data ($L^2$)

The primary theoretical contribution is the proof of convergence for initial data $u_0 \in L^2(\mathbb{T})$ (BO) and $u_0 \in L^2_+(\mathbb{T})$ (CCM).

• Theorem 1.3: Establishes that the $L^2$-norm of the error $\|u_K(t) - u(t)\|_{L^2}$ goes to zero as the truncation parameters $K, n_K(k) \to \infty$, uniformly on any bounded time interval $[-T, T]$.
• This is the first error analysis result establishing convergence in $C([0, T], L^2)$ for arbitrary $L^2$ initial data for these equations.

3. Computational Efficiency

The schemes are exact in time. Since the explicit formulas involve matrix exponentials of the truncated Lax operators, the time evolution is computed exactly (up to machine precision) without time-stepping errors. The computational cost depends only on the spatial discretization $K$ and is independent of the final time $T$, allowing for simulations over arbitrarily long times.

4. Main Results

• Convergence Theorem: For any $u_0 \in L^2$, the numerical solution $u_K(t)$ converges strongly to the true solution $u(t)$ in $L^2(\mathbb{T})$ uniformly for $t \in [-T, T]$.
• Asymptotic Conservation: The $L^2$ norm of the numerical solution converges to the $L^2$ norm of the exact solution.
• Talbot Effect Evidence: The authors apply their scheme to the BO equation with a square-wave initial profile (which is in $L^2$ but not in $H^{1/2}$). They observe the Talbot effect (a phenomenon where the solution exhibits fractal-like behavior and discontinuities at rational times).
  • This provides the first numerical evidence of the Talbot effect for the nonlinear BO equation, supported by a rigorous convergence theorem.
  • The results suggest that the nonlinear flow shares the discontinuity structure of the linear flow, a property that standard smoothing estimates (like those for KdV or NLS) cannot explain for BO.

5. Significance

• Bridging Integrability and Numerics: The paper successfully translates deep theoretical results in nonlinear integrability (explicit formulas, Lax pairs) into practical, rigorous numerical algorithms.
• Low-Regularty Analysis: It breaks the barrier of requiring high Sobolev regularity ($s>1$) for convergence proofs, addressing the critical scaling regularity ($L^2$) where the equations are well-posed.
• Structure Preservation: The ability to design schemes that exactly conserve mass and $L^2$ norm at the discrete level is crucial for long-time simulations of integrable systems, preventing artificial energy dissipation or blow-up.
• New Phenomenological Insights: By enabling stable simulation of rough data (like square waves), the method opens the door to studying complex phenomena like the Talbot effect in nonlinear dispersive systems, which were previously inaccessible to rigorous numerical analysis.

In summary, this work provides a robust framework for simulating integrable PDEs with minimal regularity assumptions, leveraging the unique algebraic structure of the equations to achieve exact time integration and rigorous convergence guarantees.