Calogero–Moser DNLS: Integrability & Soliton Dynamics

• CM-DNLS is a one-dimensional, integrable nonlinear dispersive model derived from the Calogero–Moser system, featuring mass-critical scaling and nonlocal derivative nonlinearity.
• Its Lax pair and bi-Hamiltonian structure yield an infinite hierarchy of conservation laws that facilitate rigorous soliton resolution, scattering, and multi-soliton analysis.
• The model admits explicit ground-state solitons and exhibits quantized blow-up dynamics with precise rates, offering deep insights into nonlinear dispersive phenomena in PDEs.

The Calogero–Moser Derivative Nonlinear Schrödinger Equation (CM-DNLS) is a one-dimensional, mass-critical, completely integrable nonlinear dispersive model featuring a nonlocal derivative nonlinearity derived from the continuum limit of the Calogero–Moser many-body system. The CM-DNLS exhibits a rich array of symmetries and conservation laws, including $L^2$-critical scaling, pseudo-conformal invariance, self-duality, and a Lax pair or bi-Hamiltonian structure. The model admits explicit ground-state solitons, remarkable blow-up dynamics with quantized blow-up rates, rigorously established soliton resolution, and scattering in both focusing and defocusing regimes. Its mathematical analysis intersects harmonic analysis, integrable systems, and nonlinear dispersive PDEs, and recent work has resolved several longstanding questions concerning global well-posedness, soliton resolution, and blow-up classification.

1. Definition, Structural Form, and Symmetries

The CM-DNLS is typically written for a complex field $u:\mathbb{R}_t\times\mathbb{R}_x\to\mathbb{C}$ as

$$i\,\partial_{t}u + \partial_{x}^{2}u + 2\,D_{+}\big(|u|^{2}\big)\,u = 0,$$

where $D_{+} = -i\partial_{x}\,\Pi_{+}$ and $\Pi_{+}$ is the Fourier projection onto nonnegative frequencies: $$\Pi_{+}f = \mathcal{F}^{-1}\left[\mathbf 1_{\xi>0}\widehat f(\xi)\right]$$. The nonlocal term $D_{+}\big(|u|^{2}\big)\,u$ encodes the continuum Calogero–Moser interaction.

Key symmetries include:

• Mass-critical scaling: $$u(t,x)\mapsto \lambda^{-1/2}u(\lambda^{-2}t,\lambda^{-1}x)$$, preserving the $L^2$ norm.
• Pseudo-conformal invariance: $$u(t,x)\mapsto (1/|t|^{1/2})\exp( ix^2/4t )u(-1/t, x/t)$$.
• Galilean invariance, phase, and translation symmetries.

For analysis, the CM-DNLS often employs a gauge transform—$$v = -u\exp\big(-\frac{i}{2}\int_{-\infty}^{x}|u|^2dy\big)$$—yielding the "gauge-equivalent" equation,

$$i\partial_t v + \partial_{xx}v + |D|\big(|v|^{2}\big)v - \frac{1}{4}|v|^{4}v = 0,$$

where $|D|$ is the Fourier multiplier with symbol $|\xi|$ (Gérard et al., 2022, Jeong et al., 2024).

2. Integrability: Lax Pair and Conservation Laws

The CM-DNLS is completely integrable. On the Hardy–Sobolev space $H^s_+(\mathbb{R})$, it admits a Lax pair formulation: $$\partial_t \mathcal{L} = [\mathcal{P},\mathcal{L}],\quad \mathcal{L} = -i\partial_x - u\,\Pi_{+}\overline{u},\ \mathcal{P} = i\partial_{xx} + 2u D_+(\overline{u}) [2208.04105, 2408.12843].$$ This structure yields an infinite hierarchy of conservation laws: $$I_k(u) = \langle L_u^k u, u\rangle_{L^2},\quad k=0,1,2,\dots$$ For the gauge-transformed form, an analogous Lax pair exists using Bogomolny-type operators, yielding self-dual energy functionals and higher-order conserved quantities: $$E(v) = \frac{1}{2}\int\left| \partial_x v + \frac{1}{2}\mathcal{H}(|v|^2) v \right|^2 dx, \qquad M(v) = \int |v|^2 dx$$ where $\mathcal{H}$ denotes the Hilbert transform (Jeong et al., 2024).

3. Soliton Solutions, Multi-Solitons, and Energy Cascade

A hallmark of CM-DNLS is its explicit soliton structure:

• The unique ground state is $\mathcal{R}(x) = \sqrt{2}/(x+i)$, with critical mass $2\pi$.
• All traveling waves are classified as Galilean boosts of $\mathcal{R}$, retaining the critical mass.
• Multi-soliton solutions exist for quantized masses $2\pi N$, produced via rational Hardy potentials $u_{0}=P/Q$ with $Q(x) = \prod_{j=1}^N(x-z_j)$ and $\Im z_j<0$ (Gérard et al., 2022).

Strikingly, multi-soliton dynamics exhibit a strong energy cascade: $$\|u(t)\|_{H^s} \sim |t|^{2s},\qquad t\to\infty,\ s>0,$$ with unbounded Sobolev norm growth for all $s>0$ (Gérard et al., 2022).

4. Blow-Up Phenomena and Quantized Dynamics

Unlike classical DNLS, CM-DNLS admits finite-time blow-up even for chiral initial data. Several dynamical regimes are established:

A. Chiral finite-time blow-up: Smooth initial data in the chiral class $L^2_+$ near the soliton mass yield blow-up at a slower, quantized rate $\lambda(t)\sim(T-t)^{2}$ rather than the pseudo-conformal $(T-t)$ rate (Kim et al., 2024).

B. Quantized blow-up regime: For any integer $L\geq1$, smooth (even/radial) data generate finite-time blow-up at rates $\lambda(t)\sim(T-t)^{2L}$, precisely matching a discrete quantization (Jeong et al., 2024).

C. Complete classification: For initial data in $H^{2L+1}$, any single-soliton blow-up must be either quantized with a rate $\lambda(t)\sim(T-t)^{2k}$ for some $1\leq k\leq L$, or "exotic" with $\lambda(t)\lesssim (T-t)^{2L+3/2}$ (Jeong et al., 12 Jan 2026). The approach utilizes modulation decomposition, the tower of nonlinear adapted derivatives, and a finite recursive system of ODEs governing the scaling parameter.

5. Soliton Resolution and Asymptotic Dynamics

Recent work has rigorously established soliton resolution for CM-DNLS: any solution either blows up or decomposes, as $t\to T$,

$$u(t) - \sum_{j=1}^N [\mathcal{R}]_{\lambda_j,\gamma_j,x_j} \longrightarrow z^* \quad\text{in }L^2,$$

with $N$ dictated by the initial mass, modulation parameters converging, and the remainder $z^*$ reflecting the "dispersive tail" (Kim et al., 2024). This resolution is achieved without radial symmetry, leveraging an "energy bubbling" argument that exploits both mass and self-dual energy conservation.

For global solutions, soliton resolution yields scattering to a free solution or an asymptotic multi-soliton profile plus dispersion.

6. Scattering, Zero-Dispersion Limit, and Explicit Formulae

The defocusing CM-DNLS is globally well-posed in $L^\infty\cap H^2\cap\{|u|^2-1\in L^2\}$ and supports scattering to linear solutions for data in weighted chiral Hardy–Sobolev spaces $H_{+}^{1,\alpha}$, $\alpha>1/4$ (Chen, 9 Nov 2025, Chen, 25 Feb 2025). The analysis is made explicit via a Gérard-type formula and the construction of a distorted Fourier transform associated to the Lax operator. For zero-dispersion limits, the solution converges weakly to a multivalued profile matching the inviscid Burgers–Hopf system, satisfying a maximum principle and exhibiting dispersive shock phenomena (Badreddine, 2024).

7. Periodic, Breather, and Traveling Wave Solutions

CM-DNLS admits a broad family of periodic and breather solutions on both zero and nonzero backgrounds. Periodic solutions (cn-type) and their stability, explicit breather constructions parameterized by determinant formulas, and the analysis of their Lax spectra are developed via Hirota's bilinear methods (Chen et al., 26 Jan 2025). These solutions are relevant in contexts such as stratified fluids and provide a bridge between integrability and physical applications.

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References:

• (Gérard et al., 2022) The Calogero--Moser Derivative Nonlinear Schrödinger Equation
• (Badreddine, 2024) Zero dispersion limit of the Calogero-Moser derivative NLS equation
• (Kim et al., 2024) Construction of smooth chiral finite-time blow-up solutions to Calogero--Moser derivative nonlinear Schrödinger equation
• (Kim et al., 2024) Soliton resolution for Calogero--Moser derivative nonlinear Schrödinger equation
• (Jeong et al., 2024) Quantized blow-up dynamics for Calogero--Moser derivative nonlinear Schrödinger equation
• (Chen et al., 26 Jan 2025) Traveling periodic waves and breathers in the nonlocal derivative NLS equation
• (Chen, 9 Nov 2025) Scattering of the defocusing Calogero--Moser derivative nonlinear Schrödinger equation
• (Jeong et al., 12 Jan 2026) Classification of single-bubble blow-up solutions for Calogero--Moser derivative nonlinear Schrödinger equation
• (Chen, 25 Feb 2025) The defocusing Calogero--Moser derivative nonlinear Schrödinger equation with a nonvanishing condition at infinity