Soliton Synchronization with Randomness: Rogue Waves and Universality

TL;DR

This work analyzes the $N$-soliton solutions of the focusing nonlinear Schrödinger equation to reveal a universal collision signature. By combining deterministic and stochastic inverse-scattering techniques (Darboux dressing and Riemann-Hilbert methods), it shows that a synchronized multi-soliton collision yields a local $\operatorname{sinc}$-profile at the collision point when velocities are well separated and amplitudes are matched, and this persists under i.i.d. sub-exponential amplitudes, establishing a universal central peak. The authors prove a precise leading-order linear interaction formula with a controllable nonlinear correction, and they derive Central Limit Theorems for fluctuations in both near-field and far-field regimes, with universal limiting distributions (Gaussian and Hoyt) that are independent of the amplitude distribution. This provides rigorous justification for rogue-wave formation as a coherent, universal outcome of soliton interactions and links dilute soliton gases to universal collision dynamics with probabilistic limits. The results have implications for rogue-wave modeling in optics and hydrodynamics, and for understanding nonlinear wave coherence in integrable systems under randomness.

Abstract

We consider an $N$-soliton solution of the focusing nonlinear Schrödinger equations. We give conditions for the synchronous collision of these $N$ solitons. When the solitons velocities are well separated and the solitons have equal amplitude, we show that the local wave profile at the collision point scales as the $\operatorname{sinc}(x)$ function. We show that this behaviour persists when the amplitudes of the solitons are i.i.d. sub-exponential random variables. Namely the central collision peak exhibits universality: its spatial profile converges to the $\operatorname{sinc}(x)$ function, independently of the distribution. We derive Central Limit Theorems for the fluctuations of the profile in the near-field regime (near the collision point) and in the far-regime.

Figures (5)

• Figure 1: A comparison of equal spaced vs randomly spaced velocities in Corollary \ref{['cor:sinc']}. a) plot of the scaled $N$-soliton solution $\left|\frac{1}{N} \psi_N\left(\frac{2X}{N V},1\right)\right|$ at collision time (blue curve) with parameters $N = 50$, $\mu = 2, V=50$, and equally spaced velocities $v_k = kV$, $k=1,\dots, N$, compared with the envelope $\mu \operatorname{sech}\left(\frac{2\mu X}{N V}\right)$ (red curve, see \ref{['eq:psiN.rescaled']} with $T=0$)) for $|X| < 2500$. b) comparison between the same function (blue curve) and $\frac{2\sin X}{X}$ (red curve) for $|X| < 50$. c) and d): same setup as in the first two panels, but with perturbed velocities $v_k = (k + \nu_k)V$, where $\nu_k$ are i.i.d. random variables uniformly distributed on $[-\tfrac{1}{5}, \tfrac{1}{5}]$.
• Figure 2: Top: the function $|\psi_3(x,t)|$ with $V=\Delta=20$ and $\mu=2$ at times $t=.7$ (yellow), $t=1$ (blue), and $t=1.3$ (red) for $|x|<20$. Middle: comparison between $|\psi_3(x,1)|$ (blue) and a shifted and scaled sinc profile (red) for $|x|<2$. Bottom: 3D graph of $|\psi_3(x,t)|$ for $|x|<10$ and $|t-1|<0.2$. The setting is the one analyzed in Corollary \ref{['cor:sinc']} with $v_k=k V$.
• Figure 3: Numerical simulation (red) of the solution of the fNLS equation compared to the theoretical prediction (dashed black) \ref{['eq:sinc_univeral']}. The number of solitons $N=400$ and $\Delta = 200$. Here we are in the near-field, and we have both rescaled the $x$ axis, and divided the solution by $\mu_{\mathcal{D}} N$. The amplitudes $\mu_j$'s are sampled according to a $\chi(2)$-distribution (top left), a $\text{Beta}_{2,2}$ distribution (top right), a uniform $(0,1)$ distribution (bottom left) and an exponential distribution with parameter $\lambda=1$ (bottom right). To realize this picture, we average over $1000$ trials.
• Figure 4: Fluctuation of $\psi_N(x,t)$ with respect to the average solution, $N=1200$, $\Delta = 500$, 200,000 trials. Top panel: $\mu_j$'s are i.d.d. $\text{Beta}_{2,2}$ distribution, left side the fluctuations of $|\psi_N(x,t)|$ , right side the one of $\operatorname{Re}(\psi_N(x,t))$ and $\operatorname{Im}(\psi_N(x,t))$. Bottom panel: $\mu_j$'s are i.i.d. uniform distribution in $(0,1)$, left side the fluctuations of $|\psi_N(x,t)|$ , right side the one of $\operatorname{Re}(\psi_N(x,t))$ and $\operatorname{Im}(\psi_N(x,t))$.
• Figure 5: Macroscopic profile: numerical simulation (red) of $|\psi_N(x,1)|$, theoretical prediction (dashed black) \ref{['eq:clt_envelope_mod']} and the envelope $|\omega_{\mathcal{D}}(x)|$ (blue) \ref{['eq:omega_D']}. The number of solitons $N=400$ and $\Delta = 200$. The $\mu_j$'s are sampled according to a $\chi(2)$-distribution (top left), a $\text{Beta}_{2,2}$ distribution (top right), a uniform $(0,1)$ distribution (bottom left) and an exponential distribution with parameter $\lambda=1$ (bottom right). To realize this picture, we averaged over $1000$ trials.

Theorems & Definitions (36)

• Proposition 1.1
• Theorem 1.2
• Remark 1.1
• Corollary 1.3
• Proposition 1.4: Universal $\textrm{sinc}$-profile
• Theorem 1.5: Central Limit Theorem in the near-field regime
• Remark 1.2
• Remark 1.3
• Theorem 1.6: Central Limit Theorem for the global profile at collision time
• Lemma 2.2