Large-space and Large-time Asymptotics for the Focusing Nonlinear Schrödinger Soliton Gas
Abstract
We investigate the large-space and large-time asymptotic behavior of a soliton gas for the focusing nonlinear Schrödinger equation. The soliton gas is constructed as the continuum limit of pure N-soliton solutions as N→∞, with the discrete spectrum confined to two segments Σ1 and Σ2. In particular, our framework does not require the discrete spectrum to be confined to the imaginary axis. By combining the nonlinear steepest descent method with an appropriate g-function mechanism, we show that, as x→−∞, the soliton gas is asymptotically described by a finite-gap elliptic solution with constant coefficients. In the large-time regime t→+∞, we assume that the endpoint F lies on the trajectory of H(ξ) with ξ=x/2t∈(−E1−√2E2,−E1), namely, F=H(ξ̂), ξ̂∈ (−E1−√2E2,−E1). Under this assumption, we prove that the solution exhibits distinct asymptotic behaviors in different regions of the variable ξ=x/2t. More precisely, there exist an exponentially decaying region ξ∈(−E1,+∞), a modulated elliptic-wave region ξ∈(ξ̂,−E1), and an unmodulated elliptic-wave region ξ∈(−∞,ξ̂).
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BibTeX
@misc{yan2026largespace,
title = {{Large-space and Large-time Asymptotics for the Focusing Nonlinear Schrödinger Soliton Gas}},
author = {Dedi Yan and Xianguo Geng and Jiao Wei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.21091},
eprint = {2605.21091},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.21091},
}
RIS
TY - UNPB
TI - Large-space and Large-time Asymptotics for the Focusing Nonlinear Schrödinger Soliton Gas
AU - Dedi Yan
AU - Xianguo Geng
AU - Jiao Wei
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2605.21091
UR - https://arxiv.org/abs/2605.21091
AB - We investigate the large-space and large-time asymptotic behavior of a soliton gas for the focusing nonlinear Schrödinger equation. The soliton gas is constructed as the continuum limit of pure N-soliton solutions as N→∞, with the discrete spectrum confined to two segments Σ_1 and Σ_2. In particular, our framework does not require the discrete spectrum to be confined to the imaginary axis. By combining the nonlinear steepest descent method with an appropriate g-function mechanism, we show that, as x→−∞, the soliton gas is asymptotically described by a finite-gap elliptic solution with constant coefficients. In the large-time regime t→+∞, we assume that the endpoint F lies on the trajectory of H(ξ) with ξ=x/2t∈(−E_1−√2E_2,−E_1), namely, F=H(ξ̂), ξ̂∈ (−E_1−√2E_2,−E_1). Under this assumption, we prove that the solution exhibits distinct asymptotic behaviors in different regions of the variable ξ=x/2t. More precisely, there exist an exponentially decaying region ξ∈(−E_1,+∞), a modulated elliptic-wave region ξ∈(ξ̂,−E_1), and an unmodulated elliptic-wave region ξ∈(−∞,ξ̂).
ER -