Derives long-time asymptotics of a full arbitrary-genus dark soliton gas for defocusing NLS, yielding an N-dimensional Riemann-theta finite-gap solution with O(t^{-1}) or O(t^{-1/2}) errors in different sectors.
Large-space and Large-time Asymptotics for the Focusing Nonlinear Schr\"{o}dinger Soliton Gas
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abstract
We investigate the large-space and large-time asymptotic behavior of a soliton gas for the focusing nonlinear Schr\"odinger equation. The soliton gas is constructed as the continuum limit of pure $N$-soliton solutions as $N\to\infty$, with the discrete spectrum confined to two segments $\Sigma_1$ and $\Sigma_2$. 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 $x\to-\infty$, the soliton gas is asymptotically described by a finite-gap elliptic solution with constant coefficients. In the large-time regime $t\to+\infty$, we assume that the endpoint $F$ lies on the trajectory of $H(\xi)$ with $\xi=\frac{x}{2t}\in(-E_1-\sqrt{2}E_2,-E_1)$, namely, $F=H(\hat{\xi})$, $\hat{\xi}\in (-E_1-\sqrt{2}E_2,-E_1)$. Under this assumption, we prove that the solution exhibits distinct asymptotic behaviors in different regions of the variable $\xi=\frac{x}{2t}$. More precisely, there exist an exponentially decaying region $\xi\in(-E_1,+\infty)$, a modulated elliptic-wave region $\xi\in(\hat{\xi},-E_1)$, and an unmodulated elliptic-wave region $\xi\in(-\infty,\hat{\xi})$.
fields
nlin.SI 3years
2026 3verdicts
UNVERDICTED 3representative citing papers
Long-time asymptotics for the full Camassa-Holm soliton gas are obtained from a limiting RH problem with two reflection coefficients, producing elliptic-function leading terms in three regions.
Derives explicit leading-order large-time asymptotics for a new KdV soliton gas with two nonzero reflection coefficients, expressed via Jacobi elliptic functions on a hyperelliptic surface.
citing papers explorer
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Long-time asymptotics of a full arbitrary-genus dark soliton gas for the defocusing nonlinear Schrodinger equation
Derives long-time asymptotics of a full arbitrary-genus dark soliton gas for defocusing NLS, yielding an N-dimensional Riemann-theta finite-gap solution with O(t^{-1}) or O(t^{-1/2}) errors in different sectors.
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Long-time Asymptotics of a Full Camassa-Holm Soliton Gas
Long-time asymptotics for the full Camassa-Holm soliton gas are obtained from a limiting RH problem with two reflection coefficients, producing elliptic-function leading terms in three regions.
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Large-time asymptotics of a new KdV soliton gas
Derives explicit leading-order large-time asymptotics for a new KdV soliton gas with two nonzero reflection coefficients, expressed via Jacobi elliptic functions on a hyperelliptic surface.