arXiv subjects

Dedi Yan

Publications and source records attributed to Dedi Yan.

5 recordsLinked to original sources

arXiv2026

Long-time asymptotics of a full arbitrary-genus dark soliton gas for the defocusing nonlinear Schrodinger equation

We introduce a full arbitrary-genus dark soliton gas for the defocusing nonlinear Schr\"odinger equation with finite-density boundary conditions. Starting from a generalized meromorphic Riemann--Hilbert problem with two alternating residue families on each unit-circle arc, we derive an exact thermodynamic limit whose jump matrix contains two nonzero continuum densities. The limiting Riemann--Hilbert problem is uniquely solvable. In contrast with the half dark-soliton gas, every spectral arc of the full gas carries both oscillatory exponentials. We analyze the resulting problem by the Deift--Zhou nonlinear steepest-descent method on a fixed genus-$N$ spectral curve. The moving point in each mixed sector is a stationary factorization-switching point, not a branch point. The active arc is split into two parts and opened crosswise, while lenses are opened around every remaining arc. After removal of exponentially small lens jumps, the model contour therefore retains all $N$ spectral arcs in every self-similar sector. A quotient-curve zero-counting argument proves strict monotonicity of the characteristic velocity and the global ordering of all endpoint velocities, so the self-similar line is divided into $2N+1$ nonempty sectors. The leading term is an $N$-dimensional Riemann-theta finite-gap solution. The error is $O(t^{-1})$ in the $N+1$ pure sectors and $O(t^{-1/2})$ in the $N$ mixed sectors, uniformly away from the critical rays.

nlin.SI↗

arXiv2026

Large-time asymptotics of a new KdV soliton gas

We study the large-time asymptotic behavior of a new KdV soliton gas. We first introduce a pure-soliton Riemann--Hilbert(RH) problem with $2N$ poles and two different types of residue conditions. We show that, as $N\to\infty$, this discrete problem converges to primitive-potential RH problem introduced by Dyachenko, Zakharov, and Zakharov, and the jump matrix of this soliton gas RH problem has two nonzero reflection coefficients. To analyze the large-time behavior, we apply the Deift--Zhou nonlinear steepest descent method together with an appropriate $g$-function mechanism. Through a sequence of transformations, the original RH problem is reduced to explicitly solvable model problems on an associated hyperelliptic Riemann surface. This allows us to derive an explicit leading-order asymptotic formula for the solution in terms of Jacobi elliptic function. The result provides a rigorous asymptotic description of a new KdV soliton gas and extends the available analysis beyond the previously studied case $r_2\equiv 0$.

nlin.SI↗

arXiv2026

Long-time Asymptotics of a Full Camassa-Holm Soliton Gas

We investigate the long-time asymptotics of a full soliton gas for the Camassa--Holm equation. The analysis starts from a pure-soliton Riemann--Hilbert (RH) problem with $2N$ poles and two distinct types of residue conditions. We prove that, as $N\to\infty$, this discrete RH problem converges to a limiting soliton gas RH problem whose jump matrix contains two nonzero reflection coefficients. In this sense, the limiting problem gives a full soliton gas model for the Camassa--Holm equation, in contrast to the previously studied half soliton gas models, whose jump matrices involve only one nonzero reflection coefficient. The limiting RH problem is analyzed by the Deift--Zhou nonlinear steepest descent method. The presence of two nonzero reflection coefficients requires two different types of triangular factorizations of the jump matrix and leads to a more delicate $g$-function mechanism. The main difficulty lies in the construction of suitable $g$-functions adapted to the Camassa--Holm phase, together with the precise control of their behavior near the distinguished point $k=i/2$ and at infinity. Depending on the location of the spectral endpoints $\eta_1$ and $\eta_2$, different $g$-function mechanisms arise. In this paper, we focus on Case I and derive the long-time asymptotic formulas in three elliptic-wave regions of the self-similar plane. In each region, the leading term is given by a finite-gap elliptic function, while in the central region the first correction is of order $\mathcal O(t^{-1/2})$ and involves parabolic cylinder functions.

nlin.SI↗

arXiv2026

Large-space and Large-time Asymptotics for the Focusing Nonlinear Schr\"{o}dinger Soliton Gas

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 $\xi\in(-\infty,\hat{\xi})$.

nlin.SI↗

arXiv2026

Large-space and large-time asymptotics for the mKdV soliton gas with any odd genus

We study the large-space and large-time asymptotic behavior of the soliton gas of genus $2n-1$ for the mKdV equation with $n\in \mathbb{N}_+$. As $x \to +\infty$, we show that the large-space asymptotics of the mKdV soliton gas can be expressed with the Riemann-theta function of genus $2n-1$. For large $t$, based on the nonlinear steepest descent method and $g$-function approach, we establish a global large-time asymptotic description of the mKdV soliton gas. The half-plane $\{(x,t):-\infty 0\}$ is divided into $2n+1$ separated regions. In each region, the large-time asymptotics of the mKdV soliton gas is given by using the Riemann-theta functions and uniform error estimation.

nlin.SI↗