We analytically study the long time and large space asymptotics of a new broad class of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov. These solutions are characterized by a Riemann–Hilbert problem which we show arises as the limit \(N\rightarrow + \infty \) of a gas of N-solitons. We show that this gas of solitons in the limit \(N\rightarrow \infty \) is slowly approaching a cnoidal wave solution for \(x \rightarrow - \infty \) up to terms of order \(\mathcal {O} (1/x)\), while approaching zero exponentially fast for \(x\rightarrow +\infty \). We establish an asymptotic description of the gas of solitons for large times that is valid over the entire spatial domain, in terms of Jacobi elliptic functions.

我们通过分析研究了由Dyachenko，Zakharov和Zakharov提出的KdV方程的一类新的宽解的长时间和大空间渐近性。这些解决方案的特征在于一个黎曼–希尔伯特问题，我们证明它是由N-孤子气体的极限\（N \ rightarrow + \ infty \）引起的。我们证明了在极限\（N \ rightarrow \ infty \）中的这种孤子气体正在逐步接近\（x \ rightarrow-\ infty \）的正弦波解，直到阶数\（\ mathcal {O}（ 1 / x）\），而\（x \ rightarrow + \ infty \）的指数快速接近零。根据雅可比椭圆函数，我们建立了对孤子气体的渐近描述，该描述在整个空间范围内都是有效的。