Rigorous Asymptotics of a KdV Soliton Gas
Rigorous Asymptotics of a KdV Soliton Gas
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DOI：
10.1007/s00220-021-03942-1
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发表时间：
2021-04-25
影响因子：
2.4
通讯作者：
McLaughlin, K. D. T-R
中科院分区：
文献类型：
--
作者：
Girotti, M.;Grava, T.;McLaughlin, K. D. T-R
我们解析地研究了由Dyachenko，Zakharov和Zakharov引入的KdV方程一类新的广义解的长时间和大空间渐近性。这些解的特征是一个Riemann-Hilbert问题，我们表明这个问题是由N个孤子组成的气体的极限N-+无穷大引起的。我们证明了这种孤子气体在极限N->无穷大时正缓慢地逼近x->无穷大到O(1/x)阶项的抛物线波解，而当x->+无穷大时则以指数形式快速逼近零。我们用雅可比椭圆函数建立了在整个空间域中有效的大时间孤子气体的渐近描述。
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 ->+infinity of a gas of N-solitons. We show that this gas of solitons in the limit N ->infinity is slowly approaching a cnoidal wave solution for x ->-infinity up to terms of order O(1/x), while approaching zero exponentially fast for x ->+infinity. 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.