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→ + ∞ of a gas of N-solitons. We show that this gas of solitons in the limit N→ ∞ is slowly approaching a cnoidal wave solution for x→ - ∞ up to terms of order O(1 / x) , while approaching zero exponentially fast for x→ + ∞. 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.
Rigorous Asymptotics of a KdV Soliton Gas / Girotti, M.; Grava, T.; Jenkins, R.; Mclaughlin, K. D. T. R.. - In: COMMUNICATIONS IN MATHEMATICAL PHYSICS. - ISSN 0010-3616. - 384:2(2021), pp. 733-784. [10.1007/s00220-021-03942-1]
Rigorous Asymptotics of a KdV Soliton Gas
Grava, T.
;
2021-01-01
Abstract
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→ + ∞ of a gas of N-solitons. We show that this gas of solitons in the limit N→ ∞ is slowly approaching a cnoidal wave solution for x→ - ∞ up to terms of order O(1 / x) , while approaching zero exponentially fast for x→ + ∞. 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.| File | Dimensione | Formato | |
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