2018/07/31 by M. Girotti, Manuela Girotti, T. Grava +5
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Class (philosophy) #Korteweg–de Vries equation #Limit (mathematics) #Nonlinear Waves and Solitons #Nonlinear system #Order (exchange) #Quantum Mechanics and Non-Hermitian Physics #Soliton #Space (punctuation) #Zero (linguistics) #math-ph #math.AP #math.MP #nlin.PS #nlin.SI
paper · pdf · doi:10.1007/s00220-021-03942-1
42 pages, 7 figures. To appear in Comm. Math. Physics
openalex created_date 2018/07/10 · arxiv created 2021/03/21 · arxiv updated 2021/03/23 · openalex publication_date 2021/04/25 · openalex updated_date 2026/08/06
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→ + ∞ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>→</mml:mo> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> of a gas of N -solitons. We show that this gas of solitons in the limit N→ ∞ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> is slowly approaching a cnoidal wave solution for x → - ∞ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>→</mml:mo> <mml:mo>-</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> up to terms of order \mathcal O (1/x) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , while approaching zero exponentially fast for x→ +∞ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>→</mml:mo> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> . 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.