Asymptotic linear stability of Benney–Luke line solitary waves in 2D. (7th August 2017)

• Record Type: Journal Article
• Authors: Mizumachi, Tetsu; Shimabukuro, Yusuke
• Is Part Of: Nonlinearity. Volume 30:Number 9(2017:Sep.)
• Page Start: 3419
• Page End: 3465
• Publication Date: 2017-08-07
• Subjects: line solitary wave -- transverse linear stability -- long wave models; Primary 35B35 -- 37K45 -- Secondary 35Q35; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aa7cc7
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: In this paper, we study transverse linear stability of line solitary waves to the two-dimensional Benney–Luke equation which arises in the study of small amplitude long water waves in 3D. In the case where the surface tension is weak or negligible, we find a curve of resonant continuous eigenvalues of the linearized operator in a neighborhood of λ = 0 . Time evolution of these resonant continuous eigenmodes is described by a 1D damped wave equation in the transverse variable and it gives a linear approximation of the local phase shifts of modulating line solitary waves. In exponentially weighted space whose weight function increases in the direction of the motion of the line solitary wave, the other part of solutions to the linearized equation decays exponentially as t → ∞ .