Neumann problem for the capillary wave equation. (September 2017)

• Record Type: Journal Article
• Authors: Campos, B. Juarez; Kaikina, E.; Paredes, Hector F. Ruiz
• Is Part Of: Nonlinear analysis. Volume 160(2017)
• Page Start: 108
• Page End: 134
• Publication Date: 2017-09
• Subjects: Cubic nonlinear fractional Schrödinger equation -- Initial–boundary value problem -- The global-in-time analysis -- Asymptotics of solutions; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2017.05.008
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the initial–boundary value problem IBV problem for the cubic capillary wave equation i u t + | u | 2 u = | ∂ x | 3 2 u, t > 0, x > 0 ; u ( x, 0 ) = u 0 ( x ), x > 0, u x ( 0, t ) = h ( t ), t > 0, where | ∂ x | 3 2 u = 1 2 π ∫ 0 ∞ sign ( x − y ) | x − y | u y y ( y ) d y . We prove the global in time existence of solutions of IBV problem for nonlinear Fractional Schrödinger equation with inhomogeneous Neumann boundary conditions. Also we are interested in the study of the asymptotic behavior of solutions. Our approach to get well-posedness of nonlinear problems is based on the study of a linear theory and then using the fixed point argument. To get a linear theory we propose general method based on Riemann–Hilbert approach and theory Cauchy type integral equations. To get smooth solutions in L ∞ we modify a method based on the factorization for the free Fractional Schrödinger evolution group.