Open AccessExistence and stability of periodic travelling-wavesolutions of the Benjamin equation
Author(s) -
Jaime Angulo Pava,
Borys Alvarez Samaniego
Publication year - 2005
Publication title -
communications on pure and applied analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.077
H-Index - 42
eISSN - 1553-5258
pISSN - 1534-0392
DOI - 10.3934/cpaa.2005.4.367
Subject(s) - traveling wave , stability (learning theory) , mathematical analysis , hilbert transform , physics , function (biology) , differential equation , periodic function , nonlinear system , mathematical physics , mathematics , quantum mechanics , machine learning , computer science , statistics , spectral density , evolutionary biology , biology
A family of steady periodic water waves in very deep fluids when the surface tension is present and satisfying the following nonlinear pseudo-differential equation u(t) + uu(x) + u(xxx) + lHu(xx) = 0, known as the Benjamin equation, is shown to exist. Here H denotes the periodic Hilbert transform and l is an element of R. By fixing a minimal period we obtain, via the implicit function theorem, an analytic curve of periodic travelling-wave solutions depending on the parameter l. Moreover, by making some changes in the abstract stability theory developed by Grillakis, Shatah, and Strauss, we prove that these travelling waves are nonlinearly stable to perturbations with the same wavelength