A DimensionBreaking Phenomenon for Water Waves with Weak Surface Tension
(2016) In Archive for Rational Mechanics and Analysis 220(2). p.747807 Abstract
It is well known that the waterwave problem with weak surface tension has smallamplitude line solitarywave solutions which to leading order are described by the nonlinear Schrödinger equation. The present paper contains an existence theory for threedimensional periodically modulated solitarywave solutions which have a solitarywave profile in the direction of propagation and are periodic in the transverse direction; they emanate from the line solitary waves in a dimensionbreaking bifurcation. In addition, it is shown that the line solitary waves are linearly unstable to longwavelength transverse perturbations. The key to these results is a formulation of the water wave problem as an evolutionary system in which the transverse... (More)
It is well known that the waterwave problem with weak surface tension has smallamplitude line solitarywave solutions which to leading order are described by the nonlinear Schrödinger equation. The present paper contains an existence theory for threedimensional periodically modulated solitarywave solutions which have a solitarywave profile in the direction of propagation and are periodic in the transverse direction; they emanate from the line solitary waves in a dimensionbreaking bifurcation. In addition, it is shown that the line solitary waves are linearly unstable to longwavelength transverse perturbations. The key to these results is a formulation of the water wave problem as an evolutionary system in which the transverse horizontal variable plays the role of time, a careful study of the purely imaginary spectrum of the operator obtained by linearising the evolutionary system at a line solitary wave, and an application of an infinitedimensional version of the classical Lyapunov centre theorem.
(Less)
 author
 Groves, M. D.; Sun, S. M. and Wahlén, E. ^{LU}
 organization
 publishing date
 20160501
 type
 Contribution to journal
 publication status
 published
 subject
 in
 Archive for Rational Mechanics and Analysis
 volume
 220
 issue
 2
 pages
 61 pages
 publisher
 Springer
 external identifiers

 Scopus:84958876005
 ISSN
 00039527
 DOI
 10.1007/s0020501509413
 language
 English
 LU publication?
 yes
 id
 76e277517ca343f7a2a4951eabb3970b
 alternative location
 https://arxiv.org/abs/1411.2475
 date added to LUP
 20160510 09:30:01
 date last changed
 20160909 19:02:52
@misc{76e277517ca343f7a2a4951eabb3970b, abstract = {<p>It is well known that the waterwave problem with weak surface tension has smallamplitude line solitarywave solutions which to leading order are described by the nonlinear Schrödinger equation. The present paper contains an existence theory for threedimensional periodically modulated solitarywave solutions which have a solitarywave profile in the direction of propagation and are periodic in the transverse direction; they emanate from the line solitary waves in a dimensionbreaking bifurcation. In addition, it is shown that the line solitary waves are linearly unstable to longwavelength transverse perturbations. The key to these results is a formulation of the water wave problem as an evolutionary system in which the transverse horizontal variable plays the role of time, a careful study of the purely imaginary spectrum of the operator obtained by linearising the evolutionary system at a line solitary wave, and an application of an infinitedimensional version of the classical Lyapunov centre theorem.</p>}, author = {Groves, M. D. and Sun, S. M. and Wahlén, E.}, issn = {00039527}, language = {eng}, month = {05}, number = {2}, pages = {747807}, publisher = {ARRAY(0xb3393d0)}, series = {Archive for Rational Mechanics and Analysis}, title = {A DimensionBreaking Phenomenon for Water Waves with Weak Surface Tension}, url = {http://dx.doi.org/10.1007/s0020501509413}, volume = {220}, year = {2016}, }