TY - UNPB
T1 - A global investigation of solitary-wave solutions to a two-parameter model for water waves
AU - Champneys, AR
AU - Groves, MD
N1 - Additional information: Preprint of a paper later published by Cambridge University Press, (1997) Journal of Fluid Mechanics, 342, pp. 199-229. ISSN 0022-1120
Terms of use: ©1997 Cambridge University Press
PY - 1996
Y1 - 1996
N2 - The model equation (2r''''/15)-(br'')+(ar)+(3r^2/2)-((r')^2/2)+[rr']'=0 arises as the equation for solitary-wave solutions to a fifth-order long-wave equation for gravitycapillary water waves. Being Hamiltonian, reversible and depending upon two parameters, it shares the structure of the full steady water-wave problem. Moreover, all known analytical results for local bifurcations of solitary-wave solutions to the full water-wave problem have precise counterparts for the model equation.
At the time of writing two major open problems for steady water waves are attracting particular attention. The first concerns the possible existence of solitary waves of elevation as local bifurcation phenomena in a particular parameter regime; the second, larger, issue is the determination of the global bifurcation picture for solitary waves. Given that the above equation is a good model for solitary waves of depression, it seems natural to study the above issues for this equation; they are comprehensively treated in this article.
The equation is found to have branches of solitary waves of elevation bifurcating from the trivial solution in the appropriate parameter regime, one of which is described by an explicit solution. Numerical and analytical investigations reveal a rich global bifurcation picture including multi-modal solitary waves of elevation and depression together with interactions between the two types of wave. There are also new orbit-flip bifurcations and associated multi-crested solitary waves with non-oscillatory tails
AB - The model equation (2r''''/15)-(br'')+(ar)+(3r^2/2)-((r')^2/2)+[rr']'=0 arises as the equation for solitary-wave solutions to a fifth-order long-wave equation for gravitycapillary water waves. Being Hamiltonian, reversible and depending upon two parameters, it shares the structure of the full steady water-wave problem. Moreover, all known analytical results for local bifurcations of solitary-wave solutions to the full water-wave problem have precise counterparts for the model equation.
At the time of writing two major open problems for steady water waves are attracting particular attention. The first concerns the possible existence of solitary waves of elevation as local bifurcation phenomena in a particular parameter regime; the second, larger, issue is the determination of the global bifurcation picture for solitary waves. Given that the above equation is a good model for solitary waves of depression, it seems natural to study the above issues for this equation; they are comprehensively treated in this article.
The equation is found to have branches of solitary waves of elevation bifurcating from the trivial solution in the appropriate parameter regime, one of which is described by an explicit solution. Numerical and analytical investigations reveal a rich global bifurcation picture including multi-modal solitary waves of elevation and depression together with interactions between the two types of wave. There are also new orbit-flip bifurcations and associated multi-crested solitary waves with non-oscillatory tails
KW - multi-crested solitary waves
KW - orbit-flip bifurcations
KW - water waves
KW - solitary wave solutions
M3 - Working paper
BT - A global investigation of solitary-wave solutions to a two-parameter model for water waves
ER -