The model equationformula herearises as the equation for solitary-wave solutions to a fifth-order
long-wave equation
for gravity–capillary 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.
A Variational Reduction and the Existence of a Fully Localised Solitary Wave for the Three-Dimensional Water-Wave Problem with Weak Surface Tension
