Abstract. We consider a class of pseudodifferential evolution equations of the form ut + (n(u) + Lu)x = 0, in which L is a linear smoothing operator and n is at least quadratic near the origin; this class includes in particular the Whitham equation. A family of solitary-wave solutions is found using a constrained minimisation principle and concentration-compactness methods for noncoercive functionals. The solitary waves are approximated by (scalings of) the corresponding solutions to partial differential equations arising as weakly nonlinear approximations; in the case of the Whitham equation the approximation is the Korteweg-deVries equation. We also demonstrate that the family of solitary-wave solutions is conditionally energetically stable.
This paper contains a rigorous existence theory for three-dimensional steady gravity-capillary finite-depth water waves which are uniformly translating in one horizontal spatial direction x and periodic in the transverse direction z. Physically motivated arguments are used to find a formulation of the problem as an infinite-dimensional Hamiltonian system in which x is the time-like variable, and a centre-manifold reduction technique is applied to demonstrate that the problem is locally equivalent to a finite-dimensional Hamiltonian system. General statements concerning the existence of waves which are periodic or quasiperiodic in x (and periodic in z) are made by applying standard tools in Hamiltonian-systems theory to the reduced equations.A critical curve in Bond number–Froude number parameter space is identified which is associated with bifurcations of generalized solitary waves. These waves are three dimensional but decay to two-dimensional periodic waves (small-amplitude Stokes waves) far upstream and downstream. Their existence as solutions of the water-wave problem confirms previous predictions made on the basis of model equations.
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.
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