A global investigation of solitary-wave solutions to a two-parameter model for water waves

Abstract

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.

When gap solitons become embedded solitons: a generic unfolding

“…A completely analogous result is valid when the local bifurcation of x = 0 is described by the normal forms (5) or (6). This case is non-generic in our general setup.…”

“…In contrast, energy decreasing perturbations cause the solitary wave to decay algebraically away. The existence of embedded solitons has been established in a number of physical models including generalised 5th-order Korteweg-de Vries (KdV) equations [20,4,6,33], coupled KdV equations [17], in nonlinear Schrödinger (NLS) equations with higher-order derivatives [3,16] and in various coupled NLS-type equations arising in nonlinear optics [34,10,8].…”

“…In the same manner as before we can turn the diagrams into a theorem. When the local bifurcation is described by (6) …”

“…where a prime denotes differentiation with respect to t. Homoclinic orbits to the origin of (8) were intensively studied in [6], using a combination of analytical and numerical techniques.…”

Multiple homoclinic orbits in conservative and reversible systems

“…Several models are known that are both Hamiltonian and reversible, and possess a homoclinic bellows. Let us indicate one example from the study of solitary waves in a fifth order equation for gravity-capillary water waves [ChaGro97,WagCha02], u t + 2 15 u xxxxx − bu xxx + 3uu x + 2u x u xx + uu xxx = 0.…”