Semi-stability of embedded solitons in the general fifth-order KdV equation

Evolution of perturbed embedded solitons in the general Hamiltonian fifth-order Korteweg--de Vries (KdV) equation is studied. When an embedded soliton is perturbed, it sheds a one-directional continuous-wave radiation. It is shown that the radiation amplitude is not minimal in general. A dynamical equation for velocity of the perturbed embedded soliton is derived. This equation shows that a neutrally stable embedded soliton is in fact semi-stable. When the perturbation increases the momentum of the embedded soliton, the perturbed state approaches asymptotically the embedded soliton, while when the perturbation reduces the momentum of the embedded soliton, the perturbed state decays into radiation. Classes of initial conditions to induce soliton decay or persistence are also determined. Our analytical results are confirmed by direct numerical simulations of the fifth-order KdV equation

A plethora of generalised solitary gravity-capillary water waves

The present study describes, first, an efficient algorithm for computing capillary-gravity solitary waves solutions of the irrotational Euler equations with a free surface and, second, provides numerical evidences of the existence of an infinite number of generalised solitary waves (solitary waves with undamped oscillatory wings). Using conformal mapping, the unknown fluid domain, which is to be determined, is mapped into a uniform strip of the complex plane. In the transformed domain, a Babenko-like equation is then derived and solved numerically.Comment: 20 pages, 7 figures, 45 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

On asymptotically equivalent shallow water wave equations

The integrable 3rd-order Korteweg-de Vries (KdV) equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves. However, at quadratic order, this asymptotic expansion produces an entire {\it family} of shallow water wave equations that are asymptotically equivalent to each other, under a group of nonlinear, nonlocal, normal-form transformations introduced by Kodama in combination with the application of the Helmholtz-operator. These Kodama-Helmholtz transformations are used to present connections between shallow water waves, the integrable 5th-order Korteweg-de Vries equation, and a generalization of the Camassa-Holm (CH) equation that contains an additional integrable case. The dispersion relation of the full water wave problem and any equation in this family agree to 5th order. The travelling wave solutions of the CH equation are shown to agree to 5th order with the exact solution

Higher-dimensional extended shallow water equations and resonant soliton radiation

The higher order corrections to the equations that describe nonlinear wave motion in shallow water are derived from the water wave equations. In particular, the extended cylindrical Korteweg-de Vries and Kadomtsev-Petviashvili equations—which include higher order nonlinear, dispersive, and nonlocal terms—are derived from the Euler system in (2+1) dimensions, using asymptotic expansions. It is thus found that the nonlocal terms are inherent only to the higher dimensional problem, both in cylindrical and Cartesian geometry. Asymptotic theory is used to study the resonant radiation generated by solitary waves governed by the extended equations, with an excellent comparison obtained between the theoretical predictions for the resonant radiation amplitude and the numerical solutions. In addition, resonant dispersive shock waves (undular bores) governed by the extended equations are studied. It is shown that the asymptotic theory, applicable for solitary waves, also provides an accurate estimate of the resonant radiation amplitude of the undular bore