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

Abstract

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.

Stability analysis of embedded solitons in the generalized third-order nonlinear Schrödinger equation

“…14 Recently, moving discrete breathers were linked to embedded solitons as well. This semistability holds not only for isolated embedded solitons, 27,30,31,34 but also for continuous families of embedded solitons. Because embedded solitons are in resonance with the continuous spectrum, one tends to expect that these solitons will continuously leak energy through continuous-wave radiation and thus break up eventually.…”

Continuous Families of Embedded Solitons in the Third‐Order Nonlinear Schrödinger Equation

“…Unfortunately, this speculation turns out to be incorrect. Our numerical simulations show that these solitons are still semi‐stable, just like isolated embedded solitons in other physical systems [5,13,14,18]. In other words, the freedom of arbitrary energy of embedded solitons in the TNLS equation is not sufficient to stabilize these solitons.…”

“…If embedded solitons are linearly stable, nonlinearly they could be semi‐stable, i.e., whether they persist or break up, depends on the type of initial perturbations imposed. This semi‐stability property has been established rigorously for isolated embedded solitons in Hamiltonian systems [5,14,18,22,23]. For the TNLS equation, embedded solitons exist as continuous families.…”

Radiationless Higher-Order Embedded Solitons

“…They almost stabilize if the perturbation increases their energies, but they are rapidly dispersed if their energy is decreased. Other solitary waves which respond in this way to perturbations have also been found in other systems, [43][44][45][46][47] and waves of this type are frequently referred to as semi-stable solutions.…”