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