Dissipationless hydrodynamics regularized by dispersion describe a number of physical media including water waves, nonlinear optics, and Bose--Einstein condensates. As in the classical theory of hyperbolic equations where a nonconvex flux leads to nonclassical solution structures, a nonconvex linear dispersion relation provides an intriguing dispersive hydrodynamic analogue. Here, the fifth order Korteweg--de Vries (KdV) equation, also known as the Kawahara equation, a classical model for shallow water waves, is shown to be a universal model of Eulerian hydrodynamics with higher order dispersive effects. Utilizing asymptotic methods and numerical computations, this work classifies the long-time behavior of solutions for step-like initial data. For convex dispersion, the result is a dispersive shock wave (DSW), qualitatively and quantitatively bearing close resemblance to the KdV DSW. For nonconvex dispersion, three distinct dynamic regimes are observed. For small amplitude jumps, a perturbed KdV DSW with ...
Abstract We consider a fifth-order KdV equation, where the fifth-order derivative term is multiplied by a small parameter. It has been conjectured that this equation admits a non-local solitary wave solution which has a central core and an oscillatory tail either behind or in front of the core. We prove that this solution cannot be exactly steady, and instead the amplitude of the central core decays due to the energy flux generated in the oscillatory tail. The decay rate is calculated in the limit as the parameter tends to zero. In order to verify the analytical results, we have developed a high-precision spectral method for numerical integration of this equation. The analytical and numerical result show good agreement.  Abstract We investigate the stability of a solitary wave solution of the Korteweg-de Vries equation when a fifth order spatial derivative term is added. We show that the solution ceases to be strictly localized but develops an infinite oscillating tail and we compute the amplitude of the latter. 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. © 2002 Elsevier Science B.V. All rights reserved. In dispersive wave systems with dispersion relations such that the phase speed attains an extremum at a finite wavenumber, a rich variety of solitary waves that feature decaying oscillatory tails i... Steady solutions of a generalized Korteweg-de Vries equation which has an additional fifth order derivative term are investigated on the basis of a numerical calculation. It is found that either compressive or rarefactive solitary waves are possible to exist according as the dispersion is negative or positive and that the solitary waves take the oscillatory structure when the coefficient of the fifth order derivative term dominates over that of the third order one.