Sufficient conditions are given for stability of solitary-wave solutions of model equations for one-dimensional long nonlinear waves. These conditions differ from others which have appeared previously in that they are phrased in terms of positivity properties of the Fourier transforms of the solitary waves. Their use leads to simplified proofs of existing stability results for the Korteweg-de V...

We consider a high-order nonlinear Schrödinger (HNLS) equation with thirdand fourth-order dispersions, quintic non-Kerr terms, self steepening, and self-frequency-shift effects. The model applies to the description of ultrashort optical pulse propagation in highly nonlinear media. We propose a complex envelope function ansatz composed of single bright, single dark and the product of bright and ...

We study the interaction of small amplitude, long wavelength solitary waves in the Fermi-Pasta-Ulam model with general nearest-neighbor interaction potential. We establish global-in-time existence and stability of counter-propagating solitary wave solutions. These solutions are close to the linear superposition of two solitary waves for large positive and negative values of time; for intemediat...

We prove that the collision of two solitary waves of the BBM equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space. We show precise estimates of the nonzero residue due to the collision. Moreover, we give a precise description of the collision phenomenon (change of size of the solitary waves and shifts in their trajectories). To prove these re...

Two-dimensional gravity waves travelling under an ice sheet are studied. The flow is assumed to be potential. Weakly nonlinear solutions are derived and fully nonlinear solutions are calculated numerically. Periodic waves and generalized solitary waves are studied.

We consider the linearized instability of 2D irrotational solitary water waves. The maxima of energy and the travel speed of solitary waves are not obtained at the highest wave, which has the 120 degree angle at the crest. Under the assumption of non-existence of secondary bifurcation which is con…rmed numerically, we prove linear instability of solitary waves which are higher than the wave of ...

We show that a non-thermal electron distribution may change the nature of ion acoustic solitons. If the ions are assumed to respond as a fluid to perturbations in the potential, with no significant trapping in a potential well, then a thermal plasma only supports a solitary waves with a density peak. However, with a suitable distribution of non-thermal particles, solitary waves with both densit...

Linearly polarized solitary waves, arising from the interaction of an intense laser pulse with a plasma, are investigated. Localized structures, in the form of exact numerical nonlinear solutions of the one-dimensional Maxwell-fluid model for a cold plasma with fixed ions, are presented. Unlike stationary circularly polarized solitary waves, the linear polarization gives rise to a breather-type...

We study the interaction of small amplitude, long-wavelength solitary waves in the Fermi–Pasta–Ulam model with general nearest-neighbour interaction potential. We establish global-in-time existence and stability of counterpropagating solitary wave solutions. These solutions are close to the linear superposition of two solitary waves for large positive and negative values of time; for intermedia...

A generalization of criticality – called secondary criticality – is introduced and applied to finite-amplitude Stokes waves. The theory shows that secondary criticality signals a bifurcation to a class of steady dark solitary waves which are biasymptotic to a Stokes wave with a phase jump in between, and synchronized with the Stokes wave. We find the that the bifurcation to these new solitary w...