In this work, treating an artery as a prestressed thin-walled elastic tube and the blood as an inviscid fluid, the interactions of two nonlinear waves propagating in opposite directions are studied in the longwave approximation by use of the extended PLK (Poincaré-Lighthill-Kuo) perturbation method. The results show that up to O(k3), where k is the wave number, the head-on collision of two soli...

The LL equation 1 has aroused considerable interest among physicists and mathematicians. For the one-dimensional case, there have been many contributions to the study of the soliton solution, the interaction of solitary waves, and other properties of the solitary waves 2–4 . However, we point out here that it will be a more challenging task study the high-dimensional dynamics 5, 6 about LL equa...

Motivated by the successful synthesis of several molecular quantum spin rings we are investigating whether such systems can host magnetic solitary waves. The small size of these spin systems forbids the application of a classical or continuum limit. We therefore investigate whether the time-dependent Schrödinger equation itself permits solitary waves. Example solutions are obtained via complete...

We review recent results on global attractors of U(1)-invariant dispersive Hamiltonian systems. We study several models based on the Klein-Gordon equation and sketch the proof that in these models, under certain generic assumptions, the weak global attractor is represented by the set of all solitary waves. In general, the attractors may also contain multifrequency solitary waves; we give exampl...

This paper is devoted to the study of solitary waves and solitons whose existence is related to the ratio energy/charge. These solitary waves are called hylomorphic. This class includes the Q-balls, which are spherically symmetric solutions of the nonlinear Klein-Gordon equation (NKG), as well as solitary waves and vortices which occur, by the same mechanism, in the nonlinear Schroedinger equat...

In this paper, we study the stability and evolution of solitary waves in perturbed generalized nonlinear Schrödinger (NLS) equations. Our method is based on the completeness of the bounded eigenstates of the associated linear operator in L2 space and a standard multiple-scale perturbation technique. Unlike the adiabatic perturbation method, our method details all instability mechanisms caused b...

Saddle-node bifurcations of solitary waves in generalized nonlinear Schrödinger equations with arbitrary forms of nonlinearity and external potentials in arbitrary spatial dimensions are analyzed. First, general conditions for these bifurcations are derived. Second, it is shown analytically that the linear stability of these solitary waves does not switch at saddle-node bifurcations, which is i...

We consider a two-dimensional generalization of the Benjamin-Ono equation and prove that it admits solitary-wave solutions that are analytic functions. We find the optimal decay rate at infinity of these solitary waves.

We consider systems of equations which arise in modelling strong interactions of weakly nonlinear long waves in dispersive media. For a certain class of such systems, we prove the existence and stability of localized solutions representing coupled solitary waves travelling at a common speed. Our results apply in particular to the systems derived by Gear and Grimshaw and by Liu, Kubota, and Ko a...

Abstract In coastal seas and straits, the interaction of barotropic tidal currents with the continental shelf, seamounts or sills is often observed to generate large-amplitude, horizontally propagating internal solitary waves. Typically these waves occur in regions of variable bottom topography, with the consequence that they are often modeled by nonlinear evolution equations of the Korteweg-de...