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 present a method to prove nonlinear instability of solitary waves in dispersive models. Two examples are analyzed: we prove the nonlinear long time instability of the KdV solitary wave (with respect to periodic transverse perturbations) under a KP-I flow and the transverse nonlinear instability of solitary waves for the cubic nonlinear Schrödinger equation.

Solitons or solitary waves are large-amplitude excitations with finite spatial widths 15 which are found in certain nonlinear systems and which can propagate over long distances without changing their shapes. Solitary waves also have the property that, although they may interact strongly with other solitary waves, they emerge from the interaction intact. The first report of the observation of a...

We consider asymptotic stability of a small solitary wave to supercritical 2-dimensional nonlinear Schrödinger equations iut +∆u = V u± |u|u for (x, t) ∈ R × R, in the energy class. This problem was studied by Gustafson-NakanishiTsai [14] in the n-dimensional case (n ≥ 3) by using the endpoint Strichartz estimate. Since the endpoint Strichartz estimate fails in 2-dimensional case, we use a time...

We study the Boussinesq equation from the point of view of a multipletime reductive perturbation method. As a consequence of the elimination of the secular producing terms through the use of the Korteweg–de Vries hierarchy, we show that the solitary–wave of the Boussinesq equation is a solitary–wave satisfying simultaneously all equations of the Korteweg–de Vries hierarchy, each one in an appro...

The joint influence of the polariton effect and Kerr-like nonlinearity on propagation of optical pulses is studied. The existence of different families of envelope solitary wave solutions in the vicinity of polariton gap is shown. The properties of solutions depend strongly on the carrier wave frequency. In particular, solitary waves inside and outside of the polariton gap exhibit different vel...

We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the generalized KP-BBM equation. A number of explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain periodic wave solutions, kink wave solutions, unbounded wave solutions, blow-up wave sol...

In this paper, the topological (dark) as well as non-topological (bright) soliton solutions of nonlinear evolution equations are obtained by the solitary wave ansatz method. Under some parameter conditions, exact solitary wave solutions are obtained. Note that, it is always useful and desirable to construct exact solutions especially soliton-type (dark, bright, kink, anti-kink, etc...) envelope...