We present a KdV-like 2-parameter equation ut + (3(1− δ)u+ (δ + 1)xx ux )ux = εuxxx. By using the dynamical system method, existence of different traveling wave solutions are discussed, including smooth solitary wave solution of with bell type, solitary wave solutions of valley type and peakon wave solution of valley type. Numerical integration are used to shown the different types of solutions...

A detailed mathematical analysis is undertaken of solitary wave solutions of a system of coupled NLS equations describing second harmonic generation in optical materials with (2) nonlinearity. The so called bright-bright case is studied exclusively. The system depends on a single dimension-less parameter which includes both wave and material properties. Using variational methods, the rst rigoro...

ABSTRACT The exp(-Ф(η))-expansion method is an ascending method for obtaining exact and solitary wave solutions for nonlinear evolution equations. In this article, we implement the exp(-Ф(η))-expansion method to build solitary wave solutions to the fourth order Boussinesq equation. The procedure is simple, direct and useful with the help of computer algebra. By using this method, we obtain soli...

Internal solitary waves commonly observed in the coastal ocean are oftenmodeled by a nonlinear evolution equation of the Korteweg–de Vries type. Because these waves often propagate for long distances over several inertial periods, the effect of Earth’s background rotation is potentially significant. The relevant extension of the Kortweg–de Vries is then the Ostrovsky equation, which for interna...

The interaction of solitary waves on shallow water is examined to fourth order. At first order the interaction is governed by the Korteweg–de Vries (KdV) equation, and it is shown that the unidirectional assumption, of right-moving waves only, is incompatible with mass conservation at third order. To resolve this, a mass conserving system of KdV equations, involving both rightand left-moving wa...

Conventionally, bright solitary wave solutions can be obtained in self-focusing nonlinear Schrödinger equations with attractive self-interaction. However, when selfinteraction becomes repulsive, it seems impossible to have bright solitary wave solution. Here we show that there exists symbiotic bright solitary wave solution of coupled nonlinear Schrödinger equations with repulsive self-interacti...

The long-time asymptotics is analyzed for finite energy solutions of the 1D discrete Schrödinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group U(1). For initial states close to a solitary wave, the solution converges to a sum of another solitary wave and dispersive wave which is a solution to the free Schrödinger equation. T...

We study the motion of solitary-wave solutions of a family of focusing generalized nonlinear Schrödinger equations with a confining, slowly varying external potential, V (x). A Lyapunov-Schmidt decomposition of the solution combined with energy estimates allows us to control the motion of the solitary wave over a long, but finite, time interval. We show that the center of mass of the solitary w...

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...