Periodic and solitary travelling-wave solutions of an extended reduced Ostrovsky equation are investigated. Attention is restricted to solutions that, for the appropriate choice of certain constant parameters, reduce to solutions of the reduced Ostrovsky equation. It is shown how the nature of the waves may be categorized in a simple way by considering the value of a certain single combination ...

In this work, we construct the lumped Galerkin approach based on cubic B-splines to obtain the numerical solution of the generalized regularized long wave equation. Applying the von Neumann approximation, it is shown that the linearized algorithm is unconditionally stable. The presented method is implemented to three test problems including single solitary wave, interaction of two solitary wave...

We investigate envelope solitary waves on square with two degrees of freedom and nonlinear nearest and next-nearest neighbor interactions. We consider solitary waves which are localized in the direction of their motion and periodically modulated along the perpendicular direction. In the quasi{monochromatic approximation and low{amplitude limit a system of two coupled nonlinear Schrr odinger equ...

A new class of solitary waves arises in the solution of nonlinear wave equations with constant impedance and no dispersive terms. These solitary waves depend on a balance between nonlinearity and a dispersion-like effect due to spatial variation in the sound speed of the medium. A high-order homogenized model confirms this effective dispersive behavior, and its solutions agree well with those o...

Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localised packet and which preserves this localisation in time. A soliton is a solitary wave which exhibits some strong form of stability so that it has a particle-like behaviour. In this paper we show a new mechanism which might produce solitary waves and solitons for a large class of equations, such a...

We consider asymptotic stability of a small solitary wave to supercritical 1-dimensional nonlinear Schrödinger equations iut + uxx = V u ± |u| u for (x, t) ∈ R × R, in the energy class. This problem was studied by Gustafson-Nakanishi-Tsai [16] in the 3-dimensional case using the endpoint Strichartz estimate. To prove asymptotic stability of solitary waves, we need to show that a dispersive part...

We present new results for the time reversal of nonlinear pulses traveling in a random medium, in particular for solitary waves. We consider long water waves propagating in the presence of a spatially random depth. Both hyperbolic and dispersive regimes are considered. We demonstrate that in the presence of properly scaled stochastic forcing the solution to the nonlinear (shallow water) conserv...

Let d ≥ 4 and let u be a global solution to the focusing masscritical nonlinear Schrödinger equation iut + ∆u = −|u| 4 d u with spherically symmetric H x initial data and mass equal to that of the ground state Q. We prove that if u does not scatter then, up to phase rotation and scaling, u is the solitary wave eQ. Combining this result with that of Merle [15], we obtain that in dimensions d ≥ 4...

Two-dimensional free-surface flow over a spatially periodic channel bed topography is examined using a steady periodically forced Korteweg–de Vries equation. The existence of new forced solitary-type waves with periodic tails is demonstrated using recently developed non-autonomous dynamical-systems theory. Bound states with two or more co-existing solitary waves are also identified. The solutio...

Orbital and asymptotic stability for 1-soliton solutions to the Toda lattice equations as well as small solitary waves to the FPU lattice equations are established in the energy space. Unlike analogous Hamiltonian PDEs, the lattice equations do not conserve momentum. Furthermore, the Toda lattice equation is a bidirectional model that does not fit in with existing theory for Hamiltonian system ...