This paper considers the propagation of shallow-water solitary and nonlinear periodic waves over a gradual slope with bottom friction in the framework of a variablecoefficient Korteweg–de Vries equation. We use the Whitham averaging method, using a recent development of this theory for perturbed integrable equations. This general approach enables us not only to improve known results on the adia...

In this paper, a generalized Camassa–Holm equation is studied by using the integral bifurcation method. Many travelling waves such as peaked compacton, compacton, peaked solitary wave, solitary wave and kink-like wave are found. In some parameter conditions, exact parametric representations of these travelling waves in explicit form and implicit form are obtained. 2008 Elsevier Inc. All rights ...

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 ...

In this part, we prove that the solitary wave solutions investigated in part I are extended as analytic functions in the complex plane, except at most countably many branch points and branch lines. We describe in detail how the limiting behavior of the complex sin-gularities allows the creation of non-analytic solutions with corners and/or compact support. This is the second in a series of two ...

The amplitude of solitary waves significantly affects the properties of the waves by changing their degree of nonlinearity. The period of solitary waves e.g. decreases nonlinearly with amplitude. Observations of solitary waves propagating under similar stratification conditions, but with different amplitudes, were compared to identify how the wave period depends on wave amplitude under field co...

Abstract In this paper we consider a coupled KdV system of Boussinesq type and its symmetric version. These systems were previously shown to possess generalized solitary waves consisting of a solitary pulse that decays symmetrically to oscillations of small, constant amplitude. We solve numerically the periodic initial-value problem for these systems using a high order accurate, fully discrete,...

In this paper we consider a coupled KdV system of Boussinesq type and its symmetric version. These systems were previously shown to possess Generalized Solitary Waves consisting of a solitary pulse that decays symmetrically to oscilations of small, constant amplitude. We solve numerically the periodic initial-value problem for these systems using a high order accurate, fully discrete, Galerkin-...

Due to the horizontal variability of oceanic hydrology (density and current stratification) and the variable depth over the continental shelf, internal solitary waves transform as they propagate shorewards into the coastal zone. If the background variability is smooth enough, a solitary wave possesses a soliton-like form with varying amplitude and phase. This stage is studied in detail in the f...

[1] A sequence of three internal solitary waves of elevation were observed propagating shoreward along a near-bottom density interface over Oregon’s continental shelf. These waves are highly turbulent and coincide with enhanced optical backscatter, consistent with increased suspended sediments in the bottom boundary layer. Nonlinear solitary wave solutions are employed to estimate wave speeds a...

We consider the nonlinear Dirac equation in one dimension (the massive Gross-Neveu model). We explicitly construct solitary wave solutions, and then study the linearization of the equation at a solitary wave. We present numerical simulations and justify them with explicit construction of some of the eigenfunctions. Then we present a WKB-based argument which justifies (but does not prove) the sp...