Soliton equations are infinite-dimensional integrable systems described by nonlinear evolution equations. As one of the soliton equations, long wave equation takes on profound significance of theory and reality. By using the method of nonlinearization, the relation between long wave equation and second-order eigenvalue problem is generated. Based on the nonlinearized Lax pairs, Euler-Lagrange f...

In this paper we produce numerical, genuinely three-dimensional, hexagonal traveling wave solutions of the Euler equations for water waves using a surface integral formulation derived by Craig and Sulem. These calculations are free from the requirements of either long wavelength or two-dimensionality, both of which are crucial to the KdV and KP scaling regimes, and we produce hexagonal travelin...

In this manuscript, a new hybrid technique viz Sawi transform-based homotopy perturbation method is implemented to solve one-dimensional shallow water wave equations. general, the quantities involved with such equations are commonly assumed be crisp, but parameters in actual scenario may imprecise/uncertain. Therefore, fuzzy uncertainty introduced as an initial condition. The main focus of stud...

We consider shallow-water flow past a broad bottom ridge, localized in the flow direction, using the framework of the forced Su–Gardner (SG) system of equations, with a primary focus on the transcritical regime when the Froude number of the oncoming flow is close to unity. These equations are an asymptotic long-wave approximation of the full Euler system, obtained without a simultaneous expansi...

in this paper, a method for finding an approximate solution of a class of two-dimensional nonlinear volterra integral equations of the first-kind is proposed. this problem is transformedto a nonlinear two-dimensional volterra integral equation of the second-kind. the properties ofthe bivariate shifted legendre functions are presented. the operational matrices of integrationtogether with the produ...

This article examines the accuracy for large times of asymptotic expansions from periodic homogenization wave equations. As usual, ϵ denotes small period coefficients in equation. We first prove that standard two scale expansion provides an accurate approximation exact solution t order − 2 + δ any > 0. Second, longer times, we show a different algorithm, is called criminal because it mixes p...

We construct infinitely accurate approximate solutions to systems of hyperbolic partial differential equations which model short wavelength dispersive nonlinear phenomena. The principal themes are the following. ~1! The natural framework for the study of dispersion is wavelength e solutions of systems of partial differential operators in e]. The natural e-characteristic equation and e-eikonal e...

Once upon a time, a spirited horse was dragging a boat along a peaceful canal of Northern Scotland. Suddenly, a ferocious dog charged from the bushes. The horse took fright and ran off, drawing the boat after him. To the horse's surprise, the boat offered almost no resistance at this high speed. A clever naval engineer, who chanced to witness the scene, pursued the matter through numerous exper...

This paper presents a comparison between variational iteration method (VIM) and modfied variational iteration method (MVIM) for approximate solution a system of Volterra integral equation of the first kind. We convert a system of Volterra integral equations to a system of Volterra integro-di®erential equations that use VIM and MVIM to approximate solution of this system and hence obtain an appr...

Equatorial long-wave theory applies where a small horizontal aspect ratio between meridional and zonal length scales is assumed. In an idealized setting, the theory suggests that (i) meridional wind is small, (ii) geostrophic balance holds in the meridional direction, and (iii) inertio-gravity waves are small in amplitude or ‘‘filtered out.’’ In this paper a spectral data analysis method is use...