On Linearized Korteweg-de Vries Equations

Alfredo Villanueva

doi:10.5539/jmr.v4n1p2

Abstract

Korteweg-de Vries equations (KdV) provide a way of modeling waves on shallow water surfaces. These equations, begun by John Scott Russell in 1834 through observation and experiment, are a type of nonlinear differential equations. Originating with constant coefficients, they now include time-dependent coefficients, modeling ion-acoustic waves in plasma and acoustic waves on a crystal lattice, and there is even a connection with the Fermi-Pasta-Ulam problem. Most of the solutions are given by solitons or by numerical approximations. In this work we study a linearized KdV equation with time-dependent coefficients (including fifth-order KdV) by using a special ansatz substitution.

Highlights

Korteweg-de Vries equations are nonlinear dispersive partial differential equations and their solutions are what are called solitons
In this work we study a linearized Korteweg-de Vries equations (KdV) equation with time-dependent coefficients by using a special ansatz substitution
The aim of this paper is to use an ansatz substitution to find soliton-like solutions of the linearized KDV and fifth-KDV equations with time-dependent coefficients

Summary

Introduction

Korteweg-de Vries equations are nonlinear dispersive partial differential equations and their solutions are what are called solitons (solitary waves). This solitary wave was first observed and studied by Scott Russell in 1834, and he describes his first encounter in Russell deduced empirically the velocity c of this solitary wave by the formula c2 = g(a + h), where g is gravity, a is the amplitude, and h is the depth of the water It was Boussinesq and Lord Rayleigh who showed that the soliton u(x, t) has the profile: ψ(x, t) = aS ech2(β(x − ct)). The aim of this paper is to use an ansatz substitution to find soliton-like solutions of the linearized KDV and fifth-KDV equations with time-dependent coefficients. This substitution lead to an ordinary differential equations system that has been solved in (Cordero-Soto, 2008); they have used this ansatz in (Suazo, 2010; Suazo, 2011) where they work a quadratic and nonlinear quadratic Hamiltonian

On the Linearized of KdV Equations with Time-Dependent Coefficients

Conclusion