Exact Solutions of a High-Order Nonlinear Wave Equation of Korteweg-de Vries Type under Newly Solvable Conditions

Weiguo Rui

doi:10.1155/2014/714214

Abstract

By using the integral bifurcation method together with factoring technique, we study a water wave model, a high-order nonlinear wave equation of KdV type under some newly solvable conditions. Based on our previous research works, some exact traveling wave solutions such as broken-soliton solutions, periodic wave solutions of blow-up type, smooth solitary wave solutions, and nonsmooth peakon solutions within more extensive parameter ranges are obtained. In particular, a series of smooth solitary wave solutions and nonsmooth peakon solutions are obtained. In order to show the properties of these exact solutions visually, we plot the graphs of some representative traveling wave solutions.

Highlights

In this work, we will study the following high-order nonlinear wave equation of Korteweg-de Vries type: ηt + ηx + αηηx + βηxxx + ρ1α2η2ηx+ αβ (ρ2ηηxxx + ρ3ηxηxx) + ρ4α3η3ηx (1)+ α2β (ρ5η2ηxxx + ρ6ηηxηxx + ρ7ηx3) = 0.This is an important model of water wave derived by Fokas [1] in 1995, where α = 3A/2, β = B/6, and 0 < α, β ≪ 1.Obviously, (1) is a very complex partial deferential equation, it has nine parameters α, β, ρi, (i = 1, 2, . . . , 7), and contains both high-order derivative terms and multinonlinear terms
By using the integral bifurcation method together with factoring technique, we study a water wave model, a high-order nonlinear wave equation of KdV type under some newly solvable conditions
By using the integral bifurcation method together with factoring technique, we investigate exact traveling wave solutions of (1) within more extensive parameter ranges

Summary

Introduction

30βρ2ρ5 10βρ5 (a) When ((3ρ4 − 10ρ5)/ρ2ρ5) > 0 and ρ4 ≠ 0, substituting (32) into the first equation of (9) to integrate, we obtain a smooth solitary wave solution as follows: η (x, t) (1) When ρ1ρ5 > 0 and the equation R1(φ) = 0 has four real roots φ1, φ2, φ3, and φ4, respectively, taking φ(0) = φ1, φ2, φ3, φ4 as initial value and integrating (38), we obtain four kinds of exact traveling wave solutions of implicit function type as follows:

Results

Conclusion