Some remarks on the stability and instability properties of solitary waves for the double dispersion equation

S Erbay,H A Erbay,A Erkip

doi:10.3176/proc.2015.3.09

Abstract

In this article we give a review of our recent results on the instability and stability properties of travelling wave solutions of the double dispersion equation u(tt) - u(xx) + au(xxxx) - bu(xxtt) = (vertical bar u vertical bar(p-1) u)(xx) for p > 1, a >= b > 0. After a brief reminder of the general class of nonlocal wave equations to which the double dispersion equation belongs, we summarize our findings for both the existence and orbital stability/instability of travelling wave solutions to the general class of nonlocal wave equations. We then state (i) the conditions under which travelling wave solutions of the double dispersion equation are unstable by blow-up and (ii) the conditions under which the travelling waves are orbitally stable. We plot the instability/stability regions in the plane defined by wave velocity and the quotient b/a for various values of p.

Highlights

The present paper provides an overview of results obtained in [1,2] for travelling wave solutions of the double dispersion equation utt − uxx + auxxxx − buxxtt = −(|u|p−1u)xx, (1)where u = u(x,t) is a real-valued function, a, b are positive real constants with a ≥ b, and p > 1
In this article we give a review of our recent results on the instability and stability properties of travelling wave solutions of the double dispersion equation utt − uxx + auxxxx − buxxtt = −(|u|p−1u)xx for p > 1, a ≥ b > 0
After a brief reminder of the general class of nonlocal wave equations to which the double dispersion equation belongs, we summarize our findings for both the existence and orbital stability/instability of travelling wave solutions to the general class of nonlocal wave equations

Summary

Introduction

In this article we give a review of our recent results on the instability and stability properties of travelling wave solutions of the double dispersion equation utt − uxx + auxxxx − buxxtt = −(|u|p−1u)xx for p > 1, a ≥ b > 0. We state (i) the conditions under which travelling wave solutions of the double dispersion equation are unstable by blow-up and (ii) the conditions under which the travelling waves are orbitally stable. The present paper provides an overview of results obtained in [1,2] for travelling wave solutions of the double dispersion equation utt − uxx + auxxxx − buxxtt = −(|u|p−1u)xx, (1)

Results

Conclusion