Orbital Stability of Solitary Waves to Double Dispersion Equations with Combined Power-Type Nonlinearity

Natalia Kolkovska,Nikolai Kutev,Milena Dimova

doi:10.3390/math9121398

Abstract

We consider the orbital stability of solitary waves to the double dispersion equation utt−uxx+h1uxxxx−h2uttxx+f(u)xx=0,h1>0,h2>0 with combined power-type nonlinearity f(u)=a|u|pu+b|u|2pu,p>0,a∈R,b∈R,b≠0. The stability of solitary waves with velocity c, c2<1 is proved by means of the Grillakis, Shatah, and Strauss abstract theory and the convexity of the function d(c), related to some conservation laws. We derive explicit analytical formulas for the function d(c) and its second derivative for quadratic-cubic nonlinearity f(u)=au2+bu3 and parameters b>0, c2∈0,min1,h1h2. As a consequence, the orbital stability of solitary waves is analyzed depending on the parameters of the problem. Well-known results are generalized in the case of a single cubic nonlinearity f(u)=bu3.

Highlights

We study orbital stability of the solitary waves to the double dispersion equation utt − u xx + h1 u xxxx − h2 uttxx + f (u) xx = 0, h1 > 0, h2 > 0, x ∈ R, t ∈ R+
In the first part of the present paper, we investigate orbital stability of solitary waves to double dispersion Equation (1) with combined power-type nonlinearity (3) and velocity c2 < 1
In Theorem 5(i ), we exhaustively investigate the orbital stability of solitary waves for single cubic nonlinearity (p = 3), generalizing the result in [31]

Summary

Introduction

In some applications (for example, in the propagation of a longitudinal strain wave in an isotropic compressible elastic rod, see in [1,2,3]) the coefficients a and b may have different signs, depending on the material of the rod This motivates us to study the stability of solitary waves for other sign conditions of the coefficients a, b in the quadratic-cubic nonlinearity (4). In the first part of the present paper, we investigate orbital stability of solitary waves to double dispersion Equation (1) with combined power-type nonlinearity (3) and velocity c2 < 1. Note that in our previous paper [28], as well as in the second part of the present one, we thoroughly investigate the quadratic-cubic nonlinearity and the velocities c of the solitary waves satisfying c2 < 1 (see conditions (A) and (B) in Theorem 1). The proofs of stability results are given in the Appendix A

Preliminaries

Discussion

Conclusions