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

Abstract

We consider the orbital stability of solitary waves to the double dispersion equation \( u_{tt} - u_{xx} + h_1 u_{xxxx} - h_2u_{ttxx} + f(u)_{xx} = 0, h_1 > 0, h_2 > 0 \) with combined power-type nonlinearity \( f(u) = a \left| u \right|^pu + b \left| u \right|^{2p}u, \ \ p > 0, \ \ a \in \mathbb{R}, \ \ b \in \mathbb{R},\ \ b \not = 0. \) The stability of solitary waves with velocity \( c, \ c^2 < 1 \) 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 \( f(u) = au^2 + bu^3 \) and parameters \( b > 0, \ c^2 \in \left[ 0, \ min \left( 1, \ \frac{h_1}{h_2} \right) \right) \). 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 \( f(u) = bu^3 \).