Double Dispersion Equation Combined Power-Type Nonlinearity Solitary Waves Orbital Stability
Issue Date:
16-Jun-2021
Publisher:
MDPI
Citation:
Kolkovska, N.; Dimova, M.; Kutev, N. Orbital Stability of Solitary Waves to Double Dispersion Equations with Combined Power-Type Nonlinearity. Mathematics, 2021, 9, 1398. https://doi.org/10.3390/math9121398
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 \( f(u) = bu^3 \).