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

We consider the orbital stability of solitary waves to 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 velocity c, c2<1 is proved by means Grillakis, Shatah, and Strauss abstract theory convexity function d(c), related some conservation laws. derive explicit analytical formulas for d(c) its second derivative quadratic-cubic f(u)=au2+bu3 parameters b>0, c2∈0,min1,h1h2. As a consequence, analyzed depending on problem. Well-known results are generalized in case single cubic f(u)=bu3.