Orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equations

This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equations $\left\{\begin{array}{l}i\varepsilon_{t}+\varepsilon_{xx}=n\varepsilon+\alpha|\varepsilon|^{2}\varepsilon,\\n_{t}=(|\varepsilon|^{2})_{x}, x\in R.\end{array} \right.$ Firstly, we show that there exist a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period $L$ for the generalized Long-Short wave equations. Then, combining the classical method proposed by Benjamin, Bona et al., and detailed spectral analysis given by using Lame equation and Floquet theory, we show that the dnoidal type periodic wave solution is orbitally stable by perturbations with period $L$. As the modulus of the Jacobian elliptic function $k\rightarrow 1$, we obtain the orbital stability results of solitary wave solution with zero asymptotic value for the generalized Long-Short equations. In particular, as $\alpha=0$, we can also obtain the orbital stability results of periodic wave solutions and solitary wave solutions for the long-short wave resonance equations. The results in the present paper improve and extend the previous stability results of long-shore wave equations and its extension equations.