Stability and instability of the standing waves for the Klein-Gordon-Zakharov system in one space dimension

The orbital instability of standing waves for the Klein-Gordon-Zakharov system has been established in two and three space dimensions under radially symmetric condition, see Ohta-Todorova (SIAM J. Math. Anal. 2007). In the one space dimensional case, for the non-degenerate situation, we first check that the Klein-Gordon-Zakharov system satisfies Grillakis-Shatah-Strauss' assumptions on the stability and instability theorems for abstract Hamiltonian systems, see Grillakis-Shatah-Strauss (J. Funct. Anal. 1987). As to the degenerate case that the frequency $|\omega|=1/\sqrt{2}$, we follow Wu (ArXiv: 1705.04216, 2017) to describe the instability of the standing waves for the Klein-Gordon-Zakharov system, by using the modulation argument combining with the virial identity. For this purpose, we establish a modified virial identity to overcome several troublesome terms left in the traditional virial identity.