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

摘要

> 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 by Ohta‐Todorova in 2007. In the one space dimensional case, for the nondegenerate 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 | ω | = 1 / 2 , we follow the recent splendid work of Wu (2017) to prove 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.