Abstract: The qualitative analysis method was applied to the dynamical system，the possibility of existence of kink and anti-kink solitary waves in microstructured solids and their existence conditions were analyzed with the help of the 1D higher-order nonlinear wave model was established in previous articles. The results indicate that the kink and anti-kink solitary waves can exist in higher-order nonlinear and anomalous dispersion microstructured solids under appropriate conditions. The existence conditions of the 2 solitary waves were explicitly given in here. The higher-order microscale nonlinear effect on the kink and anti-kink solitary waves was analyzed with the numerical method. The numerical analysis shows that the kink and anti-kink solitary waves become more and more steep (or gentle) with the negative reinforcing (or positive reinforcing) of higher-order microscale nonlinear effect while the amplitudes of the 2 solitary waves remain unchanged．

Key words: existence condition