Kink solitons of the double-quadratic model in the presence of an external spatially inhomogeneous force

Dynamics of kink solitons of the double-quadratic model in the presence of an external, spatially inhomogeneous force $F\left(x\right)\mathrm{}$ is considered. Stability analysis of such solutions is performed. It is shown that similarly as for kinks of the ${\phi }^4$ equation there are differences between a stability condition for the kink treated as a point particle and the kink treated as a spatially extended object. These differences occur when the force $F\left(x\right)\mathrm{}$ possesses more than one zero and follow from interactions of kink wings (i.e., parts of kinks that are far from a kink center) with additional zeros of the force $F\left(x\right)\mathrm{}$.