Figure 1.

(a) Numerically obtained stable pinned 1-pulse solution U⁺_p supported by the scalar version of (1.3) with one impurity located at the origin. In particular, n=1 and x₁=0 in (1.3). In addition, , and G₁(U)=−U²+4U−1. (b) Close-up of (a) around the impurity. The dots indicate the numerically obtained pinned 1-pulse solution, while the solid curve represents the asymptotically constructed leading order pinned 1-pulse solution. (c) The leading order magnitude of the pinned 1-pulse solutions supported by the model as function of the system parameter μ obtained from the existence condition (1.5) of theorem 1.2. The solid curve represents stable pulse solutions U⁺_p as obtained from the stability condition related to (1.6) of theorem 1.2. The dashed curve represents unstable pulse solutions U⁻_p also supported by the model. The dot on the stable curve at indicates the emergence/disappearance of a point eigenvalue out of/into the essential spectrum, see remark 2.1 and §2a(i) for more details.