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We investigate the effect of spatial inhomogeneity on perfectly periodic, self-organized striped patterns in spatially extended systems. We demonstrate that inhomogeneities select a specific translate of the striped patterns and induce algebraically decaying, dipole-type farfield deformations. Phase shifts and leading order terms are determined by effective moments of the spatial inhomogeneity. Farfield decay is proportional to the derivatives of the Green’s function of an effective Laplacian. Technically, we use mode filters and conjugacies to an effective Laplacian to establish Fredholm properties of the linearization in Kondratiev spaces. Spatial localization in a contraction argument is gained through the use of an explicit deformation ansatz and a subtle cancellation in Bloch wave space. 

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the DP equation on the real line, extending our previous work on their spectral stability [J. Math. Pures Appl. (9) 142 (2020), pp. 298–314]. The main difficulty stems from the fact that the natural energy space is a subspace of L 3 L^3 , but the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the L 2 L^2 -norm, resulting in L 3 L^3 higher-order nonlinear terms in the augmented Hamiltonian. But the usual coercivity estimate is in terms of L 2 L^2 norm for DP equation, which cannot be used to control the L 3 L^3 higher order term directly. The remedy is to observe that, given a sufficiently smooth initial condition satisfying some mild constraint, the L ∞ L^\infty orbital norm of the perturbation is bounded above by a function of its L 2 L^2 orbital norm, yielding the higher order control and the orbital stability in the L 2 ∩ L ∞ L^2\cap L^\infty space. 

Experiments with diblock co-polymer melts display undulated bilayers that emanate from defects such as triple junctions and endcaps, [8]. Undulated bilayers are characterized by oscillatory perturbations of the bilayer width, which decay on a spatial length scale that is long compared to the bilayer width. We mimic defects within the functionalized Cahn-Hillard free energy by introducing spatially localized inhomogeneities within its parameters. For length parameter \begin{document}$$ \varepsilon\ll1 $$\end{document}, we show that this induces undulated bilayer solutions whose width perturbations decay on an \begin{document}$$ O\!\left( \varepsilon^{-1/2}\right) $$\end{document} inner length scale that is long in comparison to the \begin{document}$ O(1) $$\end{document}$ scale that characterizes the bilayer width.