Given a unit vector v∈R3\documentclass[12pt]{minimal}
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\begin{document}$${\textbf {v}}\in {\mathbb {R}}^3$$\end{document} and λ∈R\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \in {\mathbb {R}}$$\end{document}, a translating λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}-soliton is a surface in R3\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^3$$\end{document} whose mean curvature H satisfies H=⟨N,v⟩+λ\documentclass[12pt]{minimal}
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\begin{document}$$H=\langle N,{\textbf {v}}\rangle +\lambda $$\end{document}, where N is the Gauss map of the surface. In this paper, we extend the phenomenon of instability of Plateau–Rayleigh for translating λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}-solitons of cylindrical type, proving that long pieces of these surfaces are unstable. Specifically, we will provide explicit bounds on the length of these unstable surfaces in terms of λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document} and the amplitude of the generating curve. It will be also proved that a graphical translating λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}-soliton is a minimizer of the weighted area in a suitable class of surfaces with the same boundary and the same weighted volume.
