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We assume that there are$ N\\geq 1$ distinct heteroclinic orbits connecting $a_-$ to $a_+$ represented by maps $ u_1,\\ldots,u_N$ that minimize the one-dimensional energy $J_R(u) =\\int_R(\\frac{\\vert u^\\prime\\vert^2}{2}+W(u))ds$. We first consider the problem of characterizing the minimizers $u:R^n\\rightarrow R^m$ of the energy $\\mathcal{J}_\\Omega(u) =\\int_\\Omega(\\frac{\\vert\\nabla u\\vert^2}{2}+W(u))dx$. Under a nondegeneracy condition on $ u_1,\\ldots,u_N $ and in two space dimensions, we"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1609.05306","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-09-17T09:26:48Z","cross_cats_sorted":[],"title_canon_sha256":"38b5d780782b838dc91c4948d7232c4b5d4c038cf4dc3959451125fe34cafc68","abstract_canon_sha256":"3769c53b6b6d3f47f871e2e9b5ee2d304795055a40aac5fe7c7a5663edf86067"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:26.815503Z","signature_b64":"m56jFXDqniJf+JR+Rv6W5y0vwSNZu3Yz5mbISuaKf0bRB3gvKlIrrCQ+sWXAfdSM7U9rRMnKQGizGLLWmdk6BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2a9f7c7c0a6bf61a43002f9f5102912e173532b91b83e1a83549f6f435d8b083","last_reissued_at":"2026-05-18T01:04:26.814908Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:26.814908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Layered solutions to the vector Allen-Cahn equation in $ R^2$. 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