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We show that, generically, at the values of where the Vakhitov-Kolokolov stability criterion breaks down, a pair of real eigenvalues (one positive, one negative) appears from the origin, leading to the linear instability of corresponding solitary waves.\n  As an auxiliary result, we state the virial i"},"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":"1107.1763","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-07-09T06:48:00Z","cross_cats_sorted":["hep-th","math-ph","math.MP","nlin.PS"],"title_canon_sha256":"543480b25f0127a5d4d3f4f1655cb289802b219d39c427689b087cea92efa58a","abstract_canon_sha256":"17bc2fd84208084d462aed5cc14132d9ef02a2962ed9dd1e70d3c6902b92e072"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:35.851047Z","signature_b64":"0hkTyZ5nF5/b0s9CzBMiYEBVQIj9iEzWG5ZMdQFsP6g9NEu6U09cT5CXBVUCAFU8MItonwiRCYSqB0k04TRUBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3acefc6ec0b75e93ac3f444cce01ef95445bb1536f55b569b3fb39c8cda4915a","last_reissued_at":"2026-05-18T04:15:35.850616Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:35.850616Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the meaning of the Vakhitov-Kolokolov stability criterion for the nonlinear Dirac equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math-ph","math.MP","nlin.PS"],"primary_cat":"math.AP","authors_text":"Andrew Comech","submitted_at":"2011-07-09T06:48:00Z","abstract_excerpt":"We consider the spectral stability of solitary wave solutions \\phi(x)e^{-i\\omega t} to the nonlinear Dirac equation in any dimension. 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