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This problem arises in the study of stability of solitons for certain nonlinear equations (e.g., the short pulse equation and the generalized Bullough-Dodd equation). We show how to apply the standard approach in the situation under consideration and as a result we provide a formula for the instability index in terms of certain spectral characteristics of the 1-D Schr\\\"odinger op"},"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":"1712.01702","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2017-12-05T15:17:55Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"28f576bba41edc6cab32952619a73b55e37ac5fb34740152c1868c4cd049d6da","abstract_canon_sha256":"0807bab8d0da539e54434f5c9694635b43fc95f0b7a96d77e2ebf159dbf41c67"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:09.044000Z","signature_b64":"8v6XqNayyzjVWQ2S3IttgzXJz31V5FllIe0OjX/qDKxaNdY1Gx5mRwijHhGGcEUr1agyZeNzbbNi72Xe8DSnBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cfe6a430dcb73081e17484dd31561cdbf33845b4b8eb99081a5ec3276cd64af6","last_reissued_at":"2026-05-18T00:12:09.043530Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:09.043530Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Hamiltonian-Krein index for a non-self-adjoint spectral problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.SP","authors_text":"Aleksey Kostenko, Noema Nicolussi","submitted_at":"2017-12-05T15:17:55Z","abstract_excerpt":"We investigate the instability index of the spectral problem $$ -c^2y'' + b^2y + V(x)y = -\\mathrm{i} z y' $$\n  on the line $\\mathbb{R}$, where $V\\in L^1_{\\rm loc}(\\mathbb{R})$ is real valued and $b,c>0$ are constants. This problem arises in the study of stability of solitons for certain nonlinear equations (e.g., the short pulse equation and the generalized Bullough-Dodd equation). 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