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The results of the instability of standing-wave solutions reveal a balance between the frequency $\\omega $ of wave and the power of nonlinearity $p $ for any fixed $ b > 0. $"},"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":"0809.2035","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2008-09-11T16:05:47Z","cross_cats_sorted":[],"title_canon_sha256":"f438839d0b6b0129c7cfc4db7bca8317909bf12245cf929383be71763e7324fb","abstract_canon_sha256":"3187914d68623b9713eac8bff53309b03aab416647c2245adb5425c2c7d86a54"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:38:17.509195Z","signature_b64":"Tnivd8bYv+tPcfIHJx3bHGsU4FdheCRIElfNrrxo4hBFeTfD7+7HpmGJCosgeEWGAwRAM9uFcI2U5xsa2f71Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d2443411ea288e4ce2890f3b726dbd5ef21b86ce95df3311258a989c61936490","last_reissued_at":"2026-05-18T04:38:17.508702Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:38:17.508702Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Instability of Standing Waves to the Inhomogeneous Nonlinear Schr\\\"odinger Equation with Harmonic Potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jianqing Chen, Yue Liu","submitted_at":"2008-09-11T16:05:47Z","abstract_excerpt":"We study the instability of standing-wave solutions $e^{i\\omega t}\\phi_{\\omega}(x)$ to the inhomogeneous nonlinear Schr\\\"{o}dinger equation $$i\\phi_t=-\\triangle\\phi+|x|^2\\phi-|x|^b|\\phi|^{p-1}\\phi, \\qquad \\in\\mathbb{R}^N, $$ where $ b > 0 $ and $ \\phi_{\\omega} $ is a ground-state solution. 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