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We focus on standing wave solutions, namely solutions of the form $\\psi (x,t)=e^{i\\alpha t}u(x)$, for some $\\alpha \\in \\R$. This ansatz yields the fourth-order elliptic equation \\begin{equation*} %\\tag{\\protect{*}}\\label{4nlsstar} \\gamma \\Delta^2 u -\\beta \\Delta u +\\alpha u =|u|^{2\\sigma} u. \\end{eq"},"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":"1710.09775","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-10-26T15:56:25Z","cross_cats_sorted":[],"title_canon_sha256":"a7cc2403613c61c2efa2dc4ba9ac326c597fcbc5b56ebab576acaa3083fa679a","abstract_canon_sha256":"5087664eebc96936d32b9ed6ebec14882b26d18a725e12788b62780ecfd44bce"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:19.634742Z","signature_b64":"NADSHNZPeGRKkWWBa1e/NW8bW/M6F66yz7MoCNB0+5c8K+pZZzamZeCydHDd8dlQ+uQxEWo3yjJvWsns95OSDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0906d77a187bce58ff5634d5248b48c64944ee747b6441ab62cee2d173e21492","last_reissued_at":"2026-05-18T00:05:19.634230Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:19.634230Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Orbitally stable standing waves of a mixed dispersion nonlinear Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Denis Bonheure, Ederson Moreira dos Santos, Jean-Baptiste Cast\\'eras, Robson Nascimento","submitted_at":"2017-10-26T15:56:25Z","abstract_excerpt":"We study the mixed dispersion fourth order nonlinear Schr\\\"odinger equation \\begin{equation*} %\\tag{\\protect{4NLS}}\\label{4nls} i \\partial_t \\psi -\\gamma \\Delta^2 \\psi +\\beta \\Delta \\psi +|\\psi|^{2\\sigma} \\psi =0\\ \\text{in}\\ \\R \\times\\R^N, \\end{equation*} where $\\gamma,\\sigma>0$ and $\\beta \\in \\R$. We focus on standing wave solutions, namely solutions of the form $\\psi (x,t)=e^{i\\alpha t}u(x)$, for some $\\alpha \\in \\R$. This ansatz yields the fourth-order elliptic equation \\begin{equation*} %\\tag{\\protect{*}}\\label{4nlsstar} \\gamma \\Delta^2 u -\\beta \\Delta u +\\alpha u =|u|^{2\\sigma} u. \\end{eq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.09775","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1710.09775","created_at":"2026-05-18T00:05:19.634312+00:00"},{"alias_kind":"arxiv_version","alias_value":"1710.09775v2","created_at":"2026-05-18T00:05:19.634312+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.09775","created_at":"2026-05-18T00:05:19.634312+00:00"},{"alias_kind":"pith_short_12","alias_value":"BEDNO6QYPPHF","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_16","alias_value":"BEDNO6QYPPHFR72W","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_8","alias_value":"BEDNO6QY","created_at":"2026-05-18T12:31:08.081275+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BEDNO6QYPPHFR72WGTKSJC2IYZ","json":"https://pith.science/pith/BEDNO6QYPPHFR72WGTKSJC2IYZ.json","graph_json":"https://pith.science/api/pith-number/BEDNO6QYPPHFR72WGTKSJC2IYZ/graph.json","events_json":"https://pith.science/api/pith-number/BEDNO6QYPPHFR72WGTKSJC2IYZ/events.json","paper":"https://pith.science/paper/BEDNO6QY"},"agent_actions":{"view_html":"https://pith.science/pith/BEDNO6QYPPHFR72WGTKSJC2IYZ","download_json":"https://pith.science/pith/BEDNO6QYPPHFR72WGTKSJC2IYZ.json","view_paper":"https://pith.science/paper/BEDNO6QY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1710.09775&json=true","fetch_graph":"https://pith.science/api/pith-number/BEDNO6QYPPHFR72WGTKSJC2IYZ/graph.json","fetch_events":"https://pith.science/api/pith-number/BEDNO6QYPPHFR72WGTKSJC2IYZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BEDNO6QYPPHFR72WGTKSJC2IYZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BEDNO6QYPPHFR72WGTKSJC2IYZ/action/storage_attestation","attest_author":"https://pith.science/pith/BEDNO6QYPPHFR72WGTKSJC2IYZ/action/author_attestation","sign_citation":"https://pith.science/pith/BEDNO6QYPPHFR72WGTKSJC2IYZ/action/citation_signature","submit_replication":"https://pith.science/pith/BEDNO6QYPPHFR72WGTKSJC2IYZ/action/replication_record"}},"created_at":"2026-05-18T00:05:19.634312+00:00","updated_at":"2026-05-18T00:05:19.634312+00:00"}