{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:XTV2CWQWRRKXVKE4MHU6LE6WBY","short_pith_number":"pith:XTV2CWQW","canonical_record":{"source":{"id":"2006.02809","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-06-04T12:14:49Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"ae31e565117c1782b0487273248639377b4815c918a96e8a0e75930232f8d624","abstract_canon_sha256":"049e641171c5978382d25fe54e7e93a9e8757ffaf3654fd0cc5b753f3a7a0d07"},"schema_version":"1.0"},"canonical_sha256":"bceba15a168c557aa89c61e9e593d60e0102fb6cfc646199d5ea4efaf8ba28f1","source":{"kind":"arxiv","id":"2006.02809","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.02809","created_at":"2026-07-05T01:08:00Z"},{"alias_kind":"arxiv_version","alias_value":"2006.02809v1","created_at":"2026-07-05T01:08:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.02809","created_at":"2026-07-05T01:08:00Z"},{"alias_kind":"pith_short_12","alias_value":"XTV2CWQWRRKX","created_at":"2026-07-05T01:08:00Z"},{"alias_kind":"pith_short_16","alias_value":"XTV2CWQWRRKXVKE4","created_at":"2026-07-05T01:08:00Z"},{"alias_kind":"pith_short_8","alias_value":"XTV2CWQW","created_at":"2026-07-05T01:08:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:XTV2CWQWRRKXVKE4MHU6LE6WBY","target":"record","payload":{"canonical_record":{"source":{"id":"2006.02809","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-06-04T12:14:49Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"ae31e565117c1782b0487273248639377b4815c918a96e8a0e75930232f8d624","abstract_canon_sha256":"049e641171c5978382d25fe54e7e93a9e8757ffaf3654fd0cc5b753f3a7a0d07"},"schema_version":"1.0"},"canonical_sha256":"bceba15a168c557aa89c61e9e593d60e0102fb6cfc646199d5ea4efaf8ba28f1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:08:00.217649Z","signature_b64":"eJU0coGGV97CIo3KtJit2DABpwXUIchmeMlHigZ1/GhG/Cq7pjOLTxfe9Qc4igH6SDqnv0etJGTG5qMWKQXwCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bceba15a168c557aa89c61e9e593d60e0102fb6cfc646199d5ea4efaf8ba28f1","last_reissued_at":"2026-07-05T01:08:00.217177Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:08:00.217177Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2006.02809","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:08:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"G5/gzZLqwaGWjRx1bE040Dc6jJ7dXc0M1PsSDJ3th7uVVrqk7iX5xw8LgfP87oc4YlForG1dDG5bM8qkP+z7Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T08:13:51.271338Z"},"content_sha256":"bc26353d456c67bc5b2c332360dd2f9c894f53149e654c07acef6921b38aea6b","schema_version":"1.0","event_id":"sha256:bc26353d456c67bc5b2c332360dd2f9c894f53149e654c07acef6921b38aea6b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:XTV2CWQWRRKXVKE4MHU6LE6WBY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The double-power nonlinear Schr\\\"odinger equation and its generalizations: uniqueness, non-degeneracy and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.AP","authors_text":"Mathieu Lewin, Simona Rota Nodari","submitted_at":"2020-06-04T12:14:49Z","abstract_excerpt":"In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form $\\Delta u+g(u)=0$. Our result applies in particular to the double power non-linearity where $g(u)=u^q-u^p-\\mu u$ for $p>q>1$ and $\\mu>0$, which we discuss with more details. In this case, the non-degeneracy of the unique solution $u_\\mu$ allows us to derive its behavior in the two limits $\\mu\\to0$ and $\\mu\\to\\mu_*$ where $\\mu_*$ is the threshold of existence. This gives the uniqueness of energy minimizers at fixed mass in certain regimes. We also make a co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.02809","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2006.02809/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:08:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BeCYUzdYa8vXW2xgI49br5whJWFQUGdEh3D5K+HzFQO1IQJSkQH+I+lW7ohq/eQvnt8egqcuVfvA9HEDq0EWCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T08:13:51.271823Z"},"content_sha256":"d3f1e0bc035cf97cdfb7f41fb6d96806eecc6566d6e5b74481e34f75260aa4cd","schema_version":"1.0","event_id":"sha256:d3f1e0bc035cf97cdfb7f41fb6d96806eecc6566d6e5b74481e34f75260aa4cd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XTV2CWQWRRKXVKE4MHU6LE6WBY/bundle.json","state_url":"https://pith.science/pith/XTV2CWQWRRKXVKE4MHU6LE6WBY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XTV2CWQWRRKXVKE4MHU6LE6WBY/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-04T08:13:51Z","links":{"resolver":"https://pith.science/pith/XTV2CWQWRRKXVKE4MHU6LE6WBY","bundle":"https://pith.science/pith/XTV2CWQWRRKXVKE4MHU6LE6WBY/bundle.json","state":"https://pith.science/pith/XTV2CWQWRRKXVKE4MHU6LE6WBY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XTV2CWQWRRKXVKE4MHU6LE6WBY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:XTV2CWQWRRKXVKE4MHU6LE6WBY","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"049e641171c5978382d25fe54e7e93a9e8757ffaf3654fd0cc5b753f3a7a0d07","cross_cats_sorted":["math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-06-04T12:14:49Z","title_canon_sha256":"ae31e565117c1782b0487273248639377b4815c918a96e8a0e75930232f8d624"},"schema_version":"1.0","source":{"id":"2006.02809","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.02809","created_at":"2026-07-05T01:08:00Z"},{"alias_kind":"arxiv_version","alias_value":"2006.02809v1","created_at":"2026-07-05T01:08:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.02809","created_at":"2026-07-05T01:08:00Z"},{"alias_kind":"pith_short_12","alias_value":"XTV2CWQWRRKX","created_at":"2026-07-05T01:08:00Z"},{"alias_kind":"pith_short_16","alias_value":"XTV2CWQWRRKXVKE4","created_at":"2026-07-05T01:08:00Z"},{"alias_kind":"pith_short_8","alias_value":"XTV2CWQW","created_at":"2026-07-05T01:08:00Z"}],"graph_snapshots":[{"event_id":"sha256:d3f1e0bc035cf97cdfb7f41fb6d96806eecc6566d6e5b74481e34f75260aa4cd","target":"graph","created_at":"2026-07-05T01:08:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2006.02809/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form $\\Delta u+g(u)=0$. Our result applies in particular to the double power non-linearity where $g(u)=u^q-u^p-\\mu u$ for $p>q>1$ and $\\mu>0$, which we discuss with more details. In this case, the non-degeneracy of the unique solution $u_\\mu$ allows us to derive its behavior in the two limits $\\mu\\to0$ and $\\mu\\to\\mu_*$ where $\\mu_*$ is the threshold of existence. This gives the uniqueness of energy minimizers at fixed mass in certain regimes. We also make a co","authors_text":"Mathieu Lewin, Simona Rota Nodari","cross_cats":["math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-06-04T12:14:49Z","title":"The double-power nonlinear Schr\\\"odinger equation and its generalizations: uniqueness, non-degeneracy and applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.02809","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:bc26353d456c67bc5b2c332360dd2f9c894f53149e654c07acef6921b38aea6b","target":"record","created_at":"2026-07-05T01:08:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"049e641171c5978382d25fe54e7e93a9e8757ffaf3654fd0cc5b753f3a7a0d07","cross_cats_sorted":["math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-06-04T12:14:49Z","title_canon_sha256":"ae31e565117c1782b0487273248639377b4815c918a96e8a0e75930232f8d624"},"schema_version":"1.0","source":{"id":"2006.02809","kind":"arxiv","version":1}},"canonical_sha256":"bceba15a168c557aa89c61e9e593d60e0102fb6cfc646199d5ea4efaf8ba28f1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bceba15a168c557aa89c61e9e593d60e0102fb6cfc646199d5ea4efaf8ba28f1","first_computed_at":"2026-07-05T01:08:00.217177Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:08:00.217177Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eJU0coGGV97CIo3KtJit2DABpwXUIchmeMlHigZ1/GhG/Cq7pjOLTxfe9Qc4igH6SDqnv0etJGTG5qMWKQXwCw==","signature_status":"signed_v1","signed_at":"2026-07-05T01:08:00.217649Z","signed_message":"canonical_sha256_bytes"},"source_id":"2006.02809","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bc26353d456c67bc5b2c332360dd2f9c894f53149e654c07acef6921b38aea6b","sha256:d3f1e0bc035cf97cdfb7f41fb6d96806eecc6566d6e5b74481e34f75260aa4cd"],"state_sha256":"627f9c49ac091f56e3412681dfa057ccc6c91d4687a1d8ccc1469573a3bc7fe1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PR66TP4JobjgAhb4KLZBRZcNzkSpIcTfPxFA4rPU4pzCvqXas6d1ZCKopWalaJXid0p2RL5z+slXeMnHKEwPDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T08:13:51.275705Z","bundle_sha256":"179e6a1473beb68735c2a6a6477b7be29887449b21de36494ca03bb804a536a7"}}