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This equation has a family of solitons, and our goal is to classify the dynamics near soliton. Together with a suitable decay assumption, there are only three possibilities: (i) the solution converges asymptotically to a solitary wave, whose $H^1$ norm is of size $\\gamma^{-2/(q-1)}$, as $\\gamma\\rightarrow0$; (ii) the soluti"},"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":"1609.05146","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-09-16T17:21:57Z","cross_cats_sorted":[],"title_canon_sha256":"25cc9ab3fa786bfd623e18db4bd52f8c162a95a02901bd0657a341e54fd37799","abstract_canon_sha256":"903aa44870c6c9a9a7a3833799003bdd51f89fb167c9e16f12367480713c9644"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:11.532731Z","signature_b64":"/enLEcOLdZL6KsXK4i4AtNbBZpgnJxJp6/t4jR2ZxnsKR1gzuvsbTf2x7a4JTlraw61H/Fa7Tjr+dTY7CKebAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a960fde4d9c1e56de777e9bb0bbcffd6c0c3060cc7bdfb0119cf068179018a9f","last_reissued_at":"2026-05-18T00:08:11.532272Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:11.532272Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On asymptotic dynamics for $L^2$ critical generalized KdV equations with a saturated perturbation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Yang Lan","submitted_at":"2016-09-16T17:21:57Z","abstract_excerpt":"In this paper, we consider the $L^2$ critical gKdV equation with a saturated perturbation: $\\partial_t u+(u_{xx}+u^5-\\gamma u|u|^{q-1})_x=0$, where $q>5$ and $0<\\gamma\\ll1$. 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