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We mainly analyze the aggregation state of $N$-soliton solutions of the mCH equation expressed by the solution of the modified Riemann-Hilbert problem in the new $(y,t)$-space when the discrete spectra are located in different regions. Starting from the modified RH problem, we find that i) when the region is a quadrature domain with $\\ell=n="},"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":"2501.03485","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"nlin.SI","submitted_at":"2025-01-07T03:09:52Z","cross_cats_sorted":["math-ph","math.AP","math.MP"],"title_canon_sha256":"1ff84106aaf832f175970b82c45302dec2de44d8e14e6b1dffc7239d5fcd6bfd","abstract_canon_sha256":"445796a42c1e7132b399a64c4e20355815ca59c96696e47e1a2dd6904c408007"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:57:55.494553Z","signature_b64":"iHgpsQ1dCHmINmpmz+XWha2n7V/3129GDDWuRVfirP86IRCZ9H1OlZXFh7lW/WaZ081zHuoTbGDRefIJH2TCBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a827f8fa68167909e69680370114114bac3fed1a749ec2ba36637e9c4644d62c","last_reissued_at":"2026-07-05T09:57:55.494078Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:57:55.494078Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the $N_\\infty$-soliton asymptotics for the modified Camassa-Holm equation with linear dispersion and vanishing boundaries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.MP"],"primary_cat":"nlin.SI","authors_text":"Weifang Weng, Zhenya Yan","submitted_at":"2025-01-07T03:09:52Z","abstract_excerpt":"We explore the $N_{\\infty}$-soliton asymptotics for the modified Camassa-Holm (mCH) equation with linear dispersion and boundaries vanishing at infinity: $m_t+(m(u^2-u_x^2)^2)_x+\\kappa u_x=0,\\quad m=u-u_{xx}$ with $\\lim_{x\\rightarrow \\pm \\infty }u(x,t)=0$. We mainly analyze the aggregation state of $N$-soliton solutions of the mCH equation expressed by the solution of the modified Riemann-Hilbert problem in the new $(y,t)$-space when the discrete spectra are located in different regions. 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