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More precisely, we show that this function $u_s$ fulfills equations of the form {equation*} \\big(u_s^{(\\alpha)}u_s^{(\\beta)}\\big)(x)=\\sum_{n=0}^{2+\\alpha+\\beta}b_{\\alpha,\\beta}(n)u_s^{(n)}(x)+c_{\\alpha,\\beta}, {equation*} for any $s>0$ and for all $\\alpha,\\beta\\in\\N_0$. We give explicit expressions for the coefficients $b_{\\alpha,\\beta}(n)$ and $c_{\\alpha,\\beta}$ for given $s$.\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":"1308.0920","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-08-05T09:41:32Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"1dbbb698bbfbddd5523c8723c8f2b92cdff582f2dfdc5e40e3f9642963b132e1","abstract_canon_sha256":"a1e12f19d1f61a194739f2a430285a70f852c7bcd453f91d1d8a4c0e852032a6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:16:41.140721Z","signature_b64":"SCbM2HUC7o47cpImOWkUxGjCyLHe9tP8KVllnf1iqXNvHbUZ1yuyRW9F/7UXA8ZT2SQwpT/iUTscBAgS/UnDCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"be3ef76918429e66ee7bae2f55258c68ba89542f00ea4574bc0e9036508a8de2","last_reissued_at":"2026-05-18T03:16:41.139949Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:16:41.139949Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nonlinear differential identities for cnoidal waves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Alice Mikikits-Leitner, Michael Leitner","submitted_at":"2013-08-05T09:41:32Z","abstract_excerpt":"This article presents a family of nonlinear differential identities for the spatially periodic function $u_s(x)$, which is essentially the Jacobian elliptic function $\\cn^2(z;m(s))$ with one non-trivial parameter $s$. More precisely, we show that this function $u_s$ fulfills equations of the form {equation*} \\big(u_s^{(\\alpha)}u_s^{(\\beta)}\\big)(x)=\\sum_{n=0}^{2+\\alpha+\\beta}b_{\\alpha,\\beta}(n)u_s^{(n)}(x)+c_{\\alpha,\\beta}, {equation*} for any $s>0$ and for all $\\alpha,\\beta\\in\\N_0$. 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