[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84288-en":3,"doc-seo-84288-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84288,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Admissible Discrete Linear Propagators for High-Order Time Splittings of Rotational Nonlinear Schrödinger Equations","Study robust high-order time-splitting methods for nonlinear Schrödinger equations whose linear part combines the Laplacian with an arbitrary three-dimensional rotation operator. After Fourier pseudospectral discretization, continuous exact factorizations may not produce a self-adjoint fixed-grid linear propagator. For the original stage-wise explicit exact integrator, a quadratic even term in the local logarithm yields a state-dependent defect, affecting observed temporal accuracy. Fixed-grid admissibility is defined and two admissible unitary propagators are constructed, with experiments confirming recoverable second-, fourth-, and sixth-order behavior.","arXiv :2607 .07923v1 [math .NA] 8 Jul 2026  \nAdmissible Discrete Linear Propagators for High-Order Time Splittings of Rotational Nonlinear Schr¨odinger Equations with Arbitrary Three-Dimensional Rotation  \nTianqi Zhang 1* and Fei Xue2†  \n1* School of Data Sciences, Zhejiang University of Finance and Economics, 18 Xueyuan Street, Hangzhou, Zhejiang, 310018, China.  \n2 School of Mathematical and Statistical Sciences, Clemson University, 220 Parkway Dr., Clemson, 29634, SC, United States.  \n*Corresponding author(s). E-mail(s): [tianqiz@zufe.edu.cn](tianqiz@zufe.edu.cn) ; Contributing authors: [fxue@clemson.edu](fxue@clemson.edu) ;  \n†These authors contributed equally to this work.  \nAbstract  \nWe study robust high-order time splittings for nonlinear Schr¨odinger equations whose linear part is defined by the Laplacian and an arbitrary three-dimensional rotation operator. After Fourier pseudospectral discretization, a continuous exact factorization of the linear flow need not yield a method self-adjoint fixed-grid propagator. For the original stage-wise explicit exact integrator, we identify a quadratic even term in the local logarithm and show that its visibility is statedependent, so the observed temporal order of accuracy can depend on the initial data. We then formulate fixed-grid admissibility for discrete linear propagatorsand construct two admissible propagators for arbitrary three-dimensional rotation: a symmetrized explicit exact integrator and a palindromic generalized shear propagator. Both are unitary, first-order consistent, method self-adjoint, and have odd local logarithms. Numerical experiments verify the predicted defect mechanism and demonstrate recovery of the designed second-, fourth-, and sixth-order behavior with the admissible propagators.  \nKeywords: nonlinear Schr¨odinger equations; Bose–Einstein condensates; high-order time-splitting methods; arbitrary-angle rotation; self-adjoint discrete propagators  \nMSC Classification: 65M70 , 65P10 , 81Q05  \n1  \n1 Introduction  \nTime-splitting spectral methods are a widely used tool for time-dependent nonlinear Schr¨odinger equations (NLS), including Gross–Pitaevskii equations (GPEs) arising in Bose–Einstein condensation [1–3] . For rotating and dipolar Bose–Einstein condensates, such methods typically combine Fourier pseudospectral discretization of dispersive linear terms with exact pointwise phase updates for trapping, local nonlinear, and nonlocal dipolar interactions [4–8] . High temporal order is usually obtained by composing a symmetric second-order splitting block, following Strang splitting, symmetric composition, and geometric integration theory [9–16] . Related high-order splitting and exponential-integrator analyses for Schr¨odinger equations include [17–21] .  \nThis paper identifies and resolves a fully discrete structural obstruction to highorder time splitting for semi-discrete rotational NLS whose linear part is defined by the Laplacian and an arbitrary three-dimensional angular momentum operator,  \nLΩ = − ~~1~~2∆ − Ω · L, Ω = (Ωx , Ωy , Ωz)T ,  \nwhere L = (Lx , Ly , Lz)T , Lx = −i(y∂z − z∂y ), Ly = −i(z∂x − x∂z ), and Lz =−i(x∂y − y∂x) . The remaining terms are assumed to generate an exactly integrable real phase flow, as in rotating GPEs with contact, Hartree, or dipolar interactions. Since this nonlinear phase flow freezes the density during the substep, it is method self-adjoint. Thus, on a fixed Fourier pseudospectral grid, the realizability of highorder symmetric compositions is governed by the discrete propagator used for thesemi-discrete linear subproblem i∂t ψ = LΩ ψ .  \nIn some cases, the desired linear-propagator structure is automatic or transparent. For instance, without rotation, the flow ψ (x, t + τ) = exp(iτ∆/2)ψ(x, t) is diagonal in Fourier space and forms a unitary reversible group on the fixed grid. For constantcoefficient matrix-valued couplings, as in spinor or spin–orbit-coupled condensate models, Fourier discretization gives inde","cbCaio4uYeweRYPo","https://ap.wps.com/l/cbCaio4uYeweRYPo","pdf",935841,7,1,34,"English","en",105,"# Abstract\n# Introduction\n## Time-splitting spectral methods and high-order composition\n## Structural obstruction from discrete rotational linear propagators\n## Discrete exact integrator and self-adjointness failure","[{\"question\":\"What problem does the paper address in high-order time-splitting for rotational NLS?\",\"answer\":\"It identifies a structural obstruction that can prevent high-order symmetric time-splittings when the discrete linear propagator for the rotated operator is not method self-adjoint on a fixed Fourier pseudospectral grid.\"},{\"question\":\"Why can a continuous exact factorization fail after discretization?\",\"answer\":\"Continuous stagewise exactness and unitarity do not guarantee that the discrete propagator satisfies the required time-reversibility or equals the exact exponential of the fixed-grid generator.\"},{\"question\":\"What are the two admissible propagators proposed for arbitrary three-dimensional rotation?\",\"answer\":\"The paper constructs a symmetrized explicit exact integrator and a palindromic generalized shear propagator, both unitary, first-order consistent, method self-adjoint, and with odd local logarithms.\"}]",1784194603,86,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"admissible-discrete-linear-propagators-for-high-order-time-splittings-of-rotational-nonlinear-schrodinger-equations","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/admissible-discrete-linear-propagators-for-high-order-time-splittings-of-rotational-nonlinear-schrodinger-equations/84288/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What problem does the paper address in high-order time-splitting for rotational NLS?","Question",{"text":76,"@type":77},"It identifies a structural obstruction that can prevent high-order symmetric time-splittings when the discrete linear propagator for the rotated operator is not method self-adjoint on a fixed Fourier pseudospectral grid.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Why can a continuous exact factorization fail after discretization?",{"text":81,"@type":77},"Continuous stagewise exactness and unitarity do not guarantee that the discrete propagator satisfies the required time-reversibility or equals the exact exponential of the fixed-grid generator.",{"name":83,"@type":74,"acceptedAnswer":84},"What are the two admissible propagators proposed for arbitrary three-dimensional rotation?",{"text":85,"@type":77},"The paper constructs a symmetrized explicit exact integrator and a palindromic generalized shear propagator, both unitary, first-order consistent, method 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